Appendix A

A Discussion of Fractal Image Compression 1

Yuval Fisherl

Caveat Emptor. Anon

Recently, fractal image compression - a scheme using fractal trans• forms to encode general images - has received considerable attention. This interest has been aroused chiefly by Michael Barnsley, who claims to have commercialized such a scheme. In spite of the popularity of the notion, scientific publications on the topic have been sparse; most articles have not contained any description of results or algorithms. Even Bamsley's book, which discusses the theme of fractal image compression at length, was spartan when it came to the specifics of image compression. The first published scheme was the doctoral dissertation of A. Jacquin, a student of Barnsley's who had previously published related papers with Barnsley without revealing their core algorithms. Other work was conducted by the author in collaboration with R. D. Boss and E. W. Jacobs3 and also in collaboration with Ben Bielefeld.4 In this appendix we discuss several schemes based on the aforementioned work by which general images can be encoded as fractal transforms.

1This work was partially supported by ONR contract N00014-91-C-0177. Other support was provided by the San Diego Supercomputing Center and the Institute for Non-Linear Science at the University of California, San Diego. 2San Diego Supercomputing Facility, University of California, San Diego, La Jolla, CA 92093. 30f the Naval Ocean Systems Center, San Diego. 40f the State University of New York, Stony Brook. 904 A A Discussion of Fractal Image Compression

Figure A.l : A portion of Lenna's hat decoded at 4 times its encoding size (left), and the original image enlarged to 4 times the size (right), showing pixelization.

The image compression scheme can be said to be fractal in several Why is it "Fractal" senses. First, an image is stored as a collection of transforms that are very Image Compression? similar to the MRCM metaphor. This has several implications. For example, just as the Barnsley fern is a set which has detail at every scale, so does the decoded image have detail created at every scale. Also, if one scales the transformations in the Barnsley fern IFS (say by multiplying everything by 2), the resulting attractor will be scaled (also by a factor of 2). In the same way, the decoded image has no natural size, it can be decoded at any size. The extra detail needed for decoding at larger sizes is generated automatically by the encoding transforms. One may wonder (but hopefully not for long) if this detail is 'real'; that is, if we decode an image of a person at larger and larger size, will we eventually see skin cells or perhaps atoms? The answer is, of course, no. The detail is not at all related to the actual detail present when the image was digitized; it is just the product of the encoding transforms which only encode the large scale features well. However, in some cases the detail is realistic at low magnifications, and this can be a useful feature of the method. For example, figure A.l shows a detail from a fractal encoding of Lenna along with a magnification of the original. The whole original image can be seen in figure A.4 (left); this is the now famous image of Lenna which is commonly used in the image compression literature. The magnification of the original shows pixelization, the dots that make up the image are clearly discernible. This is because it is magnified by a factor of 4. The decoded image does not show pixelization since detail is created at all scales. 905

Grey Scale Version of the Sierpinski Gasket

Figure A.2

Why is it Fractal An image is stored on a computer as a collection of values which indicate Image a grey level or color at each point (or pixel) of the picture. It is typical "Compression"? to use 8 bits per pixel for grey-scale images, giving 28 = 256 different possible levels of grey at each pixel. This yields a gradation of greys that is sufficient to make monochrome images stored this way look good. However, the image's pixel density must also be sufficiently high so that the individual pixels are not apparent. Thus, even small images require a large number of pixels and so they have a high memory requirement. However, the human eye is not sensitive to certain types of information loss, and so it is generally possible to store an approximation of an image as a collection of transforms using considerably less information than is required to store the original image. For example, the grey-scale version of the Sierpinski gasket in figure A.2 can be generated from only 132 bits of information using the same decoding algorithm that generated the other encoded images in this section. Because this image is self-similar, it can be stored very compactly as a collection of transformations. This is the spirit of the idea behind the fractal image compression scheme presented in the next sections. Standard image compression methods can be evaluated using their com• pression ratio; the ratio of the memory required to store an image as a collection of pixels and the memory required to store a representation of the image in compressed form. The compression ratio for the fractal scheme is hard to measure, since the image can be decoded at any scale. If we decode the grey-scale Sierpinski gasket at, say, two times its size, then we could claim 4 times the compression ratio since 4 times as many pixels would be required to store the decompressed image. For example, the decoded 906 A A Discussion of Fractal Image Compression

Graph Generated From the Lenna Image.

Figure A.3

image in figure A.l is a portion of a 5.7: l compression of the whole Lenna image. It is decoded at 4 times it's original size, so the full decoded image contains 16 times as many pixels and hence its compression ratio is 91.2: I. This may seem like cheating, but since the 4-times-larger image has detail at every scale, it really isn't.

A.l Self-Similarity in Images

The images we will encode are different than the images discussed in other parts of the book. Before, when we referred to an image, we meant a set that could be drawn in black and white on the plane, with black representing the points of the set. In this appendix, an image refers to something that looks like a black-and-white photograph. In order to discuss the compression of images, we need a mathematical Images as Graphs of model of an image. Figure A.3 shows the graph of a special function Functions z = f(x , y). This graph is generated by using the image of Lenna (see figure A.4) and plotting the grey level of the pixel at position ( x, y) as a height, with white being high and black being low. This is our model for an image, except that while the graph in figure A.3 is generated by connecting the heights on a 64 x 64 grid, we generalize this and assume that every position (x, y) can have an independent height. That is, our model of an image has infinite resolution. A.l Self-Similarity in Images 907

Thus, when we wish to refer to an image, we refer to the function f (x, y) which gives the grey level at each point (x, y). When we are dealing with an image of finite resolution, such as the images that are digitized and stored on computers, we must either average J(x, y) over the pixels of the image or insist that f (x, y) has a constant value over each pixel.

For simplicity, we assume we are dealing with square images of Normalizing Graphs of Images size 1. We require (x,y) E 12 = {(u,v) I 0:::; u,v :::; 1}, and f (x, y) E I = [0, 1]. Since we will want to use the contraction mapping principle, we will want to work in a complete metric space of images, and so we also will require that f is measurable. This is a technicality, and not a serious one since the measurable functions include the piecewise continuous functions, and one could argue that any natural image corresponds to such a function.

A Metric on Images We also want to be able to measure differences between images, and so we introduce a metric on the space of images. There are many metrics to choose from, but the simplest to use is the sup metric

o(f,g) = sup lf(x,y)- g(x,y)l. (x,y)EJ2

This metric finds the position (x, y) where two images f and g differ the most and sets this value as the distance between f and g. There are other possible choices for image models and other possible metrics to use. In fact just as before, the choice of metric determines whether the transformations we use are contractive or not. These details are important, but are beyond the scope of this appendix. Natural Images are A typical image of a face, for example figure A.4 (left) does not contain not Exactly the type of self-similarity that can be found in the Sierpinski gasket. The Self-Similar image does not appear to contain affine transformations of itself. But, in fact, this image does contain a different sort of self-similarity. Figure A.4 (right) shows sample regions of Lenna which are similar at different scales: a portion of her shoulder overlaps a region that is almost identical, and a portion of the reflection of the hat in the mirror is similar (after transforma• tion) to a part of her hat. The distinction from the kind of self-similarity we saw with ferns and gaskets is that rather than having the image be formed of copies of its whole self (under appropriate affine transformation), here the image will be formed of copies of (properly transformed) parts of itself. These parts are not identical copies of themselves under affine transforma• tion, and so we must allow some error in our representation of an image as a set of transformations. This means that the image we encode as a set of transformations will not be an identical copy of the original image but rather an approximation of it. Finally, in what kind of images can we expect to find this type of local self-similarity? Experimental results suggest that most images that one 908 A A Discussion of Fractal Image Compression

Figure A.4 : The original 256 x 256 pixel Lenna image (left) and some of its self-similar portions (right). would expect to 'see" can be compressed by taking advantage of this type of self-similarity; for example, images of trees, faces, houses, mountains, clouds, etc. However, the existence of this local self-similarity and the ability of an algorithm to detect it are distinct issues, and it is the latter which concerns us here.

A.2 A Special MRCM

In this section we describe an extension of the multiple reduction copying Partitioned MRCMs machine metaphor that can be used to encode and decode grey-scale images. As before, the machine has several dials, or variable components: Dial I: number of lens systems, Dial 2: setting of reduction factor for each lens system individually, Dial 3: configuration of lens systems for the assembly of copies. These dials are a part of the MRCM definition from chapter 5; we add to them the following two capabilities: Dial 4: A contrast and brightness adjustment for each lens, Dial 5: A mask which selects, for each lens, a part of the original to be copied. These extra features are sufficient to allow the encoding of grey scale images. The last dial is the new important feature. It partitions an image into pieces which are each transformed separately. For this reason, we call this MRCM a partitioned multiple reduction copying machine (PMRCM). By partitioning A.2 A Special MRCM 909

the image into pieces, we allow the encoding of many shapes that are difficult to encode on an MRCM, or IFS. Let us review what happens when we put an original image on the copy surface of the machine. Each lens selects a portion of the original, which we denote by D; and copies that part (with a brightness and contrast transformation) to a part of the produced copy which is denoted R,. We call the D; domains and the R; ranges. We denote this transformation by w,. The partitioning is implicit in the notation, so that we can use almost the same notation as before. Given an image f, one copying step in a machine with N lenses can be written as W(f) = w1 (f) U w2(f) U · · · U WN(f). As before the machine runs in a feedback loop; its own output is fed back as its new input again and again. A PMRCM for a Consider the 8 lens PMRCM indicated in figure A.5. The figure shows Bowtie two regions, one marked D 1 = D2 = D3 = D4 and the other marked Ds = D6 = D7 = Ds. These are the partitioned pieces of the original which will be copied by the 8 lenses. The lenses map each domain D, to a corresponding range R;, with a reduction factor of I /2. For simplicity, we assume that the contrast and brightness are not altered in this example. Figure A.6 shows three iterations of the PMRCM with three different initial images. The attractor for this system is the bow-tie figure shown in (c). This example demonstrates the utility of a PMRCM. By partitioning the original to be copied, it is very easy to encode the bow-tie image (though the astute reader will notice that this image is also possible to encode using an IFS).

o=o=o=o An 8 lens PMRCM encoding a 5 6 7 8 bowtie.

Figure A.5

Three iterations of a PMRCM with (a) three different intial images.

(b)

(c) Figure A.6 910 A A Discussion of Fractal Image Compression

We call the mathematical analogue of a PMRCM, a partitioned iterated PMRCM = PIFS function system (PIFS). A PIFS has some features in common with the networked MRCM and Bamsley's recurrent iterated function systems, but they are not at all identical. We haven't specified what kind of transformations we are allowing, and in fact one could build a PMRCM or PIFS with any transformation one wants. But in order to simplify the situation, and also in order to allow a compact specification of the final PIFS (in order to yield high compression), we restrict ourselves to transformations wi of the form

(A.l)

It is convenient to write a· vi(x, y) = [ ~ b~] [x] [e.;] Ci d, y + li Since an image is modeled as a function f(x, y), we can apply w, to an image f by wi (f) = wi ( x, y, f (x, y)). Then vi determines how the par• titioned domains of an original are mapped to the copy, while s, and o, determine the contrast and brightness of the transformation. It is always implicit, and important to remember, that each wi is restricted to Di x I. That is, wi applies only to the part of the image that is above the domain Di. This means that vi(Di) = Ri. Since we want W(J) to be an image, we must insist that UR.; = ! 2 and that RinRj = 0 when i # j. That is, when we apply W to an image, we get some single valued function above each point of the square / 2• Running the copying machine in a loop means iterating the Hutchinson operator W. We begin with an initial image fo and then iterate !J = W (Jo), h = W (J,) = W(W(J0 )), and so on. We denote the n-th iterate by fn = wn(Jo). When will W have an attractive fixed point? By the contractive mapping Fixed Points for PIFS principle, it is sufficient to have W be contractive. Since we have chosen a metric that is only sensitive to what happens in the z direction, it is not necessary to impose contractivity conditions in the x or y directions. The transformation W will be contractive when each s.; < I. In fact, the contractive mapping principle can be applied to wm (for some m), so it is sufficient for wrn to be contractive. This leads to the somewhat surprising result that there is no specific condition on the si either. In practice, it is safest to take si < I to ensure contractivity. But we know from experiments that taking si < 1.2 is safe, and that this results in slightly better encodings. When W is not contractive and wm is contractive, we call W even• Eventually tually contractive. A brief explanation of how a transformation W can be Contractive Maps eventually contractive but not contractive is in order. The map W is com• posed of a union of maps wi operating on disjoint parts of an image. The iterated transform wm is composed of a union of compositions of the form A.2 A Special MRCM 911

Since the product of the contractivities bounds the contractivity of the com• positions, the compositions may be contractive if each contains sufficiently

contractive wi1 • Thus W will be eventually contractive (in the sup metric) if it contains sufficient 'mixing' so that the contractive wi eventually dom• inate the expansive ones. In practice, given a PIFS this condition is simple to check. Suppose that we take all the si < 1. This means that when the PMRCM is run, the contrast is always reduced. This seems to suggest that when the machine is run in a feedback loop, the resulting attractor will be an insipid, contrast-less grey. But this is wrong, since contrast is created between ranges which have different brightness levels oi. So is the only contrast in the attractor between the Ri? No, if we take the vi to be contractive, then the places where there is contrast between the R, in the image will propagate to smaller and smaller scale, and this is how detail is created in the attractor. This is one reason to require that the vi be contractive. We now know how to decode an image that is encoded as a PIFS or as a PMRCM. Start with any initial image and repeatedly run the copy machine, or repeatedly apply W until we get to the fixed point foo· We will use Hutchinson's notation and denote this fixed point by foo = IWI. The decoding is easy, but it is the encoding which is interesting. To encode an image we need to figure out Ri, Di and Wi, as well as N, the number of maps w, we wish to use.

When we decode by iterating, we take an initial fo and compute f n = Decoding by Matrix Inversion WUn-I)· This can also be written as

fn(x, y) = sdn-l (vi 1 (x, y)) + o, , where i is determined by the condition (x, y) E Ri. Suppose we are dealing with an image of resolution M x M. We can write the image as a column vector, and then this equation can be written as

Jn = S Jn- 1 + 0 , where S is an M 2 x M 2 matrix with entries si that encode the vi and 0 is a column vector containing the brightness values oi. Then =snr +"n sJ-lO f n JO L..,J=l ' and if each si < c < 1 then the first term is 0 in the limit. {The con• dition si < c < 1 can be relaxed when W is eventually contractive). When I - S is invertible,

foo = L~o SJO =(I- s)- 10, where I is the identity matrix. Bielefeld pointed out that when each pixel value fn(x, y) depends on only one (or a few) other pixel values fn-l (vi 1 (x, y) ), this matrix is very sparse and can be readily inverted. 912 A A Discussion of Fractal Image Compression

A.3 Encoding Images

Suppose we are given an image f that we wish to encode. This means we want to find a collection of maps WJ, wz ... , w N with W = U~ 1Wi and f = IWI. That is, we want f to be the fixed point of the Hutchinson operator W. As in the IFS case, the fixed point equation

.f = W(f) = w 1(f) U w2(f) U · · · wN(.f) suggests how this may be achieved. We seek a partition of .f into pieces to which we apply the transforms Wi and get back .f. This is too much to hope for in general, since images are not composed of pieces that can be transformed non-trivially to fit exactly somewhere else in the image. What we can hope to find is another image f' = IWI with ti(.f', .f) small. That is, we seek a transformation W whose fixed point f' = IWI is close to, or looks like, f. In that case,

.f ~ .f' = W(J') ~ W(f) = w 1(.f) U w2(f) U · · · wN(.f).

Thus it is sufficient to approximate the parts of the image with transformed pieces. We do this by minimizing the following quantities

/i(.fn(Rixi),wi(.f)) i=l, ... ,N (A.2)

Finding the pieces Ri (and corresponding Di) is the heart of the problem. The following example suggests how this can be done. Suppose A Simple Illustrative we are dealing with a 256 x 256 pixel image at 8 bits per pixel. Example Let R1, Rz, ... , R1024 be the 8 x 8 non-overlapping sub-squares of [0, 255] x [0, 255], and let D be the collection of all 16 x 16 sub-squares. The collection D contains 241 · 241 = 58,081 squares. For each Ri search through all of D to find a Di E D which minimizes equation A.2. This domain is said to cover the range. There are 8 ways to map one square onto another, so that this means comparing 8 ·58, 081 = 464, 648 squares. Also, a square in D has 4 times as many pixels as an R~, so we must either subsample (choose 1 from each 2 x 2 sub-square of D") or average the 2 x 2 sub-squares corresponding to each pixel of R" when we minimize eqn. (A.2). Minimizing equation (A.2) means two things. First it means finding a good choice for Di (that is the part of the image that most looks like the image above Ri). Second, it means finding a good contrast and brightness setting s, and oi for Wi. For each D E D we can compute si and oi using least squares regression, which also gives a resulting root mean square (rms) difference. We then pick as Di the D E D which has the least rms difference. Two men flying in a balloon are sent off track by a strong gust of A Point about Metrics wind. Not knowing where they are, they approach a hill on which a solitary figure is perched. They lower the balloon and shout to the man on the hill, "Where are we?''. The man pauses for a long time and shouts back, just as A.3 Encoding Images 913

the balloon is leaving earshot, "You are in a balloon." So one of the men in the balloon turns to the other and says, "That man was a mathematician." Completely amazed, the second man asks, "How can you tell that?". Replies the first man, "We asked him a question, he thought about it for a long time, his answer was correct, and it was totally useless." This is what we have done with the metrics. When it came to a simple theoretical motivation, we use the sup metric which is very convenient for this. But in practice, we are happier using the rms metric which allows us to make least square computations.

Given two squares containing n pixel intensities, a1, ••. , an and Least Squares b1, ... , bn. We can seeks and o to minimize the quantity

n R = Z:::::(s · ai + o- b;) 2 . i=l This will give us a contrast and brightness setting that makes the affinely transformed a; values have the least squared distance from the b; values. The minimum of R occurs when the partial derivatives with respect to s and o are zero, which occurs when

and

In that case,

A choice of D;, along with a corresponding s; and oi, determines a map w; of the form of eqn. (A.l). Once we have the collection w1, ... , w1024 we can decode the image by estimating IWI. Figure A.7 shows four images: an arbitrary initial image fo chosen to show texture, the first iteration W(f0 ), which shows some of the texture from f 0 , W 2 (!0 ), and W 10 (!0 ). The result is surprisingly good, given the naive nature of the encoding algorithm. The original image required 65536 bytes of storage, whereas the 914 A A Discussion of Fractal Image Compression

transformations required only 3968 bytes,5 giving a compression ratio of 16.5:1. With this encoding R = 10.4 and each pixel is on average only 6.2 grey levels away from the correct value. These images show how detail is added at each iteration. The first iteration contains detail at size 8 x 8, the next at size 4 x 4, and so on.

A.4 Ways to Partition Images

The example of the last section is naive and simple, but it contains most of the ideas of a fractal image encoding scheme. First partition the image by some collection of ranges Ri. Then for each Ri seek from some collection of image pieces a D; which has a low rms error. The sets R; and Di, determine s; and oi as well as ai, bi, ci, di, ei and /; in eqn. (A.l ). We then get a transformation W = Uw; which encodes an approximation of the original image. A weakness of the example is the use of fixed size R;, since there are Quadtree Partitioning regions of the image that are difficult to cover well this way (for example, Lenna's eyes). Similarly, there are regions that could be covered well with larger Ri, thus reducing the total number of w; maps needed (and increasing the compression of the image). A generalization of the fixed size R; is the use of a quadtree partition of the image. In a quadtree partition, a square image is broken up into 4 equally sized sub-squares. Depending on some algorithmic criterion, each of these is again recursively sub-divided. An algorithm for encoding 256 x 256 pixel images based on this idea can proceed as follows. Choose for the collection D of permissible domains all the sub-squares in the image of size 8, 12, 16, 24, 32,48 and 64. Partition the image recursively by a quadtree method until the squares are of size 32. For each square in the quadtree partition, attempt to cover it by a domain that is larger. If a predetermined tolerance rms value is met, then call the square R; and the covering domain D,. If not, then subdivide the square and repeat. This algorithm works well. It works even better if diagonally oriented squares are used in the domain pool D also. Figure A.8 shows an image of a collie compressed using this scheme. In section A.5 we discuss some of the details of this scheme as well as the other two schemes discussed below. A weakness of the quadtree based partitioning is that it makes no attempt HV-Partitioning to select the domain pool D in a content dependent way. The collection must be chosen to be very large so that a good fit to a given range can be found. A way to remedy this, while increasing the flexibility of the range partition, is to use an HV-partition. In an HV-partition, a rectangular image is recursively partitioned either horizontally or vertically to form two new rectangles. The partitioning repeats recursively until some criterion is met, as before. This scheme is more flexible, since the position of the partition is variable. We

5Each transformation required 8 bits in the x and y direction to determine the position of D,, 7 bits for o,, 5 bits for s, and 3 bits to determine a rotation and flip operation for mapping D, to R,. A.4 Ways to Partition Images 915

Figure A.7 : An original image, the first, second, and tenth iterates of the encoding transformations.

can then try to make the partttiOns in such a way that they share some self-similar structure. For example, we can try to arrange the partitions so that edges in the image will tend to run diagonally through them. Then, it is possible to use the larger partitions to cover the smaller partitions with a reasonable expectation of a good cover. Figure A. I0 demonstrates this idea. The figure shows an part of an image (a); in (b) the first partition generates two rectangles, R 1 with the edge running diagonally through it, 916 A A Discussion of Fractal Image Compression

A Collie A collie (256 x 256) compressed with the quadtree scheme at 28.95: I with an rms error of 8.5.

Figure A.8

San Francisco San Francisco (256 x 256) com• pressed with the HV scheme at 7.6:1 with an rms error of 7 .I.

Figure A.9

and R2 with no edge; and in (c) the next three partitions of R 1 partition it into 4 rectangles, two rectangles which can be well covered by R 1 (since they have an edge running diagonally) and two which can be covered by R2 (since they contain no edge). Figure A.9 shows an image of San Francisco encoded using this scheme. Yet another way to partition an image is based on triangles. In the Triangular triangular partitioning scheme, a rectangular image is divided diagonally Partitioning A.5 Implementation Notes 917

The HV scheme attempts to cre• ate self-similar rectangles at differ• lst""P"-a;- 2-ti-on_....J IF::T ition• ent scales.

(a) (b) (c) Figure A.IO

Figure A.ll : A quadtree partition (5008 squares), an HV partition (2910 rectangles), and a triangular partition (2954 triangles).

into two triangles. Each of these is recursively subdivided into 4 triangles by segmenting the triangle along lines that join three partitioning points along the three sides of the triangle. This scheme has several potential advantages over the HV-partitioning scheme. It is flexible, so that triangles in the scheme can be chosen to share self-similar properties, as before. However, the artifacts arising from imperfect covering do not run horizontally and vertically, and this is less distracting. Also, the triangles can have any orientation, so we break away from the rigid 90 degree rotations of the quadtree and HV partitioning schemes. This scheme, however, remains to be fully developed and explored. Figure A. II shows sample partitions arising from the three partitioning schemes applied to the Lenna image.

A.5 Implementation Notes

Storing the Encoding To store the encoding compactly, we do not store all the coefficients in eqn. Compactly (A.l). The contrast and brightness settings are stored using a fixed number of bits. One could compute the optimal si and oi and then discretize them for storage. However, a significant improvement in fidelity can be obtained if only discretized si and oi values are used when computing the error during encoding (and eqn. (A.3) facilitates this). Using 5 bits to store s, and 7 bits 918 A A Discussion of Fractal Image Compression

to store o; has been found empirically optimal in general. The distribution of s; and o; shows some structure, so further compression can be attained by using entropy encoding. The remaining coefficients are computed when the image is decoded. In their place we store R; and D;. In the case of a quadtree partition, R; can be encoded by the storage order of the transformations if we know the size of R;. The domains D; must be stored as a position and size (and orientation if diagonal domain are used). This is not sufficient, though, since there are 8 ways to map the four comers of D; to the comers of R;. So we also must use 3 bits to determine this rotation and flip information. In the case of the HV-partitioning and triangular partitioning, the parti• tion is stored as a collection of offset values. As the rectangles (or triangles) become smaller in the partition, fewer bits are required to store the offset value. The partition can be completely reconstructed by the decoding rou• tine. One bit must be used to determine if a partition is further subdivided or will be used as an Ri and a variable number of bits must be used to specify the index of each Di in a list of all the partitions. For all three methods, and without too much effort, it is possible to achieve a compression of roughly 31 bits per w; on average. In the example of section A.3, the number of transformations is fixed. In contrast, the partitioning algorithms described are adaptive in the sense that they utilize a range size which varies depending on the local image complexity. For a fixed image, more transformations lead to better fidelity but worse compression. This trade-off between compression and fidelity leads to two different approaches to encoding an image f - one targeting fidelity and one targeting compression. These approaches are outlined in the pseudo-code below. In the code, size(R;) refers to the size of the range; in the case of rectangles, size(R;) is the length of the longest side. Another concern is encoding time, which can be significantly reduced by Optimizing Encoding employing a classification scheme on the ranges and domains. Both ranges Time and domains are classified using some criteria such as their edge-like nature, or the orientation of bright spots, etc. Considerable time savings result from only using domains in the same class as a given range when seeking a cover, the rationale being that domains in the same class as a range should cover it best.

Pseudo-Code a. Pseudo-code targeting a fidelity ec. • Choose a tolerance level ec. • Set R 1 = ! 2 and mark it uncovered. • While there are uncovered ranges R, do { • Out of the possible domains D, find the domain D, and the corresponding w, which best covers R; (i.e., which minimizes expression (A.2)). • If 8(! n (R; xI), w;(f)) < ec or size(R;) :S I'min then • Mark R, as covered, and write out the transformation w,; • else A.5 Implementation Notes 919

• Partition Ri into smaller ranges which are marked as uncov• ered, and remove Ri from the list of uncovered ranges. } b. Pseudo-code targeting a compression having N transformations. • Choose a target number of ranges Nr. • Set a list to contain R1 = ! 2 , and mark it as uncovered. • While there are uncovered ranges in the list do { • For each uncovered range in the list, find and store the domain Di E D and map wi which covers it best, and mark the range as covered. • Out of the list of ranges, find the range Rj with size ( Ri) > r min which has the largest

(i.e., which is covered worst). • If the number of ranges in the list is less than Nr then { • Partition Ri into smaller ranges which are added to the list and marked as uncovered. • Remove Rj, Wj and Dj from the list. } } • Write out all the wi in the list. Appendix B

Multifractal Measures

Carl J. G. Evertsz1 and Benoit B. Mandelbrot2

Before we generalize [fractal sets to measures], it may be recalled that, among our uses of fractal sets [to describe nature], several involve an approximation. While discussing clustered errors, we repressed our convic• tion that, between the errors, the underlying noise weakens, hut does not stop. While discussing the distribution of stars, we repressed our knowledge of the existence of interstellar matter, which is also likely to have a very irregular distribution. While discussing turbulence, we approximated it as having [nonfractal] laminar inserts. In addition, no new concept would have been needed to deal with the distribution of minerals. Between the regions where the abundance of a metal like copper justifies commercial mining, the density of this metal is low, even very low, but one does not expect any region of the world to be totally without copper. All these voids [within fractals sets] must now be filled - without, it is hoped, inordinately modifying the mental pictures we have achieved thus far. This Chapter will outline a way of reaching this goal, by assuming that various parts of the whole share the same nature. Benoit B. Mandelbrot3

1Center for Complex Systems and Visualization, University of Bremen, Postfach 330 440, D-2800 Bremen 33, Germany. 2Mathematics Department. Yale University, Box 2155 Yale Station, New Haven, CT 06520, USA and Physics Department, IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA. 3Introduction to Chapter IX of Les objets fractals: forme, hasard et dimension, 1975. A related text appears in B. B. Mandelbrot, The Fractal Geometry of Nature, 1982. 922 B Multifractal Measures

B.l Introduction

The bulk of this book is devoted to fractal sets. A set's visual expression is a region drawn in black ink against white paper (or in white chalk against a blackboard). A set's defining relation is an indicator function I(P), which can only take two values: I(P) = 1, or I(P)="true" if the point P belongs to the set S; and I(P) = 0, or I(P) ="false", if P does not belong to S. However, as stated in the 1975 quote which opens this appendix, most facts about nature cannot be expressed in terms of the contrast between "black and white", "true and false", or "1 and 0". Therefore, those aspects cannot be illustrated by sets; they demand more general mathematical ob• jects that succeed to embody the idea of "shades of grey". Those more general objects are called measures. It is most fortunate that the idea of self-similarity is readily extended from sets to measures. The goal of this appendix is to sketch the theory of self-similar measures, which are usually called multifractals. We shall include various heuristic arguments that are often used in this context, and then (section 4) describe the proper probabilistic background behind the concept of multifractal.4 Against this background, the nature of the usual heuristic steps becomes clear, their limitations and proneness to error become obvious, and the unavoidable generalizations demanded by both logic and the data become easy. However, these generalizations are beyond the scope of this appendix.

B.l.l Simple Examples of Multifractals

Consider a geographical map of a continent or island. An example of a measure 11 on such a map is "the quantity of ground water". To each subset S of the map, the measure attributes a quantity 11(S), which is the amount of ground water below S, down to some prescribed level. Now divide the map into two equally-sized pieces S 1 and S2. It will not come as a surprise if their respective ground water contents 11(SJ) and 11(S2 ) are unequal. If S 1 is subdivided further into two equally sized pieces Sn and S12, their ground water contents would again differ. This subdivision could be carried through to the size of pores in rocks, where some pores are found filled with water and others empty. This is a familiar story: some countries have more ground water than others ----> parts of a country contain more ground water than others ----> you may drill a well and find flowing water, while your neighbor finds none ----> and so on. Many other quantities exhibit the

4The probabilistic approach to multifractals was first described in two papers by B.B. Mandelbrot: Mandelbrot, B.B., Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier J. Fluid Mech. 62 (1974) 331. Mandelbrot, B.B., Multiplications a/eatoires iterees et distributions invariantes par moyenne ponderee aleatoire, I & II, Comptes Rendus (Paris): 278A (1974) 289-292 & 355-358. These papers, together with related ones, will soon be reissued in Mandelbrot, B.B., Selecta Volume N, Multifractals & 1/f Noise: 1963-76, Springer, New York. B.l Introduction 923

same behavior; that is, the quantity

J.L = the amount of ground water below S is an example of a measure which is irregular at all scales. When the irregularity is the same at all scales, or at least statistically the same, one says that the measure is self-similar, or that it is a multifractal. A Sierpinski gasket is a self-similar set, in the sense that each piece (how• ever small) is identical to the whole after some rescaling and translation; something similar holds for multifractal measures. The Chaos Game and Examples of multifractal measures have already entered in previous the Pascal Triangle chapters of this book. One is the chaos game that corresponds to an iter• ated function system or IFS that is run with unequal probabilities p 1 = 0.5, p2 = 0.3 and p 3 = 0.2 for the various reducing similarities (section 6.3), and the other is the Pascal triangle mentioned in chapter 8 and further discussed in section B.2.2. Let us take a closer look at the Sierpinski IFS. We saw that playing the chaos game long enough produces a Sierpinski gasket with fractal dimension D = log 3/ log 2. We also found that the subtriangles making up the Sierpinski gasket were visited with different probabilities - which are summarized in figures 6.22 and 6.23 for the first two levels. On very rough examination of the first figure, the subset with address 3 seems to include a smooth distribution of hitting probabilities. But closer examination reveals an irregular distribution among its subsets with addresses 31, 32 and 33. The same holds for the parts I and 2, and for closer examination of a part such as 32. We also see that if part 3={31,32,33} is blown up by a factor 2, while its component probabilities are multiplied by a factor 5, we achieve, not only a geometric fit, but also a fit of the probabilities to those at level 1 in figure 6.22. Again, the same holds for parts I and 2, with factors 2 and 3t, respectively. Hence, up to a numerical factor, the distribution of the hitting probabilities is the same in each of the subsets 1, 2 and 3. If such an exact invariance holds for all scales the overall distribution is said to be linearly self-similar. For the random IFS this invariance is exactly the one under the Markov operator defined on page 331 and discussed later on. The multifractal measure produced by the random IFS is simply the probability of hitting a subset of the triangle. The distribution of these probabilities is shown in figure B.l. The Sierpinski IFS can be interpreted as a caricature model for the above ground water example, by taking the total quantity of ground water as unity, and identifying the quantity of ground water with the hitting probability. A Measure-Generating Another way to look at the Sierpinski IFS measure is suggested by the Multiplicative Cascade figures 6.22 and 6.23 and equation (6.3). We note that, while each triangle is further fragmented into 3 subtriangles (as in figure 2.16), the measure JL (or hitting probability) is also fragmented by factors Pt = 0.5, P2 = 0.3 and p 3 = 0.2. Denote the measure of the set with address 3 by J1:1. Its 3 sub• parts with addresses 31, 32 and 33 carry the measures /L31 = PtJ1 3 , /L32 = P2J1 3 and /L33 = P3J1 3 . A single process fragments a set into smaller and smaller components according to a fixed rule, and at the same time fragments 924 B Multifractal Measures

Trinomial Measure The self-similar density of hitting probabilities of the IFS on stage 8 of the Sierpinski gasket. This is a 3-dimensional rendering of figure 6.21. The height of the function above each sub-triangle is propor• tional to number of hits in the limit of infinite game points. In order to draw this illustration we did not play the game infinitely long, but instead used the fact that this distri• bution is the 8th stage of a trinomial multiplicative cascade with mo = PI, m1 = pz and mz = p3, where the p, are the probabilities for the different contractions in the random IFS discussed in section 6.3, i.e., PI= 0.5,pz = 0.3 and P3 = 0.2.

Figure B.l

the measure of the components by another rule. Such a process is called a multiplicative process or cascade. Multiplicative processes are a very important paradigm in the theory of multifractals and play a central role in this appendix. In the language of multiplicative processes, the fragmentation ratios Pi are usually called multipliers and are denoted by rn with various indices. In the Sierpiiiski IFS, the fragmentation of the set yields a fractal. This feature is a complication, and is not essential. To avoid it, most of this appendix uses multiplicative rules that operate over an ordinary Euclidean set, usually the unit interval, but are such that the measure is fragmented non-trivially.

B.1.2 Characterization of Multifractals

Let us step back, and apply the idea of box-counting dimension to the setS supporting a measure (in the IFS example this was the Sierpiiiski gasket). One covers S with a collection of boxes of size c One evaluates the number N (E) of boxes needed to cover the object, and one finds the dimension D through the scaling relation N(E) rv cD. However, simply counting the boxes is like counting coins without caring about the denomination. When the set supporting the measure is Euclidean - as it will be in this appendix - the value of its fractal dimension only confirms that there is nothing fractal about this support. Thus, D is not sufficient to give a quantitative description of the self-similar measure supported by this set. Instead, the B.l Introduction 925

measure contained in each box must somehow be given a weight. A priori, the obvious weight would be the average density of probability in each box. In a Euclidean space of dimension E (or, more generally, in a space of embedding dimension E), the density is simply defined as t.t( S) / EE. When this density varies slowly, it can be mapped in the form of a relief, or in the form of lines of constant height. As E --+ 0, one expects this relief to tend to a limit. Furthermore, in order to characterize the irregularity of the spatial distribution of a measure, the first step - but not the last! - is to draw the familiar frequency distribution of its density. If the measure is random, one draws either the frequency distribution of the density in a sample, or its probability distribution. However, in the case of self-similar measures, this familiar process loses all meaning - simply because, as we shall see, the density itself loses all meaning. Instead, the loose notion that ordinarily leads to a density becomes embodied in a very different and more complicated quantity, log f.L(box) o: = called the coarse Holder exponent . logE This is the logarithm of the measure of the box divided by the logarithm of the size of the box. For a large class of self-similar measures, it turns out that o: is restricted to an interval [o:m;n, O:maxJ, where 0 < O:min < O:max < oo. But the study of some of the most interesting multifractal phenomena (such as turbulence or aggregation of particles into clusters) often demands O:min = 0 and, or O:max = 00. Once o: has been defined, the first step - but not the least! - is just as above: to draw the frequency distribution of o:, as follows. For each value o:, one evaluates the number NE (o:) of boxes of size E having a coarse HOlder exponent equal to o:. Now suppose that a box of side E has been selected at random among boxes whose total number is proportional to c E. The probability of hitting the value a of the coarse HOlder exponent is p,(a) = N,(a)jcE. Again, the first impulse would be to draw the distribution of this probability, but this would not be useful. In the case of interest to us, this distribution no longer tends to a limit as, E --+ 0, hence an intrinsic characteristic is to be found elsewhere. The considerations to be explored in detail in this paper will show that it is, necessary, instead, to take weighted logarithms and consider either of the functions

f,(o:) = _log NE(a), (B.l) logE or

C,(a) = -!ogp,(a). (B.2) logE As E--+ 0, both f,(a) and C,(a) tend to well-defined limits f(a) and C(a). The function C(a) is more widely applicable, but the function f(a) is more widely known. When f (a) exists one has C(a) = f(a)- E. (B.3) 926 B Multifractal Measures

The definition off (a) means that, for each o:, the number of boxes increases for decreasing E as N<(cx) ~ cf(a)_ The exponent f(a) is a continuous function of a. In the simplest cases, the graph of f(a)- often called .f(a) curve - is shaped like the mathematical symbol n, usually leaning to one side. The values of .f(a) could be interpreted loosely as a fractal dimension of the subsets of boxes of size E having coarse Holder exponent a in the limit E ---> 0. As E ---> 0, there is an increasing multitude - increasing to infinity - of subsets, each characterized by its own a, and a fractal dimension .f( o: ). This is one of several reasons for the term mult!fractal [ 191].

B.1.3 Summary This appendix restricts itself to the simplest multiplicatively generated multi• fractal measures. For the more delicate examples of multi fractal measures, it refers the reader to the literature. We have already seen a close connection between the IFS and multiplicative processes. Section B.2.2 will digress to discuss the multiplicative process that lurks behind the Pascal triangle in figures 8.14 and 8.32. There is evidence that multiplicative processes can account for many multifractal measures such as those related to the electrostatic charge distribution (the harmonic measure) on fractal bound• aries [175], wavefunctions [289] and random resistor networks [147]. But this does not mean that every self-similar measure is multiplicatively gen• erated. For example, many models for the multifractality of the dissipation field in turbulence are based on multiplicative processes, but a physical counterpart remains elusive. The very simplest multiplicatively generated self-similar measure is the binomial measure. For it, .f (o:) will be evaluated in three ways: the his• togram method [64, 65], the method of moments [191, 205] and large devi• ation theory [64, 65]. The method of moments5 is easy to use mechanically, and therefore has been applied very widely. In our first three sections, the only prerequisite is elementary analysis. Section 4 goes further, and casts self-similar measures in their proper probabilistic setting. It shows that multiplicatively generated multifractal measures are intimately related to a standard topic in probability theory, namely the behavior of sums of random variables. The method of moments and the histogram method are consequences of the theory of large deviations in such sums. While it is more technical, section 4 requires no prior knowledge of sums of random variables or of large deviation theory. The conceptual superiority of the probabilistic approach has immediate practical consequences: it is more general. It explains the nature of various mechanical manipulations, and provides tools to handle and understand self-similar measures to which the

5The earliest reference and the reference that is best known are, respectively, Frisch, U., Parisi, G, Fully developed turbulence and intermittency in Turbulence and Predictability of Geophysical Flows and Climate Dynamics, edited by Ghil, M., Benzi, R. and Parisi, G., North-Holland, New York, p. 84 (1985) Halsey, T.C., Jensen, M.H., Kadanotf, L.P., Procaccia, I. and Shraiman, B.l.: Fractal measures and their sinf?ularities: The characterization of strange sets. Phys. Rev. A 33 1141 (1986) B.2 The Binomial and Multinomial Measures 927

method of moments fails to apply. Hence, section 4 of this appendix is an introduction to more advanced literature. The reader may want to consult other books, reviews or articles about multifractals: for example references [263, 32, 90, 2, 64, 65].

B.2 The Binomial and Multinomial Measures

The introduction discussed the central role multiplicative processes play in the theory of multifractal measures. This section concerns the very simplest multiplicative process, one that generates the binomial measure. This mea• sure appears naturally in the Pascal triangle figure 8.14 and 8.32, and, with a little modification - the replacement of the binomial by the trinomial, it links to the IFS discussed in the introduction.

B.2.1 A Measure-Generating Multiplicative Cascade Exact (or linear) self-similarity of measures is best illustrated with the bi• nomial measure, (also called the Bernoulli or Besicovitch measure) [64]. In the spirit of the construction of the exactly self-similar Sierpinski gasket through a geometric cascade, this measure J.L is recursively generated with the multiplicative cascade that is schematically depicted in figure B.2. This cascade starts (k = 0) with a uniformly distributed unit of mass on the unit interval I = ! 0. = [0, 1]. The next stage ( k = 1) fragments this mass by distributing a fraction m 0 uniformly on the left half Io.o = [0, ~] of the unit interval, and the remaining fraction m 1 = l-m0 uniformly on the right half 10.1 = [ ~, 1]. At this stage, the left half carries the measure J.L( Io.o) = m 0 and the right half carries the measure J.L(l0.1) = m 1. In this process, because J.L(Io.) = J.L(lo.o) + J.L(lo.l) = mo + m1 = 1, the original measure of the unit interval is conserved; the J.L's appear like probabilities, and one says that fL is a probability measure. At the next stage (k = 2) of the cascade, the subintervals 10.0 and 10 .1 receive the same treatment as the original unit interval. That is, Io.o is split into the intervals Io.oo = [0, ~] and 10.01 = [~, !J of size 2-k, and the mass is further fragmented. Similarly, Io. 1 is split into a left half 10.10 and a right half 10. 11 • Writing J.L(lo.oo) = J.Lo.oo, and similarly for the other intervals, this second stage of the cascade yields

The condition m 0 + m 1 = I continues to insure that the original unit of mass is conserved. At the kth stage of the cascade, the mass is fragmented over the dyadic intervals ( i2-k, (i + 1)2-k], where i = 0, ... , 2k - I. Recall that a point x E [0, I] is said to have the binary expansion 0.;31{3 2 ••• /h when x = {312-1 + {322-2 + ... + f3k2-l with f3i E {0, I}. For dyadic points, like x = ~, this expansion is ambiguous, and may end with either an infinity of 928 B M ultifractal Measures

0 0 1\ /' \ 0 I 0 1 0

0 0

0 1

Figure 8.2 : The multiplicative cascade generating the binomial measure. At each stage the mass of each of the previous dyadic intervals is redistributed as follows: A fraction m 0 goes to the left half and m1 = I - mo to the right half. Here we took mo = ~ and m1 = ~- The density of the measure is shown for the first 8 stages. The scales on the coordinate axes have been kept the same throughout the figure . The actual measure of a dyadic interval is the integral of this density. For example, the measures of the 4 intervals of size ~ at stage 2 are momo, momJ , m1mo and m1m1 . B.2 The Binomial and Multinomial Measures 929

zeroes or an infinity of ones; in the present application, one must choose the former expansion. An arbitrary dyadic interval Jk = Io.(3,f3z ... fh of size 2-k consists of all points x E [0, I] whose binary expansion starts with 0.(3J(32 •.• f3k· To give an example, (31 = 0 if our interval, Jk, is in the left half J0 of the unit interval, and (31 = 1 if Jk lies in the right half 11• Similarly, (32 = 0 if Jk lies in the left half of If3,, etc. Clearly, the measure of the dyadic interval Io.f3,f3, ... f3k equals

k

II. - mnomnl ,-O.f3t (1zf33 ... f3k - ITm (3, - 0 I ' (8.4) t=l

where n 0 is the number of digits 0 in the address 0.(3 1(3 2(h ... (jk of the left end of the interval, and n 1 = k - n0 the number of digits I. Since the binomial measure of each dyadic interval of size 2-k is the result of a multiplication of k multipliers mf3, it is called a measure generated by a multiplicative process. The binomial multifractal measure is the measure /1!3 which attributes masses according to the equation (B.4) to the dyadic subintervals of the unit

interval (Note for the mathematically minded: the measures JLO.f3 1f32fh .. fh of all dyadic subintervals of the unit interval can be extended to a Borel field of subsets of [0, 1]). The multiplicative cascade is a mechanism for producing this measure. In the case m 0 = m 1 = 4 this measure reduces to the uniform (Lebesgue) measure.

B.2.2 The Pascal Triangle and the Binomial Measure

This section is a digression that can be skipped with no Joss of continuity. When viewed in a mirror, the distribution in figure 8.32 resembles closely the density of the binomial measure, as shown in figure B.2. This is not a coincidence. Remember that the height of the r 1h column in figure 8.32 equals the number h (r) of black squares in the r1h row of the Pascal triangle (mod.2) in figure 8.14. Turning the page 90° counterclockwise (so that the columns point upwards) and looking only at the black geometry of the triangle, one sees that the total number of black squares in the rows [4, 7] is twice that in the rows [0, 3], i.e., L~= 4 h(r) = 2 L~=o h(r). This factor of 2 turns out to be ubiquitous in this application. Every time one considers a block of rows [0, 2k - I], its right half [2k-l, 2k - I] contains twice as many black squares as its left half [0, 2k-I - I]. If this left half, in tum, splits into two halves of size 2k-2 , one again finds the ratio I : 2 between the left and right, etc. To conclude: up to an overall factor, the numbers of black triangles in the columns of figure 8.14 have the same structure as those of the mirrored binomial multifractal in the k = 4 stage of the multiplicative cascade shown in figure B.2. In general, one finds that 3-k h(r) is the binomial measure of the dyadic interval [r 2-k, (r +I) 2-k] for r = 0, ... 2k - 1. This highly visual analogy establishes a relationship 930 B Multifractal Measures

between the distribution h( r) of the Pascal triangle and the binomial measure f-L with multipliers mo = ~ and m1 = ~- The formal connection between the triangle and the binomial multiplica• tive process follows from an earlier result (section 8.5). Equation (8.18) states that for r = (30 + (312 + (3222 + ... f3k2k with f3i E {0, I} (that is, if f3of31 ... f3k is the binary expansion of r), one has

where n 0 is the number of O's in the expansion of r and n1 is the number

of 1's. Comparing this with equation (B.4), one finds h(r) = 3k /Lo.(3 1(32 ..• f3k, where f-L is the binomial measure with mo = ~ and m1 = ~- Note that we replaced the digit "(30" by "0." to emphasize that 0.(31(32 ... f3k stands for the dyadic subinterval of the unit interval as discussed in the previous section. It is important to mention in passing that both the binomial measure The Markov Operator and the hitting probability of the IFS on the Sierpinski gasket are invariant M(v), the Binomial measures of the Markov operator M ( v) discussed on page 331. Measure and the The binomial measure f-LB with multipliers mo and m 1 is the invariant Sierpinski IFS measure of the Markov operator M(v) = m 0 vw]1 + m 1 vw:2 1, with

w 1 (x)=~X

wz(x) = ~x + ~-

That is, M(J.LB) = f-LB. The trinomial measure in figure B.l, associated with the random Sierpinski IFS in section 6.3, is the invariant measure of M(v) = 0.5 vwj1 + 0.3 vw21 + 0.2 vw;,- 1, with

w1(x,y) = (~x, ~y)

wz(x,y) = (~x + 1, h)

w3(x, y) = (~x + ~, ~y + ~V3).

B.2.3 Self-Similarity and Singularities

The introduction briefly discussed the notion of self-similarity in the case Self-Similarity of an IFS. We now discuss this notion in the case of the binomial measure.

The measure of the arbitrarily selected interval Io.(3 1f32 ••• f3k is {-LO.f3 1f32(33 ••• rh times smaller than that of the original unit interval, which had mass I. Apart from this overall difference, the mass in both these intervals is fragmented in exactly the same way. That is, consider the mass distribution on the B.2 The Binomial and Multinomial Measures 931

interval If3 1f32f33 •.• f3k of the stage k + k' of the multiplicative cascade; then by spatially rescaling this subinterval by a factor 2k, and renormalizing its

mass by a factor (J.Lf3 1 f3zf33... f3k) -I, one recovers the mass distribution on the whole interval at the stage k' of the cascade. It is in this sense that the mass distribution (or the measure) is said to be self-similar. We now show that such self-similar measures are very singular, and discuss in more detail the notions of Holder exponents a mentioned in the Introduction. The Coarse and the For the uniform measure (the Lebesgue measure), the density is 1 every• Local Forms of the where in [0, 1]; i.e., the Lebesgue measure of an interval of size E is E. For the Holder Exponent binomial measures, the situation is altogether different. Near x = 0, equa• tion (B.4) shows that p,([O, 2-k]) = m~ = (2-k)vo with v0 = -log2 m 0 . That is, the measure in the neighborhood of 0 scales as

p,([O,E]) rv E'", with a= Vo.

The density p,j E scales like Ea-l, and if a =I 1 the limit of the density as E ----> 0 is degenerate, equal to either 0 or oo. When the measure in the £-neighborhood of a point scales as a power law in the limit E ----> 0, the exponent a of this law is called the local Holder exponent (Alternative terms, such as singularity index or singularity strength [205] are also encountered in the physics literature). That is, given a point x in the support of the measure, the local Holder exponent is defined as

a(x) =lim logp,(Bx(E)), (8.5) E-->0 logE

where Bx(E) is a ball of size E around x. When the limit fails to exist, we shall say that the local HOlder exponent is undefined. (The mathematician's more elaborate local definition replaces lim by a limsup. This replacement broadens the cases where the Holder exponent is defined locally; but this detail cannot be discussed here). In most practical applications, the limit E ----> 0, which enters in the definition of the local Holder exponent, cannot be taken. One must, instead, work with the concept of coarse (or coarse-grained) Holder exponent. This is a number attributed to each finite interval. For any box B (E) of size E, the coarse-grained Holder exponent is defined as

logp,(B(E)) a= . (8.6) logE

Thus, the concept of Holder exponent has a local and a coarse version. Both enter into the theory of multifractals, but the coarse HOlder exponent plays an especially central role. As mentioned in the introduction, a serves to label the boxes covering the set supporting a measure, thereby allowing a separate counting for each value of a. For dyadic intervals Jk of size 2-k, 932 B Multifractal Measures

equations (B.4) and (B.6) yield

(0 (J (J (J ) log flO.fJd32 ... fh a . 1 2··· k = log2-k· (8.7) log m~"rn·;··• no k -- no ----"------'- = -v0 + ---v1, -klog2 k k where

v 1 = - log2 m 1 and vo = - log2 mo. (B.S)

Let cp0 = cp0(Jk) be the fraction n0 /k of O's in the address 0.(31(32 •.. fh of interval Jk. Equation (B.7) becomes

(B.9)

The notation amin and amax is explained shortly. It follows that the coarse HOlder exponent a of an interval Jk only depends on the fraction 'Po of digits 0 in its address 0.{3!{32 ... Pk· Note that we use the same symbol a for both the local HOlder exponent (equation (8.5)) and the coarse Holder exponent (equation (B.6)). When present, the argument of the former is a point on the support of the measure, and that of the latter the address of a dyadic interval. In both cases cp0 in equation (B.9) is the fraction of zeroes in their binary address.

Without loss of generality, we can take m 1 <::: m 0 , so that v0 <::: v1 and equation (B.7) yields v0 :::; a <::: v 1• (The same restriction also applies to the local HOlder exponent). The extreme values of a are usually denoted by amin and amax, i.e.,

amin = Vo and amax = V].

This restriction on the values of the coarse Holder exponent is independent of the size 2-k of the dyadic intervals; hence, it is independent of the scale on which the fractal measure is probed. This makes a an ideal index with which to mark the boxes of any size covering the set supporting the measure. (Equation (B.22) will provide an alternative way of rewriting equation (B.7), and emphasizing the role of the coarse HOlder exponent in transforming a multiplicative process into an additive process). Keep the fraction cp0 = n 0 /k of O's in equation (B.9) constant, and let k ______, oo thereby squeezing Jk to a point. By permuting the order of appearance of the digits 0 and 1, increasingly many points x E [0, I] are found with fixed Holder exponent a(x) = cpoamin + (l-cp 0 )amax· In the case of the binomial, the extreme values 11'min = v0 and amax = v1 continue to be attained (respectively) in the left- and the right- most dyadic subinterval of the unit interval. But this is a peculiarity of the binomial measure; in general the minimal and maximum coarse and local HOlder exponents can lie anywhere in the support of the measure. B.2 The Binomial and Multinomial Measures 933

Singular Distributions A measure 11 on [0, 1] has a density p(x) in a point :1: if . J.1(Bx(t)) ( ) 11m = p X c--->0 E exists and satisfies 0 :S: p(x) < oo. If o:(x) is the local Holder exponent in x E [0, 1], then p(x) "' limE-to Ea(x)-l. In points x where o:(x) # I the density of the binomial measure is singular. Section 4.2 will show that, with probability 1, the local HOlder exponent of a randomly picked point in the support [0, 1] of the binomial measures is 1 a= (vo + vJ)/2 = -2log2(momJ).

A measure for which a= 1 occurs when, and only when, m 0 = m 1 = ~,in which case the binomial measure reduces to the uniform (Lebesgue) mea• sure. In the interesting cases, m 0 # ~, and the density is almost everywhere either 0 or oo, hence it is called singular. For reasons which will become apparent later on, a is usually denoted by o:0 or o:(O). When the local density is singular, it continues to be possible to define a E-coarse-grained density, by covering the set supporting the measure with boxes B(E) of size E and attributing the coarse density J.1(B)/EE to each box (E is the dimension of the box). Figure B.2 shows an example of a sequence of coarse-grained densities for the binomial measure.

B.2.4 The f (oo) Curve of the Binomial Measure The f (a:) curve describes the distribution of the coarse-grained Holders exponents. To introduce this function, we first compute the number Nk (ex) of intervals I"' of size 2-k with coarse HOlder exponent a. Equation (B.9) shows that the value of a: is determined by the frequency cp0 of zeroes in the expansion of the interval, and conversely, that to each a corresponds a unique cp0 (a:). Thus, the number of intervals with coarse Holder exponent a: is given by the number of ways one can distribute n 0 = cp0 ( a: )k zeros among k positions. This is the binomial coefficient

This fact explains why the term binomial is applied to the binomial distrib• ution and measure. To simplify, write z instead of cp0 , and apply Stirling's approximation k!::::::: Vhl (k/e)"'. It yields ( k) k! zk - (zk)!(k- zk)! VZk (zk)zkJk- zk (k- zk)k-zl> · Many terms cancel out, leaving 934 B Multifractal Measures

f (a:) Curve for Binomial Measure f(a) The f (a) curve for the binomial 1 measure with mo = ~ and mt = t. At each level of coarse graining of the measure, one can distinguish in• tervals with different coarse H6lder exponents a. As the coarse graining box-sizes become smaller, the num• ber N, (a) of boxes with certain a increases as N<(a) cv E-f(a)_ q = -00 / u Figure B.3

1 withg(z) = -log2 (zz(l-z) -z). Usingequation(B.9)toeliminatecp0 = z in favor of the variable a:, we find

(B.IO) with

_ _ O:max - 0: l ( O:max - 0: ) !( a: ) - og2 O:max - O:min amax - amin

- a - amin I og2 ( a - amin ) . amax - O:min amax - amin

The graph of f(a) is shown in figure B.3. Expanding f(o:) around a = ao = ( O:min + amax) /2 using the approximation In x = x - x 2 /2, yields for a:- ao « 1,

_ 2 a- ao 2 !( 0: ) -l-- ( ) (B.ll) In 2 a max - O:min

The f (a) curve has the following noticeable properties.

(a) It is defined for 0 < amin

For finite k, the above equations only hold when kcp0 is an integer h between 0 :S h :S k. Let cp~ = (h- 1)/k and rp~ = (h + 1)/k. For fixed k there is no a between a(cp~)

Another way of visualizing mu/tifractality is to consider all the points Hausdorff Dimension Versus x in the support of the measure for which the local Holder exponent Box-Counting Dimension is a (equation (8.5)). For each value of a this defines a set Aa c [0, 1]. It is not difficult to show that the box-counting dimension of each of the sets Aa of the binomial measure is 1. (See the Proof at the end of this paragraph). Hence, instead of the box-counting dimension, we need the concept of Hausdorff dimension. It is beyond the scope of this appendix to go beyond simply stating that for a class of multifractal measures (including the above binomial measure) it has been rigorously shown [268] that the Hausdorff dimension of A a is f(a) (reference [12] includes a proof in the case of the binomial measure). Proof that the box-counting dimension of A<> is 1. It suffices to show that each dyadic subinterval Jk of the unit interval contains points with any coarse Holder exponent between Ctmin and Ctmax. Take any value a' E [etmin,Ctmax]- Let the binary expansion of Jk be 0.(3!(32 ... f3k· Remember that all x E [0, 1] whose binary expansion starts with digits 0.(31(32 ... f3k lies in Jk. So any choice of an infinite sequence of digits f3k+ 1 f3k+ 2 ... such that its fraction cp0 of O's satisfies equation (8.9) for a = a', yields a point x E Jk with binary expansion f3If3z ... f3kf3k+ 1f3k+z ... having a(x) =a'.

B.2.5 Multinomial Measures and the Legendre Trans• forms

In a multinomial cascade [64, 65, 239], the base b satisfies b > 2 rather than b = 2. Each stage of the construction redistributes mass or measure over b 936 B Multifractal Measures

Rough idea of the domain of (a, b) for a multinomial multifractal with b = 4. The domain's upper bound• ary defines the function f (a). Here, all the m" are different, Ctnun =

min" ( Vo, ... 1 V/, - 1) > 0 and OCrnax = max,(vo, ... ,Vb--1) < =·

a 0 Figure B.4

equally sized intervals, following the fragmentation ratios m 0 , m 1, ... , mb-I with :L~:ci mi = I. We already encountered a trinomial multifractal (b = 3) in the Introduction, namely, an IFS-generated trinomial measure on the

Sierpinski gasket, with multipliers m 0 =PI. m 1 = p2 and m 2 = p 3 . Here we restrict ourselves to multinomial measures supported by [0, 1]. To show how the notion of f (a) generalizes to these measures, let It C [0, 1] be an arbitrary b-adic interval of size b-k, and let its base-b address be O.fM32 ... f3k with f3i E {0, I, ... , b- I}. Denote by cp the point in E = b dimensional Euclidean space whose coordinates are the frequencies 'Pi of the digits i in this address. The combinatorics of the binomial measure, and the use of the Stirling approximation can be generalized to every interval It characterized by cp. For the measure of such It and its coarse Holder exponent one obtains

For the number of such intervals one finds NA( cp) "' (b-k)-b, with

In the binomial case, a single 15 = f (a) could be deduced from the value of a, but this possibility is not available here. A given a, indeed, allows a host of possible set of values of 'Pi constrained by :L 'Pi = I and a = - :L 'Pi 1ogb mi. After the points (a, 8) corresponding to all the values of a have been combined, the result is a domain of the plane [239], as shown in black in figure B.4. There is a powerful heuristic way of replacing this domain by its upper bound. This heuristic also provides the shortest path towards the Legen• dre transforms, which are essential in the study of multifractals. (A full B.2 The Binomial and Multinomial Measures 937

mathematical justification will be given in section 4.4). The key step is to argue that, given a value of a, the 8's are dominated by the largest among them which we denote by f(a). This requires solving for the point r.p that maximizes -I: 'Pilogb'Pi, given -I: 'Pilogbmi = a, and also I: 'Pi = 1. To solve this problem, the classical method consists of using Lagrange mul• tipliers [52]. It introduces a multiplier q, with -oo < q < oo, and yields bqlogb m, mq • 'Pi="' bqlog m· = ~' L.Jj b 1 L.Jj mj and thus that

a(q) ~- ~ (L7~J) log,m; and

f(a(q)) ~- ~ ( L7~1) tog, (L7~r)

Here, the quantities I:j mj and T(q) = -Iogb I:j mj play the roles the "partition function" and the "free energy" are known to play in thermody• narrucs. In terms of T(q), the Lagrange multipliers yield 8T(q) q8T a=-- and f(a) =--T =qa-T. 8q 8q As announced, these steps replace the black domain in figure B.4 [239] by its upper boundary, which is the graph of a function f(a), which has all the properties that apply in the binomial case, except for symmetry. Knowing T(q) for all values of q, one can trace all the straight lines of equation 8q(a) =qa-T. These straight lines define f(a) as their envelope, namely f(a) = minq(qa- T). This transformation is called a Legendre transform, we shall encounter it repeatedly in contexts. In many cases, a graphical approach is illuminating. If the lines repre• sented by Dq(a) are traced in green, they merge into a green domain in the (a, 8) plane, which "surrounds" the black domain that we have considered previously. Beyond the One way to expand the notion of multiplicative process is to allow Multinomial Measures b = oo. Examples can be found in references [242, 243, 245]; one example is shown in figure B.5. The interesting thing about these measures is that although they are exactly self-similar (a piece, if expanded, looks exactly the same as the whole), the f(a) curve is very different from the symbol n encountered for the binomial. One can construct examples for which a) amin = 0 and amax = oo, b) the maximum is not quadratic and c) the maximum is not attained for one value of a but over a halftine of a values. We will briefly comment on this in section B.5. 938 B Multifractal Measures

Measure With Left-Sided f(a) An example of an exactly self• similar multiplicatively generated measure, for which a(O) = oo hence, a fortiori, arnax = oo. It fol• 0.004 lows that f (a) is defined for alllln < a. Such measures are called left• sided. The corresponding function r(q) is not defined for q < 0, and the method of moments does not adequately describe the whole mea• sure. 0.002

0 0.5 1.0 Figure B.5

B.2.6 Random Multiplicative Cascades

A second way to expand the notion of multiplicative process is to usc random multipliers. When b = 2, such a process proceeds in the same way as the binomial measure, with the crucial difference that each multiplier is the outcome of some probabilistic process such as throwing dice. Just as most fractal sets in nature are random fractal sets, the random multiplicative processes are very useful for modelling real multifractal measures like those in turbulence [233, 142, 263, 254] or diffusion limited aggregation [316, 246, 174]. For such measures, all properties of the f(a) mentioned in the last section may be violated, except that its graph always lies under the bisector f(a) = a . Such measures are described in great detail in references [64, 651. We will postpone a brief discussion of them to section B.S.

B.3 Methods for Estimating the Function f (a) from Data

It is nice to say that a measure is multifractal if it is self-similar, in the sense described for the binomial measure. In our study of the binomial, the cascade was a given, and the quantity Nk(rx)drx could be evaluated for all k and interpreted as the number of intervals of size 2- k in the kth stage of the cascade having a coarse Holder exponent between a and a+ da. But it B.3 Methods for Estimating the Function f(a) from Data 939

is important to keep in mind that behind most measures there is no obvious multiplicative cascade! When one may exist, like perhaps in turbulence or the distribution of mass in the universe, this cascade is history; the only thing present is the measure it has created. How does one find out whether a given measure is multifractal? The key is that, when only one stage k of a measure is given, one can reconstruct any previous stage h < k by coarse-graining with intervals of size 2-h. We shall examine two methods for obtaining an empirical estimate of f (a) for an arbitrary measure.

B.3.1 The Histogram Method

Given a measure f.L, the histogram method involves the following steps: (a) Coarse-grain the measure with boxes of size E. This yields a collection of boxes {B 2 (E)}~C~l, where N(E) is the total number of boxes needed to cover the set supporting the measure. (b) t-L(Bi) being the measure of box i, compute the coarse Holder exponent a, = log f.L,/ log E. (c) Make a histogram. That is, subdivide the variable a into bins of suitably small size .6.o: and estimate the number density Nc(a) by recording the number of times Nc (a ).6.o: that a specific value of the HOlder exponent falls between a and a + .6.o:. (d) Repeat step (c) for different values of coarse-graining size f. (e) Since we expect

plot - log NE (a)/ logE versus a for different values of f. This method suggests that a measure be called multifractal when the resulting plots collapse onto a curve f (a) if E is small enough. We must state that self-similar measures exist [242, 243, 245] for which the collapse to a function f (a) is extraordinarily slow, and largely irrelevant for any physically meaningful E. A test of these steps in the case of the binomial measure, and methods to accelerate the convergence are discussed in reference [253].

B.3.2 The Method of Moments

The method of moments [205] is based on a quantity called partition function (because of analogies with the partition function in the theory of equilibrium thermodynamics). It is defined as

N(t) Xq(E) = L JL{, q E R. (B.I2) t=l 940 B Multifractal Measures

For example, take the binomial measure and denote by Xq ( Ek) the partition function at coarse-graining box-size Ek = 2-k. An inspection of figure B.2 immediately yields Xq(Eo) = lq, Xq(E 1) = mZ + mi and Xqh) = (mZ + mf) 2. More generally,

Xq(Ek) = (m6 + mf)k.

Returning to equation (B.l2) in the general case, let us rewrite the mea• sures Jli of the boxes as Jli = Ea', yielding Xq (E) = 2:::(t) (Ea, )q. Mo• tivated by the results for the binomial measure, denote by N, (a )da the number of boxes, out of the total N (E) for which the coarse HOlder ex• ponent satisfies a < ai < a + da. Assume, in addition, that there exist constants amin and amax such that 0 < amin < a < amax < oo, and that N, (a) is continuous. Then the contribution of the subset of boxes with ai between a and a+ da to Xq(E), is N,(a)(E<>)qda. Instead of adding the contribution of each box i separately, integrate over da to add the contri• butions of subsets whose coarse HOlder exponent is between a and a+ da. Thus,

Xq(E) = JNe(a)(Ea)qda.

If Nf(a) "'cf(a), it follows that

Xq(E) = JEqa-f(a)da. (B.l3)

In the limit E -+ 0, the dominant contribution to the integral comes from a's close to the value that minimizes the exponent qa- f(a). If f(a) is differentiable, the necessary condition for the existence of an extremum is a oa {qa- f(a)} = 0.

For given value of q, the extremum occurs for the value a a(q) that satisfies

!!_f(a)l =q, (B.l4) oa a=a(q) and this extremum is a minimum as long as

()2 I 02 af(a) < 0. a=a(q) Thus, the function f(a) should be cap convex as in figure B.3, and for the a= a(q) where the minimum is attained the slope of f(a) is q. Note that by now we encountered three different arguments for a: i) x E [0, 1], ii) the dyadic expansion 0.{3!(32 ... f3k of an interval Jk and iii) the variable q. Only the latter two which are easily distinguished will be used in the future. B.3 Methods for Estimating the Function f(a) from Data 941

Keeping only the dominant contribution in equation (B.l3), and intro• ducing T(q) = qa(q)- f(a(q)), we find

Xq(E) rv Er(q)_ (B.l5)

For the binomial measures,

( ) 1. Iog(mZ + mi)k ( q q) T q = 1m = -log2 m 0 + m 1 , (B.l6) k--->oo log 2-k and for the multinomial measures b-1 T(q) =-1ogb L m{. (B.17) i=O Returning to the general case, it is not difficult to show that a aq T(q) = a(q). (B.l8)

This shows that f(a) can be computed from T(q), and vice versa, by the identity f(a(q)) = q a(q) - T(q). (B.l9) This relation between T(q) and f(a) has already been encountered in section B.2.5; it is called a Legendre transform. As an example, the Legendre transform of equation (B.l6) yields the f (a) of the binomial measure in equation (B.l 0). Note that equation (B.l4) and the strict cap convexity off (a) imply that a(q) is a decreasing function of q, so amin = a(oo) and amax =a( -oo); thus T(q) should be strictly cap convex. The function T(q) is sometimes written as T(q) = (q- l)Dq, where the exponents Dq [209], are called generalized dimensions. (See also the discussion in section 12.6). In practice, to compute f(a) through the partition function requires the following steps: (a) Coarse-grain the measure with a covering { Bi (E)} ;~5;) of boxes of size E and determine the corresponding box-measures /Li = p,(Bi(E)). (b) Compute the partition function in equation (B.l2) for various values of E. (c) Check whether the plots of log Xq (E) versus logE are straight lines. If they are straight, T(q) is the slope of the line corresponding to the exponent q (see equation (B.15)). (d) Form f(a) by Legendre transforming T(q) (equation (B.19)). In real applications, the above steps must be carried out numerically. The usual reason is that there is no way to get analytic expressions for lack 942 B Multifractal Measures of theoretical knowledge about the phenomena. But, even if an analytic expression for the partition function is available, it may be to difficult to find an analytic expression for T(q) or f(a). When the check described under (c) gives a straight line, the method of moments is justified, and it yields the same f(a) as the histogram method. Moreover, given that moments tend to smooth the data, while the histograms method handles raw data, the method of moments converges much faster. At this point, we must voice a serious warning. The fact that the work must proceed numerically creates a strong temptation to proceed blindly, by resorting to ready-made computer programs. Unfortunately, some programs fail to include a demanding test for the linearity postulated under (c). In• stead, they go ahead and fit a straight line "mechanically," using a criterion such as least squares fitting. While statistically objective in all cases, those fitted T(q) have no physical meaning, unless the points plotted under (c) are straight. As a matter of fact, the literature is cluttered with thoroughly bizarre f (a) curves that were obtained objectively, but mean nothing. Equation (B.15) shows that under the assumptions mentioned under equation (B.I2), the partition function scales as a function of the box -size E for all q E R. One then says that "the q1h moment" of the measure exists for all q. For example, all moments exist, for the binomial and multinomial multifractals. The method of moments suggests saying that a measure is multifractal if, and only if, the function T( q) exists for all q E R. But this definition is not satisfactory. Self-similar measures exist [242, 243] for which T(q) fails to be defined for, say, q < 0. Self-similar measures for which all moments exist should be called restricted multifractals to indicate the existence of a broader class.

B.3.3 Properties of f (a) Let us briefly review some of the characteristics of f (a) for restricted mul• tifractal measures, i.e. measures for which the method of moments is ap• plicable and the histogram method converges reasonably fast. Let A<> (E) be the subset of boxes covering the support of the measure having a coarse Holder exponent between a and o: + dcx. The total measure carried by such a subset is J'L(A"(E)) = Nc(a)E 00 da '"""cf(n)+n. Now, the total measure in all boxes is I. This implies that f (a) ::; a; the reason is that the contrary would be absurd: if there existed a value of a such that f(a) > ex, it would follow that J'L( A"(t:)) ---7 = for f ---7 0. So an f (a) always lies under the bisector. Second, consider that value of n that maximizes J'L( k" ( t:) ). If it were true for all n that f (a) < a it would follow for all a that then J'L( A" (E) ) ---7 0 as E ---7 0. That would contradict the fact that the total measure is 1. So the f (a) curve should at least have one point in common with the bisector; because f (a) is concave, this point of intersection is unique, and occurs where f' (o:) = I. This corresponds with q = 1 in the method of moments (see equation (B.14)) and this value B.4 Probabilistic Roots of Multifractals 943

of o: is denoted by o: ( 1) or o: 1 • The subset of boxes A<> 1 (E) carries all the measure in the limit E --+ 0; i.e., !-l( A 01 (E) ) --+ 1 for E --+ 0. Restating equation (B.15) as T( q) = lim€->o(log I:f:(:) !-li) /logE and using equation (B.18) one finds that

(B.20)

An equation similar to equation (B .20) can be derived for f (o: ( q)) and can be used to find f(o:) directly [157]. In particular

'\:""N(<) l o:(1) = f(o:(l)) = lim L....i=l /-li Og/Ji <--->oo logE

The quantity- I:f:(:) 1-li log 1-li is related to entropy and information. Corre• spondingly the quantity f(o:(l)) = o:(1) is called the information dimension of the measure [32, 197, 209, 85]. f (o:( 1)) is also the dimension of the set carrying "all" the measure. (This set is called the measure theoretic support of the measure). To illustrate this, let us consider the IFS discussed in the introduction. We know that the visiting process is probabilistically ruled by a trinomial multifractal measure. Let n be the number of times we play the IFS. From the above discussion we conclude that it is possible to find a subset of (3k)f(n(I)) boxes of size 3-k whose total number of visits n 1 (n) satisfies n 1 (n)jn--+ 1 for n--+ oo and k --+ oo. The set A<>(I) carries all the measure. The number of elements in the sets A<> (E) are N, ( o:) ,....., c f ( <>). The set o: with the maximum number of elements has that value of o: at which f(o:) attains its maximum. From equation (B.14) it follows that the maximum should occur for o: = o:(O). Equation (B.19) yields f(o:(O)) = -T(O) and equation (B.12) gives xo(E) = N(E) ,....., cD. Combining this with equation (B.15) yields f(o:(O)) = -T(O) = D. So the maximum value of the f(o:) curve is the box-counting dimension of the geometric support of the measure. Since the number of boxes with coarse Holder expo• nent o:(O) is infinitely larger than those with other values of the coarse Holder exponent in the limit E --+ 0, one expects a randomly picked box of size E to be of type o:(O) (see also section B.4.2). For the binomial measure, equation (B.16) gives D = -T(O) = 1, which is the dimension of the set [0, 1] supporting the measure. From the symmetry of its f (o:) curve (equation (B.lO)) it immediately follows that o:(O) = (o:min +o:max)/2. 944 B Multifractal Measures

B.4 Probabilistic Roots of Multifractals. Role of f (a) in Large Deviation Theory

At this point, many questions remain to be raised and answered. Why should an f (o:) curve exist for a self-similar measure? Its existence for the binomial measure does not guarantee its existence in general. Indeed, self• similar measures exist for which f(a) exists, but is attained very slowly and is not of direct interest. If a useful f (o:) does not always exist for self-similar measures, then what about alternative quantitative descriptions? To a large extent, these questions can be answered when the self-similar measures are generated by, or can be mapped onto multiplicative cascades. The understanding of such measures is largely based on the following basic fact about fractals. Even when a fractal is nonrandom, like the Cantor set, it becomes a random set if its origin is chosen randomly. Similarly, examine the binomial measure in a randomly chosen interval. This measure is a random variable! Therefore, as we shall see, the H6lder exponent can be expressed as a sum of random variables [232, 64, 65]. This fact, again, is true both of random and nonrandom multifractals. It throws light upon the probabilistic roots of the notion of f(o:), and in addition- this is perhaps even more important - it allows one to capture the limitations of f (o:), while providing new tools to handle more complicated self-similar measures. The properties of sums of random variables are a central topic in prob• ability theory. The next section discusses the relevance to multifractals of three theorems dealing with such sums: i) the law of large numbers, ii) the Gaussian central limit theorem and iii) the large deviations theorem. No fa• miliarity with these subjects is assumed, and the section should serve as an introduction to literature applying more advanced results from probability theory to self-similar measures.

B.4.1 Transformation of a Multiplicative Cascade into an Additive Cascade

Suppose that the dyadic interval Jk = Iorhf32 •. .fh has been picked randomly. This amounts to picking a random sequence of digits {3 1(3 2 ... f3k where each f3i is either 0 or 1 with probability 4. Indeed, suppose that Jk has equal probability to lie in the left or right half of the unit interval. For the first digit, this means that Pr{(31 = 0} = !, and similarly Pr{(31 = I} = !• Furthermore, irrespective of whether the chosen dyadic interval Jk turns out to lie in Io.o or in ! 0.1, it will again have equal probability to lie in the left or right half subinterval; that is also Pr{(32 = 0} = Pr{(32 = I} = !- Equation (B.4) has shown that the measure of randomly picked dyadic interval Jk is P,o.(3 1f32 ••. f3k = IT~=I mf3,. Because the f3i are random, either 0 or I, the variables /1>(3., are also random, either m 0 or m 1. This means that this measure p, is the product of k statistically independent values of a random variable M, which can be either m 0 or m 1 with probability ! . B.4 Probabilistic Roots of Multifractals 945

A random variable can be thought of as a "mathematical coin or Random Variables die". It has prescribed probabilities for yielding certain results when "thrown". Our convention denotes random variables by boldface capital letters. The sample value, i.e., the outcome of a throw, is denoted by a corresponding lower case letter. For example, the random variable D meant to represent a real die with six faces would have a probability distribution Pr{D = d} = 1/6 ford= 1, ... , 6.

So the measure 1-l of a randomly picked Jk is a sample value of the ran• dom variable TI~=I M, where the random multiplier M has the distribution

I Pr{M = mo} = Pr{M = mi} = -. (B.21) 2

Equation (B.6) yields the coarse Holder exponent ak of such an interval Jk in the following form, which restates equation (B.7),

(B.22)

Here we use the variables Vf3 introduced in section B.2.3 (v., -log2 rn,, i = 0, 1). Thus, the coarse HOlder exponent of a random Jk is the random variable

(B.23)

This is the average of k independently chosen sample values v of a random variable V with distribution I Pr{V = vo} = Pr{V =vi}= -. (B.24) 2 Let us rephrase the above in terms of a well-balanced coin, that is, one with equal probabilities for head and tail. A collection of k identical coins (VI, V2, ... , Vk), each with one face marked v0 and the other v 1, are tossed and the sample average t L~=I V h is computed. These averages have the same distribution as the values of the coarse Holder exponent of randomly picked dyadic intervals of size 2-k in the support [0, 1] of the binomial measure. The distribution Pr{Hk 2:: a} is very much linked to the distribution P<(a)da discussed in the introduction (equation (B.2)), and will eventually link us back with the method of moments and the histogram method. Two paths are open

1) One can count the number of coarse HOlder exponents at the k1h stage of the multiplicative cascade which are larger than a, then divide this 946 B Multifractal Measures

number by the total number 2k of boxes needed to cover the unit interval, or 2) One can also make n series of k coin tosses, for each series compute the average, then count the number of times this average is larger than o: and divide this number by n and consider the result in the limit n --+ oo. The first is essentially the histogram method. We now follow the second path using the fact that the distribution of the coarse Holder exponent at the kth level of the multiplicative cascade is the same as the distribution of the random variable Hk defined in equation (B.23). This reformulation in terms of sums of independent identically distributed random variables (equation (B.23)) allows the use of many techniques in probability theory.

B.4.2 Law of Large Numbers and the Role of o:0 as the Most Probable HOlder Exponent

Tossing the coin marked with v0 and v1 k-times, yields, say, no times the value v0 and n 1 = k - n 0 times the values v 1 • Since the probability for head and tail are equal, one expects fork--+ oo that n0 jk--+ !. and similarly for n 1• This would mean that the sample average t (n 0v0 + n 1v 1) converges as k --+ oo to the expectation 1 l EV = -vo + -VJ. 2 2 Different forms of convergence yield different forms of the law of large numbers. The weak law of large numbers guarantees such convergence when the expectation EV of V exists. The strong law of large numbers is more interesting for the theory of multifractals. It states that, almost surely (with probability 1) the sample average will converge for k --+ oo to the expectation. That is

Pr { lim ~ tVh = EV} = 1. k--+oo k h=l Using the equivalence established in the previous section between the bino• mial measure and coin tossing, this equation shows that, with probability 1, the local Holder exponent at a randomly picked point in the support of the binomial measure equals EV; that is

Pr { lim Hk = Ev} = 1. (B.25) k--+oo

Another way of obtaining this result is to use the fact that the strong law of large numbers implies that the binary expansion of a randomly picked point 0.(3J(32(33 •.. almost surely has the same frequencies of O's and 1's, that is, cp0 = 1/2 almost surely. Inserting this in equation (B.9) yields o: = EV, and equation (B.25) is recovered. B.4 Probabilistic Roots of Multifractals 947

If the law of large numbers had meant that random choice of x should yield one particular value of a with probability 1, then it should yield the value that occurs most often. Going back to the f(a) curve, this particular value of a is the one that maximizes f(a). On page 943, this special value of the coarse Holder exponent was denoted by a(O) or a0 , because it is the value of a that corresponds to q = 0 in the method of moments. Indeed, for the binomial measure we found a(O) = ( v0 + vJ) /2 in agreement with equation (B.25). This establishes a first link between f(a) and the theory of sums of random variables. The results related to the laws of large numbers only hold pointwise, that is, roughly speaking, in the limit of infinitesimal box-sizes (k ----+ oo). In most physical systems, such limits cannot be attained, therefore their properties are not especially interesting. It is clear that random selection of a large number of boxes of finite size 2-kwill not always yields coarse HOlder exponents equal to the expected value EV = !(amin + amax), but all values of the HOlder exponent between amin and a max. In other words, the deviations from the expected value become important for finite k, and their probability of occurrence must be known. The relevant information is yielded by central limit theorems and - far more importantly - by large deviation theory.

B.4.3 Gaussian Central Limit Theorem and the Shape of f(a) near o:o The Gaussian central limit theorem goes a bit beyond the law of large numbers. But it too is disappointing, its main role being that of explaining why the maximum of the f (a) is often quadratic. Actually, the term Gaussian central limit theorem covers a variety of distinct results that all conclude that deviations from the expected value have a Gaussian distribution. The specific form we need concerns the sums of independent and identically distributed random variables, such as the sum E~=l V h that enters in equation (B.23). The basic assumptions is that the random addend V is a random variable with a finite expectation EV and a finite second moment EV2 . (The binomial measure is the standard example, since both EV = 4( vo + v,) and EV2 = 4( v6 + vt) are finite). The theorem states that, in the limit k ----+ oo, the distribution of the rescaled random variable Y k = (2::~= 1 V h - kEV) / Vk converges to the Gaussian distribution with zero mean and variance u 2 = EV2 - (EV) 2 . That is,

Ek V - kEV } Jy lim Pr { h=l ~ :S y = G(x)dx, (B.26) k->oo U k -=

where the integrand

C(x) = --1 { 1exp --x2} V27T 2 948 B Multifractal Measures is the reduced Gaussian density. (A graph of this density is found on the German bill for 10 Deutsch marks, together with a portrait of Carl Friedrich Gauss). In the coin-tossing experiment that yields either v0 or v 1, the law of large numbers makes us "ideally" expect an equal number of v0 and v 1 from k tosses. That is to say, when adding k sample values of V, one expects to get the value ~vo + ~v, = kEV, so that of I:~= IVh- kEV = 0. However, equation (B.26) shows that I:~=I V h- kEV deviates from the "ideal" value 0 by an amount that scales like Vk. Let us now return to the probability density Pk (a) of the coarse Holder exponent Hk = -k L~=l vh. Writing Hk = YkiVk + EV = YkiVk + ao, we see that Hk has the variance cr2 I ,Jk, and scales like 1I ,Jk. Keeping to a finite k, the limit in equation (B.26) yields the approximation

c 1 1 a- a 0 Pk (a)da = Vk ..Jh exp {-- (~ )2} da. (crl k) 21r 2 crlvk

Here, the superscript G is meant to remind us that p~ (a) is not the actual probability density Pk(a) of the coarse Holder exponent of the binomial measure, but a Gaussian approximation that applies only for a near a 0 . Very near to the most probable value ao, equations (B.2) and (B.3) yield the approximation 2 1 2 a- ao f(a) ~ J0 (a) = 1 + -k log 2 p~(a) = 1 - - ( _ . ) 1n 2 amax amm This approximation agrees with the expansion, equation (B.ll ), around the maximum of the exact result. But as a moves away from a0 , f 0 (a) be• comes increasingly larger than the exact f(a). Outside [amin, amaxJ, the approximate f 0 (a) is grossly inadequate: the exact f (a) is not defined there, but j 0 (a) is. Further away from a 0 , j 0 (a) < 0, which is meaning• less for the binomial measure. In summary, the Gaussian central limit theorem shows that the appear• ance of a quadratic maximum in the f (a) of the binomial measure is not a coincidence. In general, it shows that a quadratic maximum contains no information other than the finiteness of the first and second moment of the logarithm of the multiplier.

B.4.4 Cramer's Large Deviations Theory and f(a) Take a random variable with EX < oo, satisfying Pr{X > EX} > 0. The Significance of Large deviation theory is concerned with very large fluctuations around the .f (a) in the Discrete expected value, namely the behavior of Finite Cases B.4 Probabilistic Roots of Multifractals 949

as a function of 8 and k. The law of large numbers tells us that in the limit k ---+ oo, Pr{k I:;~=I Xh- EX = 0} = 1. So for 8 = 0, one expects that the above quantity vanishes with speed 0. For all other 8, one expects Pr{ kI::~= I xh - EX ~ 8} to vanish for k ---+ 00. The question is, "How fast". The answer was provided by Harald Cramer in 1938 under special con• ditions that were gradually weakened by many authors. A survey (with history) is found in the entry on large deviations in reference [55]. Cramer made rigorous use of saddle point approximations that are expressed in heuristic form in the widely used justifications of the method of moments. We shall proceed in two steps: a detailed study of discrete and finite addends, made possible by a theorem in Chernoff 1948, then a quick sketch of the great case. Chernoff's theorem applies (among other cases) when the random vari• able X is discrete and finite, meaning that it can only take a finite num• ber b of values x 1 , x2, ... , x B. Thus, its distribution can be written as Pr{X =xi} =Pi, i = 1, ... , B, with I:;~ Pi = 1. The simplest example of a discrete and finite random variable is when B = b, and Pi = I /b for all i. This case corresponds to the multinomial measures discussed in section 8.2.5. Chernoff's Theorem The random variable X being discrete and finite with EX < 0, one on Large Deviations has [13]

,li.~ ~log IT{~ Xh ~ 0} ~ log hf il>( q)} where (q) is the moment generating function defined (for our needs) by (q) = E(e-qlnb X). The factor ln b has been inserted for later convenience, when b will be the base of an arbitrary multinomial measure. The Tail Distributions Chernoff's theorem can be used to compute the probability of the Coarse HOlder Exponent IT{H, 2 a} ~ Pc { ~ ~ V h ? a}

that the coarse HOlder exponent of a randomly picked interval of the bino• mial measure is larger than or equal to a for a > a 0 . Note that this is the probability of finding a deviation larger than or equal to a - a 0 to the right of the most probable value a 0 . Rewrite Pr{Hk ~a} = Pr{I:;~= 1 (Vh- a) ~ 0} and introduce the shifted random variable X= V- a, so that Pr{Hk ~ a} = Pr{I:;~=I Xh ~ 0}. This X satisfies EX= EV-a = a 0 - a < 0. Since for the binomial measure, V is a discrete random variable with B = b = 2, the same is true of X, with Pr{X = Vi - a} = ! . Chernoff's theorem applies, and yields

lim -k1 logbPr{Hk ~a}= logb{inf(q)} = rR(a). (8.27) k-+oo q 950 B Multifractal Measures

The superscript R refers to deviations to the right of the most probable value o:o; i.e., to o:o < o:. Note that the above also holds for multinomial measures, and that equation (B.27) only makes sense for o: ::; Vmax = max{ vo, v1, ••• vb-d· For later convenience, we divided both sides of the main equation in Chernoff's theorem by log b. The generating function becomes

1 b 1 b (q) = b I>-qx,lnb = b Le-q(v,-a)lnb i=l i=l 1 b = eqalnb b 2...:e-qv,lnb_ i=l

From v; = -1ogb m;, it follows that e-qv, ln b = m{. Hence,

b (q) = eqalnb ~ L m{ = eqalnb E(Mq), i=l where M is the random multiplier of equation (B.21). By simple algebra,

rR(o:) = inf {1ogb (q)} = inf { o:q +1ogb E (Mq)}. q q Let us step aside to the multinomial case discussed in section B.2.5 and in equation (B.17). In that case, p; = t for all i's, hence

Therefore, it is legitimate to generalize the definition ofT( q) to read T(q) = -logbE(Mq) -1. (B.28)

We see that rR(o:) is the Legendre transform of T(q) +I, and

rR(o:) = inf{qo:- T(q)}- J. q Chernoff's large deviation theorem reads

(B.29) =infq{qo:-T(q)}-1, o:>o:o.

For o: < o:0, all the previous steps can be redone using the shifted variable X= o:- V, and one finds

(B.30) = infq {qo:- T(q)}- 1 , o: < o:o. B.4 Probabilistic Roots of Multifractals 951

The superscript L and R in equations (8.29) and (8.30) cease to serve a purpose and will henceforth be dropped. The existence of the infimum of { qa - T ( q)} is guaranteed [ 13], and one can also show that r( a) is smooth and concave. This infimum occurs for q such that T1 (q) =a= a(q), so that r(a(q)) + 1 = qa(q)- T(q) = qT'(q)- T(q). It follows that (djda)r(a) = q, meaning that the slope of r(a) equals q. The quantity E (Mq) is the q1h moment of the random variable M. One can easily show that these moments exist for all q E R when M is a discrete finite random variable. In particular, the multinomial measure of base b yields E(Mq) = t 2::~::6 m{ < oo for all q E R. However, when M is not discrete and finite, the moments may diverge; i.e. E (M'~) = oo for certain values of q. This happens, for example, in the b = oo measure shown in figure B.S. Identity between r( a) The above properties of r( a) are very similar to those section 8.3.2 has and C(a) described for f(a), hence C(a). The only difference is that f(a) and C(a) were defined in equations (B.l) and B.2 through a number density N, and a probability density p,, respectively, while r(a) is defined through a theorem concerning tail probabilities. However, in the case of our large deviation probabilities, the densities and tail probabilities happen to have exactly the same behavior - except for corrections that vanish for large k if one takes a logarithm and divides by k. Indeed, observe that, using equations (8.29), the existence of r( a) means that

The coarse Holder exponent falls between the values a and rx + da with the probabilities

Pk(o:)da = Pr{Hk;::: a}- Pr{Hk;::: a+ da}

rv (b-k)-r(a) _ (b-k)-r(a+dn)

rv (b-k)-r(a) [1 _ (b-k)-r(a+da)H(alj.

r(a) is concave (n) and a> o:(O) puts us to the right side of the maximum, hence r(a) > r(o:+do:), and the second term in the square bracket vanishes for large k. Thus,

(8.31)

This same result is found for a < a 0 if one starts with Pr{Hk ~ o} and equation (8.30). Comparing equation (8.31) with equation (8.2) yields C(o:) = r(a). For a measure supported by a set of box-counting dimension D, the number N, (a) of boxes with coarse HOlder exponent between a and a+ da is the fraction p, (a) of the total number cD of boxes, i.e., Nk( a) = 952 B Multifractal Measures

cDpc(a). Using equations (B.l) and (B.2) yields f(a) = C(a) +D. For a measure supported by a Euclidean set of dimension E one finds the result in equation (B.3). To summarize, we have shown that

C(a) = f(a) and f(a) = C(a) +D.

This generalizes in a rigorous fashion the results that section B.2.5 has obtained for the multinomial by using Lagrange multipliers, and provides the probabilistic roots of the notions discussed in the introduction in equations (B.l) and (B.2). Cramer's large deviation theory in the continuous and/or unbounded The Continuous cases is very important and justifies a special section. But we can only and/or Unbounded state that more general Cramer type theorems exist, and that they provide a Cases full justification of the so-called thermodynamic formalism of multifractals based on the Legendre transforms.

B.S Some Applications, and Advanced Multifractals

Self-similar measures appear in a variety of natural phenomena. In fully de• Applications veloped turbulence, there is strong evidence that the rate of dissipation of ki• netic energy is multifractal [233, 142, 263, 254, 192]. Multifractal measures also play an important role in the formation of fractal patterns such as light• ning, aggregates, snowflakes (dendritic solidification) and fractal viscous fingering [32, 17, 92]. Related to these latter examples is the self-similarity of the electrostatic charge on fractal boundaries like, e.g., the Koch tree [175] or Julia sets [73, 74, 271]. The self-similarity of these measures is due to the interaction of the Laplace equation with a fractal boundary [17 5, 241]. Other occurrences of multifractals, not all of them self-similar, concern the eigenfunctions of the Schrodinger equation in disordered systems [289], the current distributions in random resistor networks [147, 138] and their hy• drodynamic analogues, the distribution of mass in the universe [158], the invariant measures on strange attractors [197, 205, 263, 90, 184] and the distribution of states in the evolution pattern of a class of cellular automata. An example of the latter is the Pascal triangle discussed in section B.2.2. In some cases, the function T(q) is defined for all q's and one has f(a) > 0 for all a. In these cases, the method of moments is sufficient. But, as time goes on, there is increasing evidence that many applications demand more advanced multifractals. Let us mention some of the most frequently encountered cases. When f.l is a nonrandom measure handled by probabilistic methods, one C(a) and always finds that C(a) > -E, hence f(a) = C(a) + E > 0. But when Negative f(a) f.l is properly random, there are values of a for which f(a) < 0. This very important possibility has recently become the subject of a significant literature [252, 270, 240, 244, 64, 65]. B.5 Some Applications, and Advanced Multifractals 953

Left-Sided Depending on the distribution of the multiplier random variable M, it can Multifractal Measures: easily happen that some of the moments E (Mq) are infinite. A typical case When the f(o:) and is that the moments do not exist for q < 0. This appears to happen (among the Cramer Plot are others) in the harmonic measure on diffusion limited aggregates [316, 148, Insufficient 246, 176]. It is also the case for the exactly self-similar multifractal measure shown in figure B.S. In cases when EV < oo but E (V2 ) = oo, the Gaussian central limit theorem does not apply and the left-side (o: < o:(O)) of the maximum of f(o:) is not quadratic. The right-hand side (o: > o:(O)) is a horizontal line with f(o:) = D, i.e. the extremum is attained in infinitely many values of a. In cases when E (V) = oo, the law of large numbers also fails to apply, and since o:(O) = E(V) = oo the whole left-side of the f(o:) is infinitely stretched. The Cramer large deviation theorems only work for deviations on the left of o:(O). Alternative central limit theorems apply to these cases. The limits they involve are not Gaussian but Levy stable [40]. This issue is also important and is discussed in references [246, 64, 65]. Bibliography

1. Books

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2. General Articles

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3. Research Articles

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Abraham, Ralph, 19 covering of the, 341 absolute value, 777 derivative criterion, 822 adaptive cut algorithm, 343 for the dynamical system, 259 addresses, 309 problem of approximating, 341 addressing scheme, 309 totally disconnected, 314 dynamics, 620 Attractor (BASIC program), 767 for JFS Attractors, 313 attractorlets, 328, 344 for Sierpinski gasket, 80, 309, 371 Avnir, D., 475 for the Cantor set, 73, 313 axiom, 360, 361, 387 language of, 3 12 of period-doubling branches, 620 Banach, Stefan, 233, 263 space of, 312 band merging, 628 three-digit, 309 band splitting, 628, 636 aggregation, 475 Barnsley's fern, 255, 256 of a zinc ion, 477 transformations, 256 Alexandroff, Pawel Sergejewitsch, 107 Barnsley, Michael F., 35, 229, 278, 280, 297, 328 alga, 357 BASIC, 60, 179 algorithm, 33, 52 DIM statement, 296 automatic, 278 LINE, 61 allometric growth, 143 PSET, 61 ammonite, 142 SCREEN, 62 anabaena catenula, 357, 358 BASIC program, 133 angle basin boundary, 758 binary expansion, 816 basin of attraction, 665, 757, 759, 775, 790, 857 periodic, 805 basin of infinity, 794 pre-periodic, 805 Beckmann, Petr, 159 approximations, 150 Benard cells, 698 finite stage, 150 Benedicks, Michael, 676 quality of the, 278 Berger, Marc A., 229 Archimedes, 153, 185 Bernoulli, Daniel, 262 arctangent series, 160 Bernoulli, Jacob, 186 area reduction, 666 bifurcation, 586, 603, 617, 641 Aristotle, 126 calculation, 604 arithmetic mean, 735 period-doubling, 610 arithmetic precision, 581 bifurcation diagram, 605 arithmetic triangle, 82 binary decomposition, 812, 851 Astronomica Nova, 40 binary representations, 176 attractive, 593, 822 binomial coefficients, 409, 420 attractor, 232, 341, 790, 820 divisibility properties, 420, 423, 433, 437 coexisting, 757 binomial measure, 331 972 Index

Birkhoff, George David, 678 Cantor set of sheets, 694 bisection, 892 Cantor, Georg, 63, 67, 107, 173, 870 blinkers, 414 capacity dimension, 202 blueprint, 238 Caratheodory, Constantin, 871, 873 body, 142 Carleson, Lennart, 676 height, 142 carry, 427,434,435,443,451 mass, 210 Cartesian coordinates, 777 Bohr, Niels, 1, 9 Casio fx-7000G, 48 Boll, Dave, 859 catalytic oxidation, 452 Bondarenko, Boris A., 407 Cauchy sequence, 265 Borel measure, 330 cauliflower, 64, 105, 137, 144, 229 Borwein, Jonathon M., 157, 161 Cayley, Sir Arthur, 773, 774 Borwein, Peter B., 157, 161 Cech, Eduard, 107 boundary crisis, 649 cell division, 357 Bouyer, Martine, 161 cellular automata, 411, 412, 454, 952 box-count power law, 724 linear, 422 box-counting dimension, 202, 212, 218, 721, 735, Cellular Automata (BASIC program), 455 740, 757 central limit theorem, 484 limitations, 722, 726 Ceulen, Ludolph van, 158 Brahe, Tycho, 39 chain rule, 889 brain function anomalies, 53 Chaitin, Gregory J., 428 branch point, 878 chaos, 6, 46, 52, 55, 59, 76, 536, 567-569, 585, branching, 397 624, 636 branching order, 116 acoustic, 754 Brent, R. P., 161 icons of, 659 broccoli, 137 inheritance of, 574 Brouwer, Luitzen Egbertus Jan, 107, 108 routes to, 585, 640 Brown, Robert, 297 chaos game, 35, 36, 298, 300, 306, 308, 328, 341, Brownian motion, 297, 476, 481 820, 822, 923 fractional, 493, 494 analysis of the, 307 one-dimensional, 491 density of points, 324 Brownian Skyline (BASIC program), 504 game point, 298 Buffon, L. Comte de, 323 statistics of the, 329 bush, 398 with equal probabilities, 315 butterfly effect, 42 Chaos Game (BASIC program), 351 chaotic transients, 647 calculator, 49, 576 characteristic equation, 177 camera, 20 Charkovsky sequence, 638 cancellation of significant digits, 786 Charkovsky, Alexander N., 638 Cantor brush, 117 chemical reaction, 656 Cantor maze, 242 circle, encoding of, 243 Cantor set, 63, 67, 172, 219, 252, 342, 364, 381, classical fractals, 123 574,623,668,705,741,828,944 climate irregularities, 53 addresses for the, 73 cloud, 212, 501 construction, 68 cluster, 461 dynamics on, 620 correlation length, 468 in complex plane, 833 dendritic, 479 program, 226 incipient percolation, 467 Cantor Set and Devil's Staircase (BASIC program), maximal size, 466 227 of galaxies, 458 Index 973

percolation, 467 test, 101 coarse HOlder exponent, 931 Conway, John Horton, 413 coarse-grained Holder, 931 correlation, 496 coarse-graining, 939 correlation dimension, 738 coast of Britain, 199 correlation length, 467, 468 box-counting dimension, 215 Coulomb's law, 801 complexity of the, 199 Courant, Richard, 683 length of the, 199 cover code, 52 open, 217 collage, 278 order of, 110 design of the, 281 Cramer, H., 952 fern, 278 Cremer, Hubert, 123 leaf, 279 crisis, 646, 708 mapping, 239 critical exponent, 648 optimization, 282 critical line, 634, 635 quality of the, 281 critical orbit, 830, 886, 889 collage theorem, 280 critical point, 829, 833, 886 Collatz, Lothar, 33 critical value, 633, 829, 847 color image, 330 critical value lines, 633 comb, 872 Crutchfield, James P., 19 compass dimension, 202, 208, 210 curdling, 224 compass settings, 192, 200 curves, 113, 208 complete metric space, 265, 268 non planar, 113 complex. argument, 778 parametrized, 370 complex conjugate, 781 planar, 113 complex division, 781 self-similar, 208 complex. number, 776 space-filling, 372 complex plane, 123 Cusanus, Nicolaus, 155 complex. square root, 784, 820, 839 cycle, 34, 35, 58, 552 complexity, 16, 38 periodic, 533 degree of, 202 complexity of nature, 135 Dase, Johann Martin Zacharias, 159 composition, 608 decay rate, 529 computer arithmetic, 533 decimal, 307 computer hardware, 162 numbers, 65 computer languages, 60 system, 307 concept of limits, 136 decimal MRCM, 307 connected, 251, 803 decoding, 260, 303 continued fraction expansions, 163, 568 images, 303 continuity, 565 method, 260 continuous, 569 dendrite, 873, 891 contraction, 234 dendritic structures, 475 factor, 266 dense, 552 mapping principle, 263, 266, 288 density of measure, 933 ratio, 318 derivative, 679, 717 contraction factor, 715 derivative criterion, 822, 858, 864, 867, 889 control parameter, 59 deterministic, 34, 46, 51 control unit, 17, 19 feedback process, 46 convection, 698, 699, 702 fractals, 299 convergence, 265 iterated function system, 302 974 Index

rendering of the attractor, 341 distribution function, 2 shape, 299 divisibility properties, 410, 421, 423, 454 strictly, 301 divisibility set, 424 Devaney, Robert L., 536, 569 DLA, 478, 952 devil's staircase, 220, 226 DNA, 353 area of the, 221 Douady, Adrien, 769, 800, 817, 853, 871 boundary curve, 225 dough, 551 program, 226 dragon, 240, 374, 384 dialects of fractal geometry, 230 dust, 72 diameter, 217 dyadic interval, 927 die, 35, 298 dynamic law, 17 biased, 322, 326 dynamic process, 265 ordinary, 298 dynamical system, 233, 658 perfect, 315 attractor for the, 259 differential equation, 678, 679, 695, 699, 759, 763 conservative, 655 numerical methods, 683 dissipative, 655 system, 685 linear, 594 diffusion limited aggregation, 477 dynamical systems theory, 233, 265 mathematical model, 479 dynamics of an iterator, 62 digits, 28 Dynkin, Evgeni B., 407 Dimension, 7 dimension, 106--108, 202 Eadem Mutata Resurgo, 186 box-counting, 218 Eckmann, Jean-Pierre, 751 correlation, 738 Edgar, Gerald, 216, 862 covering, 109 eigenfunctions, 952 fractal, 195, 441 eigenvalue, 626, 675 Hausdorff, 109, 216, 218 Einstein, Albert, 4 information, 735 electrochemical deposition, 475 Ljapunov, 739 mathematical modeling, 476 mass, 736 electrostatic field, 800 pointwise, 736 encirclement, 795, 808, 826 precision, 743 algorithm, 798 problem of, 742 energy, 655 Renyi, 736 ENIAC, 48, 333 self-similarity, 441 c:-collar, 267 disconnected, 251 equipotential, 80 I, 809, 896 displacement, 481 ergodic, 554, 726 mean square, 481 ergodicity, 525, 728 proportional, 482 erosion model, 355 distance, 263, 267, 274 error amplification, 709, 715, 751, 785 between two images, 267 error development, 581 between two sets, 267 error propagation, 41, 55, 512, 515 Euclidean, 216 escape set, 77, 125,789,791 Hausdorff, 263, 268 escape time, 648 in the plane, 274 escaping sequence, 77 of points, 274 Escher, Mauritz C., 79 distribution, 67 Ettingshausen, Andreas von, 409 bell-shaped, 482 Euclidean dimension, 202 Gaussian, 482 Euler step, 683, 716 invariant, 527, 529, 568 Euler, Leonhard, 85, 157, 168, 262 Index 975 expansion, 70, 163 fixed point, 593, 599, 608, 641, 771, 862 binary, 70, 101, 175, 549, 555, 557, 576 (un)stable, 593 binary coded decimal, 576 attractive, 593, 855, 864 continued fraction, 164 indifferent, 857, 867 decimal, 70, 425 of the IFS, 259 p-adic, 426 parabolic, 868 triadic, 70, 72 repelling, 821, 839, 864 stability, 594 factorial, 409 super attractive, 596, 855 factorization, 422, 425, 427 unstable, 603 Falconer, Kenneth, 216 fixed point equation, 167 Faraday, Michael, 753 floating point arithmetic, 535 Fatou, Pierre, 773 flow, 717, 740 feedback, 17, 57 folded band, 688 clock, 19 forest fires, 464 cycle, 20 simulation, 465 experiment, 19 Fortune Wheel Reduction Copy Machine, 30 I, 321 loop, 231, 266 Fourier, 162 machine, 17, 31 analysis, 261 feedback system, 29, 34, 35, 76 series, 262 geometric, 187 Transformaion techniques, 162 feedback systems Fourier, Jean Baptiste Joseph de, 262 class of, 266 fractal branching structures, 457 quadratic, 123 fractal dimension, 195, 202, 422 sub-class of, 266 prescribed, 458 Feigenbaum (BASIC program), 652 universal, 469 Feigenbaum constant, 590, 612, 618, 675, 707 fractal geometry, 23, 59 Feigenbaum diagram, 587, 651, 693 fractal surface construction, 497 Feigenbaum point, 582, 588, 610, 619, 622, 629 fractal surfaces, 211 Feigenbaum scenario, 675, 754 fractal, random and nonrandom, 944 Feigenbaum, Mitchell, 53, 587, 590, 612 fractals, 35, 76 Fermat, Pierre de, 82 classical, 63 fern, 229, 285 construction of basic, 149 non-self-similar, 288 gallery of historical, 131 Fibonacci, 29 FRCM, 301 -Association, 30 free energy, 937 -Quarterly, 30 friction, 655 generator, 339 Friedrichs, Kurt Otto, 683 generator formula, 340 Frobenius-Perron equation, 528 numbers, 30 Frobenius-Perron operator, 528, 529 sequence, 29, 153 Fibonacci, Leonardo, 65 Galilei, Galileo, 139 fibrillation of the heart, 53 Galle, Johann G., 38 field line, 800, 802, 813, 816, 852, 853, 871, 872 Game of Life, 413 angle, 804, 815, 854 majority rule, 415 dynamics, 816 one-out-of-eight rule, 414 figure-eight, 833, 834 parity rule, 419 final curve, 95 game point, 35, 298 final state, 510, 586, 759 Gauss map, 568 final state sensitivity, 757 Gauss, Carl Friedrich, 4, 38, 85, 157, 420, 568, 948 976 Index

Gaussian central limit theorem, 947 box-counting dimension, 721 Gaussian distribution, 482, 947 dimension, 670, 726 Gaussian random numbers, 484 information, 733 generalized dimensions, 941 information dimension, 735 generator, 90, 208 invariance, 662, 663 generic parabola, 527, 544, 631 Ljapunov exponent, 712 geometric feedback system, 187 natural measure, 733 geometric mean, 735 unstable direction, 712 geometric series, 147, 221, 612 Henon transformation, 660, 71 0 construction process, 149 decomposition, 661 Giant sequoias, 140 derivative, 713 Gleick, James, 41 Feigenbaum diagram, 675 glider, 414, 415 fixed points, 674 golden mean, 30, 153, 165, 869, 870, 882 inverse, 668 continued fraction expansion, 165 Herman, M. R., 869 Golub, Jerry, 508 Herschel, Friedrich W., 38 graphical iteration, 337, 472, 510, 583, 593 Heun's method, 766 backward, 826 hexagonal web, 88 Graphical Iteration (BASIC program), 61 Hilbert curve, 63, 388, 392 grass, 399 Hilbert, David, 63, 94, 107, 373, 387 Grassberger, Peter, 724 Hints for PC Users, 62, 134 Great Britain, 192 Hirsch, Morris W., 507 Gregory series, 158 histogram, 526, 630 Gregory, James, 156, 158 spike, 606, 633 GroBmann, Siegfried, 53, 629 histogram method, 939 group, 244 Holder condition, 217 growth, 195 Holmes, Philip, 658 allometric, 197 Holzfuss, Joachim, 754 cubic, 198 homeomorphism, 106, 570 proportional, 143 homoclinic point, 644 growth law, 142, 195 HP 28S, 49 growth rate, 43 Hubbard, John H., 769, 770, 800, 817, 853, 872 Guckenheimer, John, 658 human brain, 258 Guilloud, Jean, 160 encoding schemes, 259 guns, 414 Hurewicz, Witold, 107 Hurst exponent, 493 Hadamard, Jacques Salomon, 122 Hurst, H. E., 493 half-tone image, 329 Hutchinson distance, 331 Hamilton, William R., 837 Hutchinson equation, 436 Hanan, James, 363 Hutchinson operator, 34, 171, 238, 269, 302, 438 Hao, Bai-Lin, 658 contractivity, 270 harmonic measure, 953 Hutchinson, J., 171, 229, 233, 263 Hausdorff dimension, 202, 216, 218 hydrodynamics, 756 Hausdorff distance, 150, 263, 268 Hausdorff measure, s-dimensional, 217 ice crystals, 243 Hausdorff, Felix, 63, 107, 108, 202, 203, 216, 233, IPS, 230,328,435,437,923,930 263 fractal dimension for the attractors, 271 head size, 142 hierarchical, 284, 290, 337, 392, 442, 444 Henon, Michel, 659 image, 231 Henon attractor, 659, 663, 667 attractor image, 305 Index 977

code for the, 36, 329 quatemion, 837 color, 330 self-similarity, 822, 890 compression, 258 structural dichotomy, 833, 843 encoding, 261 Julia sets, 952 example of a coding, 258 Julia, Gaston, 63, 122, 772, 774 final, 232 JuliaSets (BASIC program), 840 half-tone, 329 initial, 238 Kadanoff, Leo P., 472 leaf, 278 Kaplan, James L., 738 perception, 258 Kaplan-Yorke conjecture, 738, 739, 741 target, 278 Kepler's model of the solar system, 38 imitations of coastlines, 499 Kepler, Johannes, 38, 39 incommensurability, 126, 163 kidney, 94, 211 indifferent, 822 kneading, 536, 551 infimum, 216 cut-and-paste, 543 information, 729 stretch-and-fold, 543, 544, 660, 688 definition, 729 substitution property, 542 power law, 734 Koch curve, 63, 89, 112, 200, 365, 380, 405 information dimension, 202, 735 construction, 91 relation to box-counting dimension, 735 Koch's Original Construction, 89 information theory, 730 length of, 92 initial value, 681 random, 400, 459 initial value problem, 681 self-similarity dimension, 205 initiator, 90 Koch Curve (BASIC program), 180 injective, 751 Koch island, 89, 149, 200, 386 input, 17, 27, 31, 34 area of the, 149 input unit, 17, 19 random, 400 interest rate, 44 Koch, Helge von, 63, 89, 145, 152 intermittency, 640, 644, 650, 708 Kolmogorow, Andrej N., 507 scaling law, 646 Kramp, Christian, 85 intersection, 106 Kummer criterion, 427, 429, 434, 435 invariance, 76 Kummer, Ernst Eduard, 133, 254, 424, 425 invariance property, 173 invariant measure, 329, 727 L-system, 354, 361, 376, 380, 402 invariant set, 823 extensions, 401 inverse problem, 261, 278 parametric, 401 irreducible, 422 stochastic, 399 isometric growth, 143 L-Systems (BASIC program), 403 iterated function system, 230 labeling, 73 iteration, 18, 42, 49, 55, 57, 544, 547, 551 Lagrange transform, 937 graphical, 58 Lagrange, Joseph Louis, 262 iterator, 17, 37, 55 Lange, Ehler, 333 language, 150, 230 Julia set, 123, 771, 775, 790, 791, 820, 823, 839, Laplace equation, 480, 952 878 Laplace, Pierre Simon, I, 323 (dis)connected, 833 Laplacian fractals, 480 by chaos game, 820 laser instabilities, 53 invariance, 822 Lauterborn, Werner, 754 iterated function system, 823 law of gravity, 40 pinching model, 817 law of large numbers, 946 978 Index leaf, 128, 278 dimension, 705 spiraling, 128 model, 702 least-squares method, 743 reconstruction, 750 Lebesgue, Henri L., 107, 109 Lorenz experiment, 48 Legendre transform, 937, 941, 952 Lorenz map, 691, 693, 703 Legendre's identity, 429 Lorenz system, 697, 699, 717, 768 Legendre, Adrien Marie, 425, 429 crisis, 707 Leibniz, Gottfried Wilhelm, 5, 17, 91, 156 intermittency, 706 lens system, 23, 26 Ljapunov exponent, 716 level set, 812, 851 Lorenz map, 702, 703 cell, 812, 816, 832, 851 periodic solution, 705 level sets, 811 physical model, 698 Lewy, Hans, 683 streamlines, 701 Li, Tien-Yien, 657 temperature profile, 701 Libchaber, Albert, 508 Lorenz, Edward N., 42, 46, 53, 512, 581, 678, 697, Liber Abaci, 29 700 Lichtenberger, Ralph, 838 1963 paper, 657 lightning, 952 Lozi attractor, 672 limit, 135 Lozi, Rene, 671 limit objects, 147 Lucas' criterion, 428 limit structure, 151 Lucas, Edouard, 425, 428, 433 boundary of the, 151 Lindemann, F., 160 Machin, John, 158, 160 Lindenmayer, Aristid, 128, 353, 355, 363, 401 Mandelbrojt, Szolem, 122 linear congruential method, 339 Mandelbrot (BASIC program), 898 linear mapping, 235 Mandelbrot set, 841, 843, 896 Liouville monster, 871 algorithm, 848, 896 Liouville number, 870 atom, 866 Liouville, Joseph, 870 binary decomposition, 851 Ljapunov dimension, 739, 741 buds, 852, 855, 866 acoustic chaos, 756 central piece, 857 Ljapunov exponent, 516, 518, 523, 568, 709, 712, dimension, 851 714, 738, 739 encirclement, 845, 848, 896 acoustic chaos, 756 equipotentials, 851 algorithm, 710, 712 field lines, 851 algorithm for time series, 751 level set, 851 continuous system, 715 pinching model, 852 from time series, 751 potential function, 851 invariance, 719 secondary, 873, 892 zero, 719 self-similarity, 874, 890 Ljapunov, Alexander Michailowitsch, 516 Mandelbrot Set Pixel Game (BASIC program), 899 local HOlder exponent, 931 Mandelbrot Test (BASIC program), 899 locally connected, 853, 871, 873 Mandelbrot, Benoit B., 63, 89, 122, 183, 493, 841, lognog diagram, 193 843 for the coast of Britain, 194 Mane, Ricardo, 748 of the Koch curve, 20 I Manhattan Project, 48 lognog diagramm, 195 mapping ratio, 20 logistic equation, 42, 45, 48, 333, 678 mappings, 235 look-up table, 412, 420, 454 affine linear, 236 Lorenz attractor, 658, 697, 698, 702, 766 linear, 235 Index 979

Margolus, Norman, 416 monster, 100 Markov operator, 330 fractal, 229 Mars, 39 of mathematics, 63 mass, 224, 727 monster spider, 117 mass dimension, 736 Monte Carlo methods, 323 mathematical category of constructions, 256 moon, 40,78 mathematics, 135 mountains, 15 in school, 293 MRCM, 23, 34, 36, 230, 233, 354, 364, 387, 820 new objectivity, 135 adaptive iteration, 342 Matsushita, Mitsugu, 475 blueprint of, 258, 278 Maxwell, James Clerk, 2 decimal, 307 May, Robert M., 42, 53 lens systems, 288 mean square displacement, 481 limit image, 288 measure mathematical description, 288 binomial, 331 networked, 284, 337, 392 multifractal, 331 MRCM Iteration (BASIC program), 294 memory, 31, 34 Mullin, Tom, 756 memory effects, 22 multi-fractal, 737 Menger sponge, 109 multifractal, 331, 736, 922, 935 construction of, 109 multifractals, 216 Menger, Karl, 107, 108, 115 Multiple Reduction Copy Machine, see MRCM metabolic rate, 210 multiplicative cascade, 927 meter, 65 multiplier, 885, 889 meter stick, 308 method of least squares, 194 natural flakes, 90 method of moments, 939 natural measure, 727 metric, 274 NCTM, 25 choice of the, 274 Neumann, John von, 333, 339, 412 Euclidean, 264 Newton's method, 28, 167, 773 Manhattan, 264 Newton, Sir Isaac, 17, 40, 91 maximum, 264 noise, 551 suitable, 276 nonlinear effects, 26 topology, 263 nonlinear physics, 2 metric space, 110, 263, 264 nonlinearity, 3 compact, 110 Norton, V. Alan, 838 complete, 265 middle-square generator, 339 object, 139 Misiurewicz point, 886, 888 one-dimensional, 112 Misiurewicz, Michal, 671, 888 scaled-up, 139 Mitchison, G. J., 358 one-step machines, 27 mixing, 58, 520, 554, 558 open, 569 mod-p condition, 429, 433, 440 open set, 565 mod-p criterion, 451 orbit, 509 modulo, 420, 422, 423 backward, 669, 826 modulus, 777 ergodic, 525 moment, 951 periodic, 509, 533, 535, 604 moments of measure, 942 organs, 94, 210 Monadology, 5 oscillator, 695 monitor, 20 output, 17, 27, 31, 34 monitor-inside-a-monitor, 20 output unit, 17, 19 980 Index

overlap, 275 Poincare map, 694 overlapping attractors, 346 point at infinity, 781, 790 point charge, 801 parabola, 57, 608, 691, 831 point of no return, 793 parameter, 18, 27 point set topology, !50 parametrization, 858 point sets, !50 partition function, 937 pointwise dimension, 736 Pascal triangle, 82, 407, 419, 447 polar coordinate, 778, 80 I a color coding, 132, 408, 433 poly-line, 593 coordinate systems, 424 polynomial, 420, 421 Pascal, Blaise, 82, 88, 254 equivalent, 570 pattern formation, 422, 952 Pontrjagin, Lew Semjenowitsch, I 07 Peano curve, 63, 94, 220, 372, 383, 394 population dynamics, 29, 42, 53 construction, 95 Portugal, 183, 199 S-shaped, 391 potential energy, 80 I space-filling, 98 potential function, 80 I, 811 Peano, Giuseppe, 63, 94, 107, 225, 372 power law, 195 pendulum over magnets, 758, 761, 774 behavior, 200 percolation, 458, 459 pre-periodic point, 886 cluster, 470 preimage, 593, 601, 606, 727, 795, 815, 820 models, 458 Principia Mathematica, 135 threshold, 464, 470 prisoner set, 125, 789, 792, 795, 800, 802, 826, 834, period, 588 844 period-doubling, 587, 592, 610, 6ll, 618, 619, 636, (dis)connected, 843 674, 675, 705, 754, 875 connected, 832 periodic orbit, 604, 628, 864, 889 probabilities, 304 derivative criterion, 864 badly defined, 334 periodic point, 523, 552, 555, 557, 564, 576, 639 choice of, 304 periodic trajectory, 705 for the chaos game, 349 periodic window, 635, 636, 708, 875 heuristic methods for choosing, 327 Perrin, Jean, 482 probability theory, 82 perturbation method, 3 problem, 281 phase transition, 467 optimization, 281 phyllotaxis, 283, 285 traveling salesman, 282 pi, 153, 160, 323 processing unit, 19, 31, 34, 42 approximations of, 158, 161 production rules, 360 Cusanus' method, 155 program, 60, 132, 179, 226, 293, 350, 503 Ludolph's number, 158 chaos game for the fern, 350 Machin's formula, 160 graphical iteration, 60 Rutherford's calculation, 159 iterating the MRCM, 293 pinching, 852 random midpoint displacement, 503 Pisa, 29 proportio divina, 30 Pisano, Leonardo, 29 Prusinkiewicz, Przemyslaw, 356, 363, 40 I pixel, 329 pseudo-random, 333 pixel game, 771, 775 Pythagoras of Samos, 126 planets, 38, 40 Pythagorean tree, 126 plant, 15 Plath, Peter, 477 quadratic, 42, 52, 59 Platonic solids, 38 dynamic law, 52 Poincare, Henri, 53, 107, 507, 585, 644, 694 quadratic equation, 60 I, 787 Index 981 quadratic iterator, 37, 520, 536, 581, 585, 651, 672, rose garden, 368 690, 710, 826 rotation, 236, 244, 882 equivalence to tent transformation, 562 Rozenberg, Grzegorz, 353 generalization to two dimensions, 660 Ruelle, David, 507, 656, 751 in low precision, 534 Runge-Kutta method, 717 quadratic law, 52 Rutherford, William, 159 quatemions, 837 saddle point, 642, 644 rabbit, 823, 838 saddle-node bifurcation, 642 rabbit problem, 32 Sagan, Carl, 162 rabbits, 29 Salamin, Eugene, 161 Ramanujan, Srinivasa, 157 Saltzman, B., 701 random, 459 sample value, 945 fractals, 459 saw-tooth transformation, 541, 555, 571 midpoint displacement, 487 scaling, 931 number generator, 48, 322 scaling factor, 138, 203, 309 process, 299 Schwarzian derivative, 616 successive additions, 498 scientific method, 5 random function, 492 self-affine, 145, 146, 223, 283 rescaled, 492 self-intersection, 94, 214 random resistor networks, 952 self-similar, 76, 283 random variable, 944 perfect, 76 randomness, 34, 297, 459 statistically, 494 Rayleigh, Lord, 754 strictly, 146, 283 reaction rate, 452 self-similarity, 95, 137, 202, 619, 636, 822, 874 real number, 776 asymptotic, 885, 890, 894 reconstruction, 658, 745, 748 at a point, 882 acoustic chaos, 755 at Feigenbaum point, 623 reduction, 236 of probability distribution, 325 reduction factor, 23, 26, 203, 236 of the Feigenbaum diagram, 588 reflection, 236, 244 statistical, 145 renormalization, 470, 710 self-similarity dimension, 202, 205 technique, 470 of the Koch curve, 205 renormalization group, 3 sensitive dependence on initial conditions, 6, 48, Renyi dimension, 736 511, 538, 551, 557, 561, 666, 703, 804 repeller, 593, 603, 820 sensitivity, 511, 524, 533, 582 derivative criterion, 822 sensitivity constant, 551 repelling, 822 series, 147 rescaling, 626 geometric, 147 rest point, 593 Semetz, Manfred, 21 0 return map, 689 set, 69 Riemann, Bernhard, 4 countable, 69 Riemannian sphere, 781 set theory, 67 Rossler, Otto. E., 686 shadowing lemma, 576 Rossler attractor, 687, 688, 766 Shanks, Daniel, 160 paper model, 690 Shannon, C., 730 reconstruction, 748, 749 Shaw, Robert, 521 Rossler system, 686, 695 shift, 75 Feigenbaum diagram, 693 binary, 101 romanesco, 137, 144 shift on two symbols, 551 982 Index

shift operator, 549 approximation of, 166 shift transformation, 554, 568, 575, 804 complex, 839 Shishikura, M., 851 of two, 166 Shrodinger equation, 952 square, encoding of, 243 Siegel disk, 868, 872 stability, 25, 507, 511, 610, 644 Siegel, Carl Ludwig, 868 stability condition, 683 Sierpinski arrowhead, 368, 371, 382 stable, 56, 57, 59 Sierpinski carpet, 81, 119, 219, 254 staircase, 221 Sierpinski fern, 291 boundary of the, 222 Sierpinski gasket, 24, 25, 36, 63, 174, 219, 244, star ships, 414 252,366,369,370,407,434,930 statistical mechanics, 2 binary characterization, 175, 434 statistical tests, 339 perfect, 24 statistics of the chaos game. 329 program, 132 Steen, Lynn Arthur, 407 relatives, 244 stereographic projection, 782 variation, 239 Stewart, H. B., 333, 407, 695 Sierpinski Gasket by Binary Addresses (BASIC pro- Stifel, Michael, 85 gram), 134 Stirling, James, 410 Sierpinski, Waclaw, 63, 78, 174 strange attractor, 255, 656, 693 similar, 23 characterization, 670 similarity, 138 coexistence, 676 similarity transformation, 23, 138, 202 dimension, 721 similitude, 23, 26 reconstruction, 745 simply connected, 251 strange attractors, 952 singularity, 466 Strassnitzky, L. K. Schulz von, 159 singularity strength, 931 stream function, 700 Skewed Sierpinski Gasket (BASIC program), 134 Stroemgren, Elis, 40 Smale, Stephen, 507, 644, 658 structures, 203 Smith, Alvy Ray, 363 basic, 285 snowflake curve, 89 branching, 457 snowflakes, 952 complexity of, 16 software, 40 dendritic, 475 space-filling, 94, 373 in space, 212 Spain, 183, 199 in the plane, 212 Sparrow, Colin, 705, 708 mass of the, 224 spectral characterization, 498 natural, 65 spiders, 112 random fractal dendritic, 458 monster, 117 self-similar, 203 order of, 115 space-filling, 94 Spira Mirabilis, 186 tree-like, 458 spirals, 185, 891, 894 Sucker, Britta, 333 Archimedean, 185 Sullivan, Dennis, 869 golden, 190, 882 Sumerian, 27 length of, 185, 189 super attractive, 596-598, 609, 855, 863, 864 logarithmic, 142, 185 super object, 112, 113 polygonal, 188 super-cluster, 470 smooth, 189 super-site, 470 square root, 126 supremum, 216 square, 244 survivors, 521 square root, 26, 28, 166 Swinney, Harry, 508 Index 983 symmetry transformation, 244 tree, 66, 242, 402 decimal number, 66 Takens, Floris, 507, 657, 748 Pythagorean, 126 Tan Lei, 879, 889 triadic numbers, 69, 75 tangent bifurcation, 640, 642, 674 triangle, encoding of, 243 target set, 809, 846 triangular construction, 497 temperature, 700 triangular lattice, 463 tent transformation, 32, 556, 571, 704 triangulation, 896 binary representation, 556 truncation, 533 equivalence to quadratic iterator, 562 turbulence, 1, 53, 952 thermodynamic formalism, 952 acoustic, 753 thermodynamics, 937, 939 turtle graphics, 376, 402 Thomae, Stefan, 53, 629 step length, 381 Thompson, J. M. T., 355, 695 turtle state, 377 three-body problem, 40 stacking of, 397 threshold radius, 794 twig, 242, 396 time delay, 748 twin Christmas tree, 240 time profile, 598, 602, 607 two-step method, 28, 31 time series, 509,581,586 Time Series (BASIC program), 582 Ulam, Stanislaw Marcin, 48, 333, 412 Toffoli, Tommaso, 416 uncertainty exponent, 757 Tombaugh, Clyde W., 38 uniform distribution, 322, 484 topological conjugacy, 569 unit disk, 802 topological dimension, 202 universal object, 115 topological invariance, 108 universality, 4, 112, 586, 590, 616, 618, 625 topological semi-conjugacy, 569, 571 of the Menger sponge, 115 topology, 106 of the Sierpinski carpet, 112 totally disconnected, 804 Universe, 952 touching point, 118, 866 unstable, 57, 59, 593 touching points, 3 12 Urysohn, Pawel Samuilowitsch, 107 trajectory Uspenski, Wladimir A., 407 periodic, 692 Utah, 199 transcendental, 870 transformation, 26, 54, 107, 138, 540 variance, 947 affine, 26, 223, 300 variational equation, 717 affine linear, 234 vascular branching, 211 Cantor's, 107 vectors, 31 for the Barnsley fern, 256 Verhulst, Pierre F., 42, 44, 45 invariance, 168 vessel systems, 94 linear, 26 vibrating plate, 753 nonlinear, 125, 820 video feedback, 19 renormalization, 471 setup, 19 shift, 549, 555 Vieta's law, 865 similarity, 138, 168, 202, 234, 300 Vieta, Fran~ois, 156 symmetry, 244 viscous fingering, 480 transition to chaos, 6 visualization, 362, 376 transitivity, 536 volume, 740 transversal, 897 Voyager II, 53 trapezoidal method, 684 trapping region, 664 Wallis, John, 156 984 Index weather model, 41 weather prediction, 46, 59 Weierstrass, Karl, 90 wheel of fortune, 34 Wilcox, Michael, 358 Wilson, Ken G., 3, 474 wire, SOl Wolf, A., 751 Wolfram, Stephen, 407, 412 worst case scenario, 259 Wrench, John W., Jr., 160

Yorke, James A., 521, 657, 738

Zu Chong-Zhi, 154 Zuse, Konrad, 412