7. Dimension and Entropy. x This section will ¯rst describe several possible de¯nitions for the concept of \dimension", and then use related methods to de¯ne the \topological entropy" of a compact dynamical system. 7A. Box Dimension. There are several possible real valued invariants for a compact metricx space X which can be called the \dimension" of X . Let us ¯rst describe two dimen- sion functions which measure how e±ciently X can be covered by balls (or boxes) of equal size. Following Falconer, I will call these the upper and lower \box-counting dimension", or briefly box dimension, of X . The presentation will depend on some elementary inequalities, which will also be needed in 7C section on topological entropy. x Remarks. This concept of metric dimension has a long history, and has been studied under a variety of names. The oldest form of the de¯nition, based on studying the volume of the ²-neighborhood of X as a function of ² , was introduced by Bouligand, generalizing earlier work by Steiner and Minkowski. (See Problem 7-d below.) More modern forms of the de¯nition were introduced by Pontryagin and Schnirelman, and later by Kolmogorov and Tihomirov. (Compare Mandelbrot 39.) It should not be confused with Hausdor® dimension or with topological dimension. Thex Hausdor® dimension, introduced by Felix Hausdor® in 1919, is also a real valued metric invariant, but measures how e±ciently X can be covered by balls of varying size. It coincides with the box dimension in many special cases, but is better behaved in general. The topological dimension, introduced by Lebesgue, Brouwer, Menger and Urysohn, is quite di®erent, being integer valued, and invariant under arbitrary homeomorphisms. These other concepts of dimension will be discussed in 7B. (For further information, see Edgar, and Falconer, as well as the classical book of Hurewiczx and Wallman.) Let X be compact metric. Suppose that we can only distinguish points of X to an accuracy of ² . How many distinct points can we distinguish? Here are three possible de¯nitions. The distance between points of X will be written as d(x; x0) . By the ²-covering number C²(X) will be meant the smallest number k 0 so that X can be covered by sets X ; : : : ; X of diameter diam(X ) ² . ¸ 1 k i · By the ²-separated number S²(X) will be meant the largest cardinality of a subset x ; : : : ; x X such that d(x ; x ) > ² for h = j . f 1 `g ½ h j 6 2 Figure 33. A set X , covered by B (X) = 13 ²-boxes. This ²-box number ½ ² B²(X) is relatively easy to compute, while the numbers S²(X) and C²(X) are more di±cult to compute. 7-1 7. DIMENSION AND ENTROPY n Now suppose that X is contained in the euclidean space . Then the ²-box number n B²(X) is de¯ned as follows. Cover by countably many disjoint \half-open boxes" box (i ; : : : ; i ) = [i ² ; (i + 1)²) [i ² ; (i + 1)²) ² 1 n 1 1 £ ¢ ¢ ¢ £ n n of edge length ² , indexed by n-tuples of integers. De¯ne B²(X) to be the number of these boxes box²(i1 ; : : : ; in) which intersect X . Evidently this is a quantity which is particularly well adapted to computer calculations. An easy compactness argument shows that the number C²(X) is always ¯nite. The number S²(X) is also ¯nite, with C (X) S (X) C (X) : (7: 1) 2² · ² · ² For given sets X X = X and x ; : : : ; x X as above, we have ` k since 1 [ ¢ ¢ ¢ [ k f 1 `g ½ · each Xi can contain at most one xj , and the left hand inequality is true since the closed ²-neighborhoods of the x are sets of diameter 2² covering X . Similarly, note that j · C (X) B (X) 2n C (X) : (7: 2) ²pn · ² · ² The left hand inequality is true since each such ²-box has diameter ²pn , and the right hand inequality since a set of diameter ² can intersect at most 2n such ²-boxes. · How rapidly do the numbers C²(X) or S²(X) or B²(X) grow as ² tends to zero? For many interesting examples, these numbers grow roughly like a power of 1=² . In other words, the logarithms log C²(X) log S²(X) log B²(X) are roughly linear functions of log(1=²) , so that the ratios ¼ ¼ log C²(X) log S²(X) log B²(X) log(1=²) ¼ log(1=²) ¼ log(1=²) converge to a common limit as ² 0 . This limit does not always exist. (See Lemma 7.2 below. ????) However we can alwa!ys consider the lim inf , which we write as log C²(X) log S²(X) dim¡(X) = lim inf = lim inf ; B ² 0 log(1=²) ² 0 log(1=²) ! ! as well as the lim sup + log C²(X) log S²(X) dimB(X) = lim sup = lim sup : ² 0 log(1=²) ² 0 log(1=²) ! ! Here the equalities on the right follow easily from formula (7: 1) above. Similarly, if X is n contained in the Euclidean space , then it follows from (7: 2) that log B²(X) + log B²(X) dimB¡(X) = lim inf ; dimB(X) = lim sup : ² 0 log(1=²) ² 0 log(1=²) ! ! Furthermore, in this case it is easy to check that dim+(X) n . In all cases, note that B · + 0 dim¡(X) dim (X) · B · B · 1 provided that X = . 6 ; n De¯nition. Whether or not X is contained in , we will call dimB¡(X) the lower + + box dimension and dimB(X) the upper box dimension of X . If dimB¡(X) = dimB(X) , then 7-2 7B. OTHER DEFINITIONS we will call this common value simply the box dimension dimB(X) . (For a well de¯ned invariant intermediate between the upper and lower box dimensions, see Problem 7-e.) In terms of information theory, the number log B²(X) can be described as the quantity of information needed to specify the coordinates a point of X to within an accuracy of ² . If we use logarithms to the base two, so as to measure information in bits, and set ² = 1=2k , then the statement that log B (X)= log (1=²) dim (X) , means that 2 ² 2 ¼ B we need roughly k dimB(X) bits of information in order to specify a point of X to an accuracy of 1=2k . |||||||||||||||||| 7B. Other De¯nitions of Dimension. The box dimension is a rather crude invariant, wellxsuited to empirical studies. The Hausdor® dimension is a more sophisticated invariant. For our purposes, it can be de¯ned as follows. De¯nition. The Hausdor® dimension dimH (X) of a (not necessarily compact) metric space X is the in¯mum of real numbers ± 0 with the following property: For any ² > 0 , X can be covered by at most countably many¸ sets A with diam(A ) < ² and with f ig i ± diam(Ai) < ² : Xi If there is no such number ± , then by de¯nition dim (X) = + . H 1 Closely associated is the concept of Hausdor® measure. For an arbitrary subset S X ½ and an arbitrary real number ± 0 , the ±-dimensional outer Hausdor® measure ´±(S) is de¯ned by the formula ¸ ± ´±(S) = lim inf diam(Ai) [0; ] : ² 0 ( ) 2 1 ! Xi Here A is to range over all ¯nite or countably in¯nite coverings of S by sets of diameter f ig less than ² . It is not di±cult to check that ´±(X) = 0 whenever ± > dimH (X) , and that ´±(X) = whenever ± < dimH (X) . Thus this concept of measure is possibly non-trivial only when1± is precisely equal to the Hausdor® dimension of X . De¯nition. Following Lebesgue, the topological dimension dim top(X) can be de¯ned as the smallest integer n such that X can be covered by arbitrarily small open sets with the property that no point belongs to more than n + 1 of these sets. (This is sometimes called the covering dimension, to distinguish it from an alternative inductive de¯nition of topological dimension, due to Menger and Urysohn. However, these two de¯nitions of topological dimen- sion always agree, provided that X is metrizable with a countable dense subset. Compare [Hurewicz-Wallman].) As an example, if X is an n-dimensional simplicial complex, then the \open star neigh- borhoods" of the vertices form a covering with this property, and by subdividing the complex we can make this covering arbitrarily ¯ne. In fact the topological dimension is precisely equal to n in this case. (Compare Problem 7-g, as well as 7.1 below.) Note that all four de¯nitions of dimension clearly have the following monotonicity prop- erty: X Y = dim(X) dim(Y ) : (7: 3) ½ ) · 7-3 7. DIMENSION AND ENTROPY The relationship between the four de¯nitions can be described as follows. Lemma 7.1. The inequalities + 0 dim (X) dim (X) dim¡(X) dim (X) · top · H · B · B · 1 are valid whenever they make sense. Furthermore, if X is a compact region in Euclidean n-space, then all four dimensions take the value n . In fact the inequality dim (X) dim¡(X) follows easily from the de¯nitions. (Com- H · B pare Problem 7-h below.) The proof that dim top(X) dimH (X) is more di±cult, and will not be given here. It is due to NÄobeling for subsets· of Euclidean space and to Szpil- rajn in general. (See [Hurewicz and Wallman] for further information.) For the proof that n dim top(X) n when X contains an open subset of , see Problem 7-g. The rest of the proof is straigh¸ tforward. ¤ If X can be described as the union of ¯nitely many smooth compact pieces, then it is not hard to show that these four de¯nitions of dimension all coincide.