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Elements of Fractal Geometry and Dynamics Yakov Pesin Vaughn Elements of Fractal Geometry and Dynamics Yakov Pesin Vaughn Climenhaga Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 E-mail address: [email protected] Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 E-mail address: [email protected] Contents Foreword: MASS and REU at Penn State University ix Lecture 1 1 a. A three-fold cord 1 b. Intricate geometry and self-similarity 1 c. Things that move (or don’t) 4 Lecture 2 7 a. Dynamical systems: terminology and notation 7 b. Examples 8 Lecture 3 14 a. A linear example with chaotic behaviour 14 b. The Cantor set and symbolic dynamics 17 Lecture 4 21 a. A little basic topology 21 b. The topology of symbolic space 22 c. What the coding map doesn’t do 23 Lecture 5 26 a. Geometry of Cantor-like sets 26 b. More general Cantor-like sets 27 c. Making sense of it all 30 Lecture 6 32 a. The definition of Hausdorff dimension 32 Lecture 7 36 a. Properties of Hausdorff dimension 36 b. Topological dimension 38 c. Comparison of Hausdorff and topological dimension 39 Lecture 8 41 a. Some point set topology 41 b. Metrics and topologies 43 Lecture 9 48 a. Topological consequences 48 b. Hausdorff dimension of Cantor-like constructions 49 Lecture 10 51 a. Completion of the proof of Theorem 21 51 Lecture 11 53 a. A competing idea of dimension 53 b. Basic properties and relationships 55 v vi CONTENTS Lecture 12 58 a. Further properties of box dimension 58 b. A counterexample 59 Lecture 13 62 a. Subadditivity, or the lack thereof 62 b. A little bit of measure theory 63 Lecture 14 65 a. Lebesgue measure and outer measures 65 b. Hausdorff measures 67 Lecture 15 69 a. Choosing an outer measure 69 b. Measures on symbolic space 70 c. Measures on Cantor sets 71 Lecture 16 73 a. Markov measures 73 b. The support of a measure 74 c. Markov measures and dynamics 76 Lecture 17 78 a. Using measures to determine dimension 78 b. Pointwise dimension 79 Lecture 18 81 a. The Non-uniform Mass Distribution Principle 81 b. Non-constant pointwise dimension 82 Lecture 19 86 a. More about the Lyapunov exponent 86 b. Fractals within fractals 87 c. Hausdorff dimension for Markov constructions 89 Lecture 20 92 a. FitzHugh-Nagumo and you 92 Lecture 21 94 a. Numerical investigations 94 b. Studying the discrete model 95 Lecture 22 97 a. Stability of fixed points 97 b. Things that don’t stand still, but do at least come back 101 Lecture 23 105 a. Down the rabbit hole 105 b. Becoming one-dimensional 107 Lecture 24 110 a. Bifurcations for the logistic map 110 b. Different sorts of bifurcations 112 Lecture 25 115 a. The simple part of the bifurcation diagram 115 b. The other end of the bifurcation diagram 116 c. The centre cannot hold—escape to infinity 118 CONTENTS vii Lecture 26 120 a. Some parts of phase space are more equal than others 120 b. Windows of stability 121 c. Outside the windows of stability 123 Lecture 27 124 a. Chaos in the logistic family 124 b. Attractors for the FitzHugh–Nagumo model 124 c. The Smale–Williams solenoid 126 Lecture 28 128 a. The Smale–Williams solenoid 128 b. Quantifying the attractor 130 Lecture 29 132 a. The non-conformal case 132 b. The attractor for the FitzHugh–Nagumo map 132 c. From attractors to horseshoes 133 Lecture 30 137 a. Symbolic dynamics on the Smale horseshoe 137 b. Variations on the horseshoe map 138 Lecture 31 141 a. Transient and persistent chaos 141 b. Continuous-time systems 141 Lecture 32 146 a. The pendulum 146 b. Two-dimensional systems 148 c. The Lorenz equations 151 Lecture 33 153 a. Beyond the linear mindset 153 b. Examining the Lorenz system 154 Lecture 34 158 a. Homoclinic orbits and horseshoes 158 b. Horseshoes for the Lorenz system 160 Lecture 35 163 a. More about horseshoes in the Lorenz system 163 Lecture 36 166 a. The Lorenz attractor 166 b. The geometric Lorenz attractor 166 Lecture 37 170 a. Random fractals 170 b. Back to the Lorenz attractor, and beyond 170 LECTURE 1 1 Lecture 1 a. A three-fold cord. The word “fractal” is one which has wriggled its way into the popular consciousness over the past few decades, to the point where a Google search for “fractal” yields over 12 million results (at the time of this writing), more than six times as many as a search for the rather more fundamental mathematical notion of “isomorphism”. With a few clicks of a mouse, and without any need to enter the jargon-ridden world of academic publications, one may find websites devoted to fractals for kids, a blog featuring the fractal of the day, photo galleries of fractals occurring in nature, online stores selling posters brightly emblazoned with computer-generated images of fractals. the list goes on. Faced with this jungle of information, we may rightly ask, echoing Paul Gauguin, “What are fractals? Where do they come from? Where do we go with them?” The answers to the second and third questions, at least as far as we are concerned, will have to do with the other two strands of the three-fold cord holding this course together—namely, dynamical systems and chaos.1 As an initial, na¨ıve formulation, we may say that the combination of dynamical systems and fractals is responsible for the presence of chaotic behaviour. For our purposes, fractals will come from certain dynamical systems, and will lead us to an understanding of certain aspects of chaos. But all in good time. We must begin by addressing the first question; “What are fractals?” b. Intricate geometry and self-similarity. Consider an oak tree in the dead of winter, viewed from a good distance away. Its trunk rises from the ground to the point where it narrows and sends off several large boughs; each of these boughs leads away from the centre of the tree and eventually sends off smaller branches of its own. Walking closer to the tree, one sees that these branches in turn send of still smaller branches, which were not visible from further away, and more careful inspection reveals a similar branching structure all the way down to the level of tiny twigs only an inch or two long.2 The key points to observe are as follows. First, the tree has a compli- cated and intricate shape, which is not well captured by the more familiar geometric objects, such as lines, circles, polygons, and so on. Secondly, we see the same sort of shape on all scales—whether we view the tree from fifty yards away or from fifty inches, we will see a branching structure in which the largest branch (or trunk) in our field of view splits into smaller branches, which then divide themselves, and so on. 1Chaos theory has, of course, also entered the popular imagination in its own right recently, thanks in part to its mention in movies such as Jurassic Park. 2All of this is still present in summer, of course, but the leaves get in the way of easy observation. 2 CONTENTS These features are shared by many other objects which we may think of as fractals—we see a similar picture if we consider the bronchial tree, the network of passageways leading into the lungs, which branches recursively across a wide range of scales. Or we may consider some of the works of the artist M. C. Escher, we see intricate patterns repeating at smaller and smaller scales. Yet another striking example may be seen by looking at a high-resolution satellite image (or detailed map) of a coastline. The boundary between land and sea does not follow a nice, simple path, but rather twists and turns back and forth; each bay and peninsula is adorned with still smaller bays and peninsulas, and given a map of an unfamiliar coast, we would be hard pressed to identify the scale at which the map was printed if we were not told what it was. The two threads connecting these examples are their complicated geom- etry and some sort of self-similarity. Recall that two geometric figures (for example, two triangles) are similar if one can be obtained from the other by a combination of rigid motions and rescaling. A fractal exhibits a sort of similarity with itself; if we rescale a part of the image to the size of the whole, we obtain something which looks nearly the same as the original. We now make these notions more precise. Simple geometric shapes, such as circles, triangles, squares, etc., have boundaries which are smooth curves, or at least piecewise smooth. That is to say, we may write the boundary parametrically as ~r(t) = (x(t), y(t)), and for the shapes we are familiar with, x and y are piecewise differentiable functions from R to R, so that the tangent vector ~r ′(t) exists for all but a few isolated values of t. By contrast, we will see that a fractal “curve”, such as a coastline, is continuous everywhere but differentiable nowhere. As an example of this initially rather unsightly behaviour, we consider the von Koch curve, defined as follows. Taking the interval [0, 1], remove the middle third (just as in the construction of the usual Cantor set), and replace it with the other two sides of the equilateral triangle for which it is the base. One obtains the piecewise linear curve at the top of Figure 1; this is the basic pattern from which we will build our fractal.
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