The Pennsylvania State University The Graduate School Department of Mathematics REGULAR AND CHAOTIC DYNAMICS OF OUTER BILLIARDS A Thesis in Mathematics by Daniel I. Genin c 2005 Daniel I. Genin Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2005 The thesis of Daniel I. Genin was reviewed and approved∗ by the fol- lowing: Sergei Tabahchnikov Professor of Mathematics Thesis Adviser Chair of Committee Mark Levi Professor of Mathematics Yakov Pesin Distinguished Professor of Mathematics Howard Weiss Professor of Mathematics Elena Katok Associate Professor of Business Administration Nigel Higson Professor of Mathematics Head of Department of Mathematics ∗Signatures are on file in the Graduate School. ABSTRACT This thesis explores two quite different types of dynamics occurring in outer billiards. First, we construct the first example of a hyperbolic outer billiard. Thus settling the question of whether chaotic outer billiards exist. The much more regular (quasi-periodic) dynamics for a certain class of quadrilaterals is considered next. Some results on complexity growth and boundedness of orbits are obtained from sym- bolic dynamical description of the mapping. In conclusion, we present numerical studies of outer billiard systems extending the the above examples. iii Contents 1 Introduction 1 1.1 History, known results and open questions . 1 1.2 Plan of exposition . 5 1.3 Definition and Properties . 7 1.4 Chaotic outer billiards . 12 1.5 Polygonal Outer Billiards . 15 1.6 Numerical results and conjectures . 18 2 Chaotic Outer Billiards 21 2.1 Introduction . 21 2.2 Area construction . 22 2.3 Table . 23 2.4 Cone field . 25 2.5 Hyperbolicity . 28 3 Polygonal Outer Billiards 43 3.1 Introduction . 43 iv 3.2 Polygonal outer billiards . 45 3.2.1 Definitions and notation . 45 3.2.2 Symbolic dynamics, vertex coding and complexity 51 3.3 Trapezoidal outer billiards . 53 3.3.1 First return map on x = 0 . 54 3.3.2 Symbolic dynamics of TX . 62 3.3.3 Dynamics for rational α . 70 3.3.4 Poincare section of T . 78 3.3.5 Boundedness of orbits . 87 3.3.6 Complexity . 91 4 Numerical Explorations 99 4.1 Introduction . 99 4.2 More chaotic outer billiards . 101 4.3 Unbounded orbits? . 106 References 113 v List of Figures 1.1 Outer billiard map . 1 1.2 Definition of outer billiard map . 7 1.3 Impact oscillator and outer billiards . 10 1.4 Sinai billiards . 12 1.5 Bunimovich billiards . 13 1.6 Defocusing . 13 1.7 Wojtkowski billiards . 14 2.1 The outer billiard table . 24 2.2 Cone construction . 27 2.3 Cone preservation . 31 2.4 Cone nesting . 32 2.5 Orbits reflecting in corners . 36 2.6 Order one orbits . 38 2.7 A point reflecting in opposite corners . 39 2.8 Order two orbit . 40 2.9 Order three orbit . 41 vi 2.10 Proof illustration . 42 3.1 L1 . 46 3.2 D for a regular pentagon . 48 3.3 Structure of T 2 . 49 3.4 Dual polygon P ∗ . 51 3.5 Trapezoid . 54 3.6 Diagram describing T + . 58 3.7 Black and white sets . 86 4.1 Regular pentagon a=0.1, 0.2 . 102 4.2 Regular pentagon a=0.3, 0.4 . 102 4.3 Irregular pentagon a=0.1, 0.3 . 103 4.4 Irregular pentagon a=0.5, 0.8 . 103 4.5 Regular pentagon . 104 4.6 Irregular pentagon . 105 4.7 Square . 105 4.8 Dynamics around (100,0) . 106 4.9 Orbit of a satellite domain . 107 4.10 Orbits of satellite domains (50,0) . 108 4.11 Dart quadrilateral . 109 4.12 2000 iterates of (1,1.001) for α = p2 . 110 4.13 Distance from the origin along x = 1 . 111 vii ACKNOWLEDGEMENTS I am grateful to my parents and grandparents for their vigorous encour- agement to complete my doctoral work and to my friends for helping me stay sane. To my advisor for teaching me how to approach and solve mathematical problems and to all others who have taught me all the mathematics I know, and even more that I still do not. And especially to the professors of the PennState Dynamical Systems Group. Thank you. viii Chapter 1 Introduction 1.1 History, known results and open questions ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ Figure 1.1: Outer billiard map The outer billiard map was first introduced by B.H. Neumann [16] and later popularized by J. Moser in the article \Is the solar system 1 stable?" [15] and his book \Stable and Random Motions in Dynam- ical Systems" [14]. Since then outer billiards have been explored by many mathematicians from many different directions. One source of questions are ordinary billiards { interesting questions about ordinary billiards can be reformulated and are often still meaningful for outer billiards. On the other hand, there are some interesting questions about outer billiards which do not make sense for ordinary billiards. For example, the original question of Moser, which to this day re- mains only partially answered, is about unbounded orbits of the outer billiard map. This question, of course, makes no sense for ordinary billiards because the phase space in the later case is compact. Moser indicated that KAM theory can be used to show that any sufficiently smooth strictly convex table has only bounded orbits (the details were worked out in the thesis of R. Douday (unpublished)), but the wider question of whether there are any tables (even not strictly convex) for which the outer billiard map has unbounded orbits remains unsolved. Kolodziej [11], and Shaidenko and Vivaldi [17] independently arrived at a result on boundedness of orbits for a certain class of polygonal outer billiard maps, which was also later reproved using a different method by Gutkin and Simanyi [8]. The exact result is discussed in Section 1.5. Tabachnikov showed that near infinity the orbits of any outer billiard are well approximated by a periodic Hamiltonian flow [21], which puts a bound on the rate of escape of unbounded orbits if such exist, and 2 suggested a non-polygonal, although still discontinuous, outer billiard for which an open set of points appears to escape to infinity. Another fruitful area of research has been the study of dynamical structures for polygonal outer billiards. This is still a relatively new area so there are many results on interesting special cases but little in a way of general theory. It is not hard to see that polygonal outer billiards are piece-wise isometries and so this area has significant con- nections with the study of interval exchanges. The most interesting re- sults in my opinion are on renormalizability of dynamics. Tabachnikov obtained a complete symbolic description for dynamics of pentagonal outer billiard [20] and as a consequence proved that its dynamics is renormalizable about certain points and that the set of non-periodic orbits has a fractal structure with dimension log 6= log(p5 + 2). Adler, Kitchens and Tresser derived a similar result for a piece-wise rotation of the torus by π=4 [1]. Outer billiards about regular odd n-gons can be shown to factor over piece-wise toral rotations so the two systems are closely related. More recently Kouptsov, Lowenstein and Vivaldi [12] used rigorous numerical computations to prove that renormaliz- ability appears for all piecewise rotations of the torus by θ if cos θ is a quadratic irrational. These results hint at a strong connection between dynamics and arithmetic properties of the polygon and seem to point in the direction of Boshernitzan's result on renormalizability of interval exchanges over quadratic fields [2]. It is plausible that similar results 3 may be proved for piecewise toral rotations and by extension for outer billiards about regular polygons. Complexity of piecewise isometries and of polygonal outer billiards in particular has also recently become a subject of intensive study. Al- though it has been known for some time that growth of complexity for this type of maps is sub-exponential, implying that metric and topo- logical entropies are zero, few exact results are known. For ordinary polygonal billiards results on sub-exponential growth were obtained by Katok [10], Galperin, Kruger and Troubetzkoy [6], and Gutkin and Haydn [7], also recently some more precise bounds for several cases were obtained by Cassaigne, Hubert and Troubetzkoy [5]. A first step in this direction for outer billiards was a recent paper by Gutkin and Tabachnikov [9] in which bounds on complexity growth are derived for a wide class of maps that includes outer billiards. In particular, they prove that for a general p-gon complexity grows at most as np+2 and it grows as n2 for lattice polygons. In view of the afore mentioned results connecting arithmetic properties