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Billiards: a Geometric and Ergodic Approach Nicolás Daniel Mart´Inez

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Billiards: a Geometric and Ergodic Approach Nicolás Daniel Mart´Inez Billiards: A Geometric and Ergodic Approach Nicol´asDaniel Mart´ınezRamos Department of Mathematics Universidad de Los Andes Supervisor Alexander Getmanenko In partial fulfillment of the requirements for the degree of Mathematician November 27, 2019 Acknowledgements As always, there are no possible words, either written or spoken which could show my gratitude to all people that in any way have helped me through my studies and my life. First of all, to my family: my parents, my brother, and my cat, for giving me a home in which I could grow up and become what I am today and what I will be in the future. All of the love, understanding and advice I have received from them guide the rest of my existence. Please, be assured, all that I will ever do in my life goes to you even if you do not even know about it. Now, I would like to thank my friends, from the university, from school, from life, from anywhere. You guys and gals are the reason living is worth all the trouble. I would like to mention all of you, but I am sure I would miss a lot of people, so, please, all of those who find a friend in me, believe me, this is for you. And of course, I would not be a student if I did not take time to properly thank all of the teachers and professors I have had the plea- sure to know and learn from, each one of you, for even the tiniest amount of knowledge you have passed down to me, deserves the most sincere gratitude. I would like to make a special mention to my thesis advisor, Alexander Getmanenko, not only for being an outstanding mentor and professor, but for taking once again the hardships of being a student along with me. To being a student: Young years for smiles and tears. Present for Future. Abstract This thesis comes from the interest to study dynamical systems which do not depend on differential equations and which allow trajectories with nondifferentiable points. At first, the main motivation was the study of laser dynamics in conics to further understand how tunneling worked in such systems, but in the process I found the rich and vast theory of Dynamical Billiards and from then on, our study focused on understanding them, as well as the tools that have been developed to study them. With this in mind, this work is divided into three parts. At first, the purpose of the document will be to properly define what a Billiard is and explore the ways in which movement can be studied. Moreover, we will study some basic properties of the movement in these systems and we will give some examples which we will explore later with more detail. Afterwards, we proceeded to explain the proof of one of the fundamen- tal tools in the study of dynamics and ergodic theory, the Oseledec's Multiplicative Theorem. This theorem ensures us that the limit by which the Lyapunov Exponents are defined exist provided some mild conditions on a transformation which preserves the measure. More- over, we exposed it using rather simple ideas and theorems derived mostly from functional and real analysis. Finally, we used the theorem proved in the previous section and our geometric analysis of the system to derive some conclusions regarding the Billiard Map. This work we have done is not only important for its pedagogic value as a first experience writing a formal mathematical document, but also as a first stepping stone into the world of Dynamics. With this particular example, we have used some of the main concepts on this branch of mathematics, namely the stable and unstable spaces, Lya- punov Exponents and hyperbolic systems. Contents 1 Basic concepts and first examples1 1.1 Definition and Construction of Billiards . .1 1.2 Motion: Flow and Map . .3 1.2.1 The Billiard Flow . .3 1.2.2 The Billiard Map . .8 1.2.3 Families of reflected paths . 15 1.3 Some examples . 19 1.3.1 Billiard in a circle . 19 1.3.2 The Ellipse . 21 1.3.3 The Stadium . 22 2 Oseledec's Multiplicative Ergodic Theorem 24 2.1 Statement of the theorem . 24 2.2 Proof of the first three conclusions . 25 2.2.1 Some theorems about measurability . 28 2.3 Proof of the forth conclusion . 30 2.3.1 A simpler setup . 30 2.3.2 Skew Products . 36 2.3.3 Conclusion . 42 v CONTENTS 3 Lyapunov Exponents for the Billiard Map 43 3.1 Understanding Lyapunov Exponents of Diffeomorphisms . 43 3.2 The Billiard Map and its Lyapunov Exponents . 46 3.2.1 A simple example . 49 3.3 A Projective Criterion for nonzero Lyapunov Exponents . 50 3.3.1 A hyperbolic example . 54 4 Appendix: Some Ergodic Theorems 56 4.1 Ergodic Theorems . 56 References 59 vi Chapter 1 Basic concepts and first examples In the following chapter we will go over the basic definitions for studying billiards. We will give some examples that illustrate the main ideas when we view it as a dynamical system. Moreover, we will take a look at some examples that will appear later in the document. 1.1 Definition and Construction of Billiards In this thesis, we will use the following setup used by Chernov and Markarian in their book [1]: Let D ⊂ R2 be a domain with a smooth or a piece-wise smooth boundary. The billiard system corresponds to the motion of a point particle inside this domain with specular reflections (angle of incidence is the angle of reflection) off the boundary δD. The definition of motion is not in itself a mathematical concept, however, to fill this gap we will shortly introduce the concepts of billiard flow and billiard map. In simple terms, the particle moves in a straight motion until the moment it collides with a point of the boundary, at that moment, the particle starts mov- ing in the direction that makes the same angle with the normal to the wall as the original direction. Moreover, we assume that this particle moves at unit speed. The previous ideas are too vague to properly work with them, then we will impose certain assumptions which will give us more structure on the systems [1]: 2 Assumption 1.1.1. Let D0 ⊂ R be an open, connected set. Our billiard domain will be D0 = D. Assumption 1.1.2. The boundary @D is a union of a finite number of Cr; r ≥ 3 compact curves. We will number these curves and denote them as Γi so that we 1 1.1 Definition and Construction of Billiards have: δD = Γ = Γ1 [···[ Γn: We assume our curves to be at least thrice differentiable to be able to work with their curvature. Assumption 1.1.3. The curves Γi can only intersect each other at their end- points. The endpoints of the curves must be handled with precaution, as at these points it may be impossible to find a tangent and therefore the specular reflection is not defined. Thus, we define the corners of the system as: Γ∗ = δΓ1 [···[ δΓn: For the last assumption, we define a parametrization of each Γi: 2 fi :[ai; bi] ! R : Furthermore, we take the intervals where our curves are defined so that: [ai; bi] \ [aj; bj] = ; for i 6= j: 2 Assumption 1.1.4. Let each Γi be parametrized by a map fi :[a; b] ! R . Then the second derivative of the fi never vanishes or is identically zero. By the previous assumption there are three possible kinds of boundaries which we will also call walls. The first, which is a flat wall, that is, a wall which is given 00 by a line and thus, fi (x) = 0. The second possible wall is the dispersing wall, that is, a wall that is concave. The name comes from the fact that in optics these kind of walls scatters nearby 00 rays of light. As the parametrization is concave we have that the vector fi (x) 6= 0 and it points outside the domain D. On the other hand, the last possible kind of wall is the focusing wall, that is convex. As before, the name comes from the fact that rays of light reflected by these walls get closer. In terms of the derivative, we can characterize these wall 00 by fi (x) 6= 0 and it points inside the domain D. As an example of the classification of these walls, consider the following image: 2 1.2 Motion: Flow and Map Figure 1.1: A billiard table with three types of wall. Note that walls 1 and 3 are flat walls as they are described by a line, wall 2 and 5 are dispersing and wall 4 is focusing. We will need the folowing definition: Definition 1.1.1. The curvature K of each wall Γi is given by: 8 0 if Γ is flat <> i 00 K = −kfi k if Γi is focusing > 00 :kfi k if Γi is focusing. 1.2 Motion: Flow and Map As it is often the case in dynamical systems, we can either study the system by a continuous motion (which we will denote flow) or a discrete motion (called map). In the following section we will briefly go over each point of view and discuss the results we can gather from them. The following section uses ideas both found in [1] and [2] 1.2.1 The Billiard Flow We will denote the position of the particle q 2 R2 and its velocity v 2 S1.
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