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AN INTRODUCTION to DYNAMICAL BILLIARDS Contents 1 AN INTRODUCTION TO DYNAMICAL BILLIARDS SUN WOO PARK 2 Abstract. Some billiard tables in R contain crucial references to dynamical systems but can be analyzed with Euclidean geometry. In this expository paper, we will analyze billiard trajectories in circles, circular rings, and ellipses as well as relate their charactersitics to ergodic theory and dynamical systems. Contents 1. Background 1 1.1. Recurrence 1 1.2. Invariance and Ergodicity 2 1.3. Rotation 3 2. Dynamical Billiards 4 2.1. Circle 5 2.2. Circular Ring 7 2.3. Ellipse 9 2.4. Completely Integrable 14 Acknowledgments 15 References 15 Dynamical billiards exhibits crucial characteristics related to dynamical systems. Some billiard tables in R2 can be understood with Euclidean geometry. In this ex- pository paper, we will analyze some of the billiard tables in R2, specifically circles, circular rings, and ellipses. In the first section we will present some preliminary background. In the second section we will analyze billiard trajectories in the afore- mentioned billiard tables and relate their characteristics with dynamical systems. We will also briefly discuss the notion of completely integrable billiard mappings and Birkhoff's conjecture. 1. Background (This section follows Chapter 1 and 2 of Chernov [1] and Chapter 3 and 4 of Rudin [2]) In this section, we define basic concepts in measure theory and ergodic theory. We will focus on probability measures, related theorems, and recurrent sets on certain maps. The definitions of probability measures and σ-algebra are in Chapter 1 of Chernov [1]. 1.1. Recurrence. Definition 1.1. Let (X,A,µ) and (Y ,B,υ) be measure spaces. A map T : X ! Y is measure-preserving if it satisfies the following two properties: 1 2 SUN WOO PARK (1) For each open set B 2 B, T −1(B) 2 A (2) µ(T −1(B))=υ(B) We will now prove the probabilistic version of Poincar´eRecurrence Theorem. Theorem 1.2 (Poincar´eRecurrence Theorem). Given a measure space (X,A,µ), let T: X ! X be a measure-preserving map. Given A 2 A, define A0 as the set of n points a 2 A such that T (a) 2 A for infinitely many n ≥ 0. Then A0 2 A and µ(A0)=µ(A). j Proof. Let Bn := fa 2 A j T (a) 2= A for all j ≥ ng. Notice that 1 A0 = An [n=1 Bn Thus, it suffices to show that Bn 2 A and µ(Bn)=0 for every n ≥ 1. Since −j Bn = An [j≥n T (A) it is clear that Bn 2 A. This implies −j −j Bn ⊂ [j≥0T (A)n [j≥n T (A) Observe that −j −n −j [j≥0T (A) = T ([j≥nT (A)) Since T is a measure-preserving map, −j −j µ([j≥0T (A)) = µ([j≥nT (A)) Thus, µ(Bn) = 0. The above theorem shows that under a measure-preserving map almost every point of a measurable set is recurrent. 1.2. Invariance and Ergodicity. Definition 1.3. Let (X,A) be a measurable space and let T : X ! X be a mea- surable map. A measure µ is T-invariant if for every A 2 A, µ(T −1(A)) = µ(A) Definition 1.4. Let T be a measure preserving map on a probability space (X,A,µ). T is ergodic if every T -invariant set has measure 0 or 1. We now state one of the most important theorems in ergodic theory. The proof of the theorem is in Chapter 2 of Chernov [1]. Theorem 1.5 (Birkhoff-Khinchin). Given a probability space (X,A,µ), let T :X ! X be a measure preserving map. If f:X ! R is in L1(X,A,µ), the limit n 1 X f~(x) := lim f(T j(x)) n!1 n + 1 j=0 exists for almost every x 2 X, and the function f(~x) is T-invariant, integrable, and Z Z f~dµ = f dµ X X The function f~ is called the time average of f. Using the theorem, we can prove the following useful proposition. The proof of the proposition is also in Chapter 2 of Chernov [1]. AN INTRODUCTION TO DYNAMICAL BILLIARDS 3 Proposition 1.6. Given a probability space (X,A,µ), the following statements are equivalent: (1) T is ergodic. (2) If f 2 Lp(X) for a positive integer p, then f is constant almost everywhere. (3) For every A,B 2 A, we have n−1 1 X lim µ(T −n(A) \ (B)) = µ(A)µ(B) n!1 n m=0 1 ~ R (4) For every f 2 L (X) we have f(x) = X f dµ almost everywhere. We will use the proposition in understanding ergodicity of irrational rotations on the circle, one of the examples in dynamical systems which is important for understanding billiard trajectories in circles. 1.3. Rotation. We now state how ergodicity and invariance of measures can be used in rational and irrational rotations on the circle. Dynamical systems deal with understanding the rules of a time-dependent mathematical object. We can describe the time-dependence of an object by iterations of a map on the object. Define a mapping T : [0,1] ! [0,1] such that T a(x) = x + a (mod 1) where x 2 [0,1] and a 2 R. Here x denotes the initial position and a denotes the angle of rotation along the circle. Then the n iterations of the map T can be expressed as n follows: T a (x) = x + na (mod 1) = m + r (mod 1) where m 2 Z and r 2 [0,1]. It is clear that T is measure-preserving. Proposition 1.7. Every orbit of the mapping T a(x) is dense in [0,1] if a 2 RnQ. n Proof. First we will show that each T a (x) is distinct. For two distinct natural i j numbers i and j assume T a(x) = T a(x) + k for some positive integer k. In other words, x + ia = x + ja + k. So, a(i − j) is a natural number. Since a is irrational, i = j, a contradiction. Divide the interval [0,1] into n intervals, each with length 1/n. Notice that the n n + 1 terms x, T a(x), :::T a (x) are distinct. Then by pigeon hole principle there i j exists an interval that includes at least two of the terms T a(x) and T a(x) for i 6= j. Thus for every positive integer n there exist two distinct natural numbers i and i j j such that jT a(x)−T a(x)j < 1/n. Now we prove that every neighborhood of an arbitrary point y in [0,1] contains m the term T a (x). Without loss of generality we can show that every neighborhood of m the initial term x contains the term T a (x). Notice that Ta is a measure-preserving map with respect to the lebesgue measure µ. In other words, the euclidean distance between two points of the interval is invariant under iterations of the map Ta; so, the following holds for natural numbers i, j, and n. i−j j i−j j i j d(Ta (x); x) = d(Ta (Ta (x);Ta (x)) = d(Ta(x);Ta (x)) < 1=n i−j Thus every neighborhood of the initial term x contains the term T a (x). Hence every orbit of Ta is dense in [0,1]. Now we prove one of the important characteristics of Ta. Theorem 1.8. T is ergodic if and only if a 2 RnQ. 4 SUN WOO PARK Proof. Assume a is irrational. Define f to be an arbitrary T -invariant function in L2(X) where X is the interval [0,1]. We claim that f is constant almost everywhere. The Fourier expansion of f in L2(X) is as follows. 1 X 2πinx cne n=−∞ Similarly, the Fourier expansion of f◦T in L2(X) is as follows. 1 1 X 2πin(x+a) X 2πina 2πinx cne = cne e n=−∞ n=−∞ Since f is T -invariant, f◦T is same as f, which implies 1 X 2πinx 2πina e cn(1 − e ) = 0 n=−∞ Since the elements of the series e2πinx form an orthonormal basis of the Hilbert 2πina 2πina space, all coefficients cn(1-e ) are zero. Since a is irrational, 1-e is not 0. Thus, every Fourier series coefficient is equal to 0. This shows that f is constant. Hence, T is ergodic by proposition 1.6. Now assume T is ergodic. We can prove by contrapositive that if a is rational, then T is not ergodic. Observe the Fourier expansion of f−f◦T in L2(X). Since a is rational, 1-e2πina is 0 for some integer n. Hence not all coefficients of the Fourier series are zero, which implies that there exists a function f in L2(X) such that f is not constant. Thus, T is not ergodic by proposition 1.6. 2. Dynamical Billiards (This section follows Chapter 4 of Chernov [1] and Chapter 4 of Tabachnikov [3].) Dynamical Billiards focuses on the characteristics of billiard trajectory in respect to time. In this section we focus on some of the simple forms of smooth billiard tables on 2-dimensional euclidean space. We start the section with an overview of dynamical billiards. Definition 2.1. A billiard table Q 2 R2 is an open bounded connected domain such that its boundary @Q is a finite union of smooth compact curves.
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