Billiards 
BIRKHOFF BILLIARDS: The motion of a free particle in a bounded region reflecting elastically at the boundary is called a billiard . Convex twodimensional convex regions in the plane define Birkhoff billiards . A Birkhoff billiard is smooth if the boundary of the table is described by arbitrarily often differentiable functions. 
EXAMPLES: 1) Ellipse: x^{2}/a^{2} + y^{2}/b^{2} = 1. 2) Polygons: eg. triangle 3) Polygons with rounded corners: eg. stadium 4) Tables of equal width: eg. Ruleaux triangle 

WHY STUDY BILLIARDS? Billiards ard beautiful and simple dynamical systems featuring many complexities of Hamiltonian systems in general. They form a limiting case of the geodesic flow and illustrate theorems in topology, geometry or ergodic theory. Some dynamical problems are related to the Dirichlet problem which still features Kac's problem "can one hear the shape of a smooth drum" (until now, known isospectral counterexamples to Kac's question are not smooth). Billiards are used to investigate the transition from quantum mechanics (the Dirichlet problem, asymptotics of eigenvalues and eigenfunctions) to classical mechanics (the geodesic flow). 
THE BILLIARD MAP: Let s be T = R (mod 1) a point on the boundary of the table and let : be the angle of the velocity vector to the tangent at s. Define . If s' is the new intersection with boundary and is defined from the impact angle . the billiard map on the annulus X = T x [1,1] is The boundaries , consist of fixed points. 
DEFINITION: INTEGRABILITY: A billiard map T:X > X is called integrable if there exists a piecewise continuous f: X > R such that each set {f(s,u)= c} is a finite union of discrete points or onedimensional curves. (Warning: this is one of many definitions of integrability for billiards). 
DEFINITION: LYAPUNOV EXPONENT. is called the Lyapunov exponent of T at (s,u). It measures sensitive dependence on initial conditions. 
DEFINITION: CHAOS. A billiard is called chaotic if the Pesin set has positive Lebesgue measure. No smooth, convex chaotic billiard is known! (Warning: the word "chaos" is used in many different ways). 
The return map of a Birkhoff billiard is an areapreserving map on the annulus. Birkhof billiards become so a natural object in ergodic theory .  By a change of variable formula this is equivalent to: Jacobian satisfies . 
Smooth strictly convex billiards have many periodic orbits: a ball which returns after p reflections back having been winding around q times around the table. These Birkhoff periodic orbits can be used to construct invariant continua of the map, so called Mather sets In some cases, these sets are Cantor sets, sometimes invariant curves.  are points on , let , where . A periodic orbit T^{k}(s,u) =(s_{k}, u_{k}) corresponds to a critical point of , a point, where the gradient vanishes. There exists a maximum and so a critical point. 
DEFINITION: CAUSTICS: 
EXAMPLES OF CAUSTICS 1) Ellipses have conformal conics as caustics, hyperbola for , ellipses for . 2) convex curve, the string construction leads to table with as caustic. 3) Curves of equal width have caustics which agree with the evolute of the table. 
INVARIANT CURVESCAUSTICS.
If
is an invariant
curve under T and v(s) is a unit vector in
the trajectory direction, P(s) point on table, curvature at s, we get a caustic
(Douady).
PROOF. T(s,u) = (s_{1},u_{1}). , angle of trajectory L(s,s_{1}) to xaxes. , where . : angle of tangent at T(s), then . Furthermore , so that . 
CURVES OF EQUAL WIDTH. angle of tangent. radius of curvature at . defines a closed convex curve if . If additionally , this is a table of equal width. 
CAUSTIC AND EVOLUTE: 
ARE THERE FRACTAL CAUSTICS? Are there caustics which are fractals? Are there fractal evolutes of convex curves of equal width? 
DEFINITION: FRACTALS. Z subset of an Euclidean space. For , define , where runs over all open covers of Z. ( , U_{j} open and ). The limit is called sdimensional Hausdorff measure of Z. It exists in because increases for . If , then h^{t}(Z)=0 for all t>s. Define the Hausdorff dimension by , . Fractals are sets Z with noninteger . 

G. Birkhoff, Acta Math, 50, 1927 Ya.G. Sinai, Introduction to ergodic theory, 1976 S. Tabachnikov. Billiards, 1995 D.V. Treshchev, V.V. Kozlov, Billiards, Transl. Math. Monog., 89 
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© 1999, Oliver Knill , dynamicalsystems.org 