Billiards with Positive Topological Entropy |
Common terms and phrases
A₁ admit positive topological angle B₁ backward direction billiard flow billiard map billiard system billiard tables boundary point bump function Cb,c Chapter collision construct convex billiard convex tables corresponding critical points D₁ defined Definition 3.5 disjoint disk dynamical systems ergodic theory euclidian exponential growth Figure geodesic flows hm¹ homeomorphism horizontal curve hyperbolic space hypothesis Im(c imaginary axis increases exponentially fast intersection iterations Lemma linear cone Lyapunov exponents MARKARIAN Markov graph metric metric space monotonicity neighbourhood normal periodic point obstacles outer boundary periodic orbits perturb Pesin regions phase space positive entropy positive topological entropy preimage Proof Proposition 3.6 prove r≤ro radius rectangles U(x saddle extremes scatterer shadow condition single inner circle six-periodic orbit six-periodic point strictly convex subset sufficiently small radii surjective system h tangent Theorem 4.1 trajectories transversal two-periodic orbits two-periodic points vectors vertex