3 pages matching geodesic flows in this book
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Convex billiards with obstacles
Topological entropy and billiards
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admit positive topological angle annulus B(mb backward direction base-point billiard flow billiard map billiard system billiard tables boundary point bump function Chapter collision complement component construct contains convex billiard convex tables corresponding critical points defined Definition 3.5 denote disjoint disk distance spheres dynamical systems ergodic theory euclidian exponential growth Figure geodesic flows graph G homeomorphism horizontal curve hyperbolic space hypothesis Im(c imaginary axis increases exponentially fast iterations Lemma linear cone Lyapunov exponents Markarian Markov graph Math metric metric space monotonicity Moreover neighbourhood normal periodic point normal two-periodic open and dense outer boundary periodic orbits perturb Pesin regions phase space positive entropy positive topological entropy possibly smaller preimage Proposition 3.6 prove radius rectangle U(y saddle extremes satisfies scatterer shadow condition single inner circle six-periodic orbit six-periodic point strictly convex sufficiently small radii surjective T2 o k tangent Theorem 4.1 trajectories two-periodic orbits two-periodic points U(xj vertex vertical