Semi-dispersing billiards with an infinite cusp

Details Semi-dispersing billiards with an infinite cusp Semi-dispersing billiards with an infinite cusp

2001-07-18

arxiv.org/abs/nlin/0107041v3

Comm. Math. Phys. 230 (2002), no. 1, 133-180

Nonlinear Sciences

Chaotic Dynamics

57 pages, 19 figures. Revised version

Scientific paper

10.1007/s00220-002-0710-7

Let $f: [0, +\infty) \to (0, +\infty)$ be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain $Q$ delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. Under certain conditions on $f$, we prove that the billiard flow in $Q$ has a hyperbolic structure and, for some examples, that it is also ergodic. This is done using the cross section corresponding to collisions with the dispersing part of the boundary. The relevant invariant measure for this Poincar\'e section is infinite, whence the need to surpass the existing results, designed for finite-measure dynamical systems.

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