## Institute for Mathematical Sciences

## Preprint ims01-13

** Marco Lenci**
* Billiards with an infinite cusp*

Abstract: Let $f: [0, +\infty) \longrightarrow (0, +\infty)$ 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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