Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities

TL;DR

This paper develops a symbolic coding framework for non-uniformly hyperbolic surface maps with discontinuities, including billiard systems, accommodating unbounded derivatives near discontinuities and non-exponentially converging orbits.

Contribution

It introduces a novel symbolic dynamics construction for non-uniform hyperbolic maps with discontinuities, extending previous methods to more general settings.

Findings

Applicable to non-uniformly hyperbolic billiards like Bunimovich stadia

Codes orbits that do not converge exponentially fast to discontinuities

Handles unbounded derivatives near discontinuities

Abstract

This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. Under natural geometrical assumptions on the underlying space $M$, we code a set of non-uniformly hyperbolic orbits that do not converge exponentially fast to $D$. The results apply to non-uniformly hyperbolic planar billiards, e.g. Bunimovich stadia.