Lyapunov exponents and nonadapted measures for dispersing billiards

TL;DR

This paper constructs nonadapted measures with positive entropy for dispersing billiards, demonstrating phase transitions in thermodynamic formalism even under restrictions to adapted or positive entropy measures.

Contribution

It introduces a method to construct nonadapted measures with positive entropy for dispersing billiards with grazing orbits, revealing phase transitions in thermodynamic formalism.

Findings

Existence of nonadapted measures with positive entropy

Phase transition in thermodynamic formalism for dispersing billiards

Construction applicable to billiards with grazing periodic orbits

Abstract

For hyperbolic systems with singularities, such as dispersing billiards, Pesin theory as developed by Katok and Strelcyn applies to measures that are "adapted" in the sense that they do not give too much weight to neighborhoods of the singularity set. The zero-entropy measures supported on grazing periodic orbits are nonadapted, but it has been an open question whether there are nonadapted measures with positive entropy. We construct such measures for any dispersing billiard with a periodic orbit having a single grazing collision; we then use our construction to show that the thermodynamic formalism for such billiards has a phase transition even when one restricts attention to adapted or to positive entropy measures.