Dolgopyat type estimates for pinched open billiard flows (Revised)

TL;DR

This paper establishes Dolgopyat type spectral estimates for open billiard flows with pinched conditions, leveraging non-degeneracy of the symplectic form and C^1 laminations to simplify analysis of Ruelle transfer operators.

Contribution

It introduces a class of open billiard flows satisfying pinching conditions, ensuring C^1 laminations and enabling Dolgopyat estimates under less restrictive assumptions.

Findings

Non-degeneracy of the symplectic form over the non-wandering set.

Existence of a class of flows with pinched conditions leading to C^1 laminations.

Derivation of Dolgopyat type spectral estimates for these flows.

Abstract

In this paper we consider open billiard flows in Euclidean spaces with C^1 (un)stable laminations over their non-wandering sets. We show that for such billiard flows the standard symplectic form satisfies a specific non-degeneracy condition over the non-wandering set. This allows to use some previous general results and obtain Dolgopyat type estimates for spectra of Ruelle transfer operators under simpler conditions. We also describe a class of open billiard flows in Euclidean spaces satisfying a certain pinching condition, which in turn implies that the (un)stable laminations over the non-wandering set are C^1.