Exponential decay of correlations for piecewise cone hyperbolic contact flows

TL;DR

This paper establishes exponential decay of correlations for a class of three-dimensional piecewise hyperbolic contact flows with singularities, a first in continuous-time dynamics with such features.

Contribution

It introduces a novel proof combining Dolgopyat's estimates and discrete-time hyperbolic dynamics techniques for continuous-time singular flows.

Findings

Proves exponential decay of correlations for piecewise hyperbolic contact flows.

First such proof for continuous-time systems with singularities.

Combines techniques from Dolgopyat and Gou"ezel's work.

Abstract

We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuous-time dynamics with singularities on a manifold. Our proof combines the second author's version of Dolgopyat's estimates for contact flows and the first author's work with Gou\"ezel on piecewise hyperbolic discrete-time dynamics. (Presentation revised.)