Area expanding C^{1+\alpha} Suspension Semiflows

TL;DR

This paper develops a functional analytic framework for analyzing mixing rates in a broad class of low-regularity suspension semiflows, including Lorenz semiflows, with discontinuous return maps and unbounded return times.

Contribution

It introduces a new approach to study rates of mixing for suspension semiflows with low regularity and discontinuities, extending existing methods to more complex dynamical systems.

Findings

Laplace transform of correlation functions admits a meromorphic extension

Established quasi-compactness of weighted transfer operators for piecewise C^{1+eta} maps

Framework applies to Lorenz semiflows and similar systems

Abstract

We study a large class of suspension semiflows which contains the Lorenz semiflows. This is a class with low regularity (merely C^{1+\alpha}) and where the return map is discontinuous and the return time is unbounded. We establish the functional analytic framework which is typically employed to study rates of mixing. The Laplace transform of the correlation function is shown to admit a meromorphic extension to a strip about he imaginary axis. As part of this argument we give a new result concerning the quasi-compactness of weighted transfer operators for piecewise C^{1+\alpha} expanding interval maps.