A renewal scheme for non-uniformly hyperbolic semiflows

TL;DR

This paper develops a renewal scheme for non-uniformly hyperbolic semiflows to derive precise correlation estimates, applicable to certain examples with unbounded flow-boxes, without relying on Markov structures.

Contribution

It introduces a renewal approach for non-uniform hyperbolic semiflows that does not require Markov structures and provides sharp correlation estimates for specific classes of examples.

Findings

Sharp estimates for correlation functions obtained.

Applicable to flow-boxes of unbounded length.

Optimal results for finite length flow-box observables.

Abstract

We investigate a renewal scheme for non-uniformly hyperbolic semiflows that closely resembles the renewal scheme developed in the discrete time case, in order to obtain sharp estimates for the correlation function. Also, the involved observables are supported on a flow-box of unbounded length. The present abstract setting does not require the use of Markov structure. However, the classes of examples covered here are rather restrictive. In these examples, it is easier to exploit the full force of the method and get optimal results for observables supported on finite length flow-boxes.