Upper and lower bounds for the correlation function via inducing with general return times

TL;DR

This paper establishes conditions for higher order asymptotics of correlation functions in non-uniformly expanding maps with general return times, extending mixing rate results to broader classes of observables and measure settings.

Contribution

It introduces new sufficient conditions for asymptotics of correlation functions in infinite measure systems, generalizing previous finite measure results to non-Markov maps.

Findings

Conditions for higher order correlation asymptotics in infinite measure

Extension of sharp mixing rate results to non-Markov maps

Application to non-Markov interval maps with indifferent fixed points

Abstract

For non-uniformly expanding maps inducing with a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining higher order asymptotics for the correlation function in the infinite measure setting. Along the way, we show that these conditions are sufficient to recover previous results on sharp mixing rates in the finite measure setting for non-Markov maps, but for a larger class of observables. The results are illustrated by (finite and infinite measure preserving) non-Markov intervals maps with an indifferent fixed point.