Lower Bounds for the Decay of Correlations in Non-uniformly Expanding Maps

TL;DR

This paper establishes polynomial lower bounds for the decay of correlations in non-uniformly expanding maps, under certain conditions related to transfer operators, weak mixing, and aperiodicity, with applications to higher-dimensional and indifferent fixed point maps.

Contribution

It provides new conditions ensuring polynomial decay of correlations for non-uniformly expanding maps, including cases with indifferent fixed points and non-Markov structures.

Findings

Polynomial decay bounds for non-uniformly expanding maps.

Conditions for aperiodicity in higher-dimensional expanding maps.

Lower bounds for maps with indifferent fixed points and unbounded distortion.

Abstract

We give conditions under which nonuniformly expanding maps exhibit lower bounds of polynomial type for the decay of correlations and for a large class of observables. We show that if the Lasota-Yorke type inequality for the transfer operator of a first return map are satisfied in a Banach space B, and the absolutely continuous invariant measure obtained is weak mixing, in terms of aperiodicity, then under some renewal condition, the maps have polynomial decay of correlations for observables in B. We also provide some general conditions that give aperiodicity for expanding maps in higher dimensional spaces. As applications, we obtain lower bounds for piecewise expanding maps with an indifferent fixed point and for which we also allow non-Markov structure and unbounded distortion. The observables are functions that have bounded variation or satisfy quasi-H\"older conditions and have their support bounded away from the neutral fixed points.