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Lower bounds for the decay of correlations in non-uniformly expanding maps

Published online by Cambridge University Press:  07 November 2017

HUYI HU  and
SANDRO VAIENTI
HUYI HU
Affiliation:
Mathematics Department, Michigan State University, East Lansing, MI 48824, USA email hu@math.msu.edu
SANDRO VAIENTI
Affiliation:
Aix-Marseille Université, CNRS, CPT, UMR 7332, Marseille, France Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France email vaienti@cpt.univ-mrs.fr
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Abstract

We give conditions under which non-uniformly 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 is satisfied in a Banach space ${\mathcal{B}}$, 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 ${\mathcal{B}}.$ 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-Hölder conditions and have their support bounded away from the neutral fixed points.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 7 , July 2019 , pp. 1936 - 1970
Copyright
© Cambridge University Press, 2017 

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