Exponential rate of decay of correlations of equilibrium states associated with non-uniformly expanding circle maps

TL;DR

This paper establishes conditions under which non-uniformly expanding circle maps exhibit exponential decay of correlations, linking the decay rate to the spectral gap of the transfer operator for observables with specific moduli of continuity.

Contribution

It identifies a new sufficient condition on the modulus of continuity that guarantees exponential decay of correlations for equilibrium states.

Findings

Exponential decay of correlations is proven under certain modulus conditions.

Spectral gap property of the transfer operator is established for these systems.

Results apply to examples like the Manneville-Pomeau family.

Abstract

In the context of expanding maps of the circle with an indifferent fixed point, understanding the joint behavior of dynamics and pairs of moduli of continuity $ (\omega, \Omega) $ may be a useful element for the development of equilibrium theory. Here we identify a particular feature of modulus $ \Omega $ (precisely $ \lim_{x \to 0^+} \sup_{\mathsf d} \Omega\big({\mathsf d} x \big) / \Omega(\mathsf d) = 0 $) as a sufficient condition for the system to exhibit exponential decay of correlations with respect to the unique equilibrium state associated with a potential having $ \omega $ as modulus of continuity. This result is derived from obtaining the spectral gap property for the transfer operator acting on the space of observables with $ \Omega $ as modulus of continuity, a property that, as is well known, also ensures the Central Limit Theorem. Examples of application of our results include the Manneville-Pomeau family