Statistical properties of nonuniformly expanding 1d maps with logarithmic singularities

TL;DR

This paper studies a family of circle maps with critical points and logarithmic singularities, proving the existence of invariant measures with strong statistical properties like exponential decay of correlations and positive variance in the CLT.

Contribution

It establishes the existence of absolutely continuous invariant measures with exponential mixing and CLT variance for maps with logarithmic singularities, extending previous results to this class.

Findings

Existence of absolutely continuous invariant measures

Exponential decay of correlations

Positive variance in the Central Limit Theorem

Abstract

For a certain parametrized family of maps on the circle, with critical points and logarithmic singularities where derivatives blow up to infinity, a positive measure set of parameters was constructed in [19], corresponding to maps which exhibit nonuniformly hyperbolic behavior. For these parameters, we prove the existence of absolutely continuous invariant measures with good statistical properties, such as exponential decay of correlations. Combining our construction with the logarithmic nature of the singularities, we obtain a positive variance in Central Limit Theorem, for any nonconstant H\"older continuous observable.