Statistical properties of one-dimensional maps with critical points and singularities

TL;DR

This paper proves that certain one-dimensional maps with critical points and singularities have a Markov tower structure, leading to exponential decay of correlations and an absolutely continuous invariant measure.

Contribution

It establishes the existence of an induced Markov tower with exponential return times for maps with multiple critical and singular points, a novel result in this context.

Findings

Existence of an induced Markov tower with exponential return times

Presence of an absolutely continuous invariant probability measure

Exponential decay of correlations for Hölder continuous observables

Abstract

We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for H\"{o}lder observations.