Statistical Properties and Decay of Correlations for Interval Maps with Critical Points and Singularities

TL;DR

This paper studies the statistical behavior of certain one-dimensional maps with critical points and singularities, establishing limit theorems and decay of correlations under mild conditions.

Contribution

It proves the central limit theorem and invariance principles for maps with critical points, extending understanding of their statistical properties.

Findings

Proves the central limit theorem for the class of maps.

Establishes decay of correlations under mild conditions.

Demonstrates an almost sure invariance principle.

Abstract

We consider a class of piecewise smooth one-dimensional maps with critical points and singularities (possibly with infinite derivative). Under mild summability conditions on the growth of the derivative on critical orbits, we prove the central limit theorem and a vector-valued almost sure invariance principle. We also obtain results on decay of correlations.