Entropy formula and continuity of entropy for piecewise expanding maps

TL;DR

This paper derives an entropy formula for certain piecewise expanding maps and establishes conditions for the continuity of entropy with respect to parameters, with applications to tent maps and higher-dimensional maps.

Contribution

It provides a new entropy formula for invariant measures of piecewise expanding maps and analyzes entropy continuity in parametrized families.

Findings

Entropy formula for invariant measures of piecewise expanding maps

Sufficient conditions for entropy continuity in parameterized families

Application to tent maps and three-dimensional homoclinic tangency maps

Abstract

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, in some parametrized families we present sufficient conditions for the continuity of that entropy with respect to the parameter. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.