Matching in a family of piecewise affine maps

TL;DR

This paper investigates how metric and topological entropy vary in a family of interval maps, revealing semi-regular behavior linked to a combinatorial property called matching.

Contribution

It demonstrates that entropy is often smooth on open dense sets in parameter space due to the matching property, providing new insights into the entropy dynamics of piecewise affine maps.

Findings

Entropy exhibits semi-regular behavior in many cases.

Smooth entropy regions are open and dense in parameter space.

Matching property explains the smoothness of entropy variations.

Abstract

We consider a class of simple one parameter families of interval maps, and we study how metric (resp. topological) entropy changes as the parameter varies. We show that in many cases the entropy displays a semi-regular behaviour, i.e. it is smooth on an open and dense set. This feature is due to a combinatorial property called matching