Entropy via multiplicity

TL;DR

This paper investigates the topological entropy of piecewise affine maps, showing how singularities and expansion rates influence entropy, and provides bounds and examples illustrating entropy behavior.

Contribution

It establishes bounds on entropy based on expansion rates and characterizes when singularities contribute, advancing understanding of entropy in piecewise affine maps.

Findings

Singularities contribute to entropy only with angular expansion

Non-expanding conformal maps have zero topological entropy

Provides estimates and examples of entropy growth in skew-products

Abstract

The topological entropy of piecewise affine maps is studied. It is shown that singularities may contribute to the entropy only if there is angular expansion and we bound the entropy via the expansion rates of the map. As a corollary we deduce that non-expanding conformal piecewise affine maps have zero topological entropy. We estimate the entropy of piecewise affine skew-products. Examples of abnormal entropy growth are provided.