Discontinuity of Topological Entropy for the Lozi Maps

TL;DR

This paper demonstrates that the topological entropy of Lozi maps can exhibit discontinuous jumps, specifically from zero to above 0.1203, revealing that entropy is not always upper semi-continuous in these systems.

Contribution

It proves the discontinuity of topological entropy for Lozi maps and extends the result to a neighborhood in parameter space, highlighting non semi-continuity.

Findings

Entropy jumps from zero to above 0.1203 at certain parameters

Discontinuity occurs along a line segment in parameter space

Topological entropy is not upper semi-continuous for Lozi maps

Abstract

Recently, Buzzi showed in the compact case that the entropy map $f\rightarrow$ $h_{top}(f)$ is lower semi-continuous for all piecewise affine surface homeomorphisms. We prove that topological entropy for the Lozi maps can jump from zero to a value above 0.1203 as one crosses a particular parameter and hence it is not upper semi-continuous in general. Moreover, our results can be extended to a small neighborhood of this parameter showing the jump in the entropy occurs along a line segment in the parameter space.