Monotonicity of the Lozi Family and the Zero Entropy Locus

TL;DR

This paper investigates the monotonicity of the entropy in the Lozi family of maps near specific parameter values and identifies parameters where the system exhibits zero entropy, combining rigorous and numerical methods.

Contribution

It extends previous results by showing entropy monotonicity in new parameter directions and provides rigorous and numerical analysis of zero entropy parameters.

Findings

Entropy is monotonic in the vertical direction near a=2.

Monotonicity extends to some other directions for 1<a≤2.

Identifies parameters with zero entropy using rigorous and numerical methods.

Abstract

Ishii and Sands showed the monotonicity of the Lozi family $\mathcal L_{a,b}$ in a $\mathcal C^1$ neighborhood of a-axis in the $a$-$b$ parameter space. We show the monotonicity of the entropy in the vertical direction around $a=2$ and in some other directions for $1<a\leq2$. Also we give some rigorous and numerical results for the parameters at which the Lozi family has zero entropy.