Heterochaos baker maps and the Dyck system: maximal entropy measures and a mechanism for the breakdown of entropy approachability

TL;DR

This paper introduces generalized heterochaos baker maps in higher dimensions, links their coding spaces to the Dyck system, and analyzes their invariant measures, revealing two maximal entropy measures and a breakdown mechanism for entropy approachability.

Contribution

It extends heterochaos baker maps to higher dimensions, connects their coding to the Dyck system, and studies their invariant measures and entropy properties.

Findings

Existence of two ergodic measures of maximal entropy.

Natural coding spaces coincide with the Dyck system.

Mechanism for the breakdown of entropy approachability.

Abstract

We introduce two parametrized families of piecewise affine maps on $[0,1]^2$ and $[0,1]^3$, as generalizations of the heterochaos baker maps which were introduced and investigated in [Y. Saiki, H. Takahasi, J. A. Yorke, Nonlinearity, 34 (2021), 5744--5761] as minimal models of the unstable dimension variability in multidimensional dynamical systems. We show that natural coding spaces of these maps coincide with the Dyck system that has come from the theory of languages. Based on this coincidence, we start to develop a complementary analysis on their invariant measures. As a first attempt, we show the existence of two ergodic measures of maximal entropy for the generalized heterochaos baker maps. We also clarify a mechanism for the breakdown of entropy approachability.