Distributions of periodic points for the Dyck shift and the heterochaos baker maps

TL;DR

This paper explores the distribution of periodic points in the Dyck shift and heterochaos baker maps, revealing how ergodic measures of maximal entropy relate to these points' asymptotic distributions.

Contribution

It establishes a connection between ergodic measures of maximal entropy and the asymptotic distribution of periodic points in both the Dyck shift and heterochaos baker maps.

Findings

Ergodic MMEs of the Dyck shift are represented by asymptotic distributions of periodic points with different multipliers.

Ergodic MMEs of heterochaos baker maps correspond to distributions of periodic points with varying unstable dimensions.

Abstract

The heterochaos baker maps are piecewise affine maps on the square or the cube that are one of the simplest partially hyperbolic systems. The Dyck shift is a well-known example of a subshift that has two fully supported ergodic measures of maximal entropy (MMEs). We show that the two ergodic MMEs of the Dyck shift are represented as asymptotic distributions of sets of periodic points of different multipliers. We transfer this result to the heterochaos baker maps, and show that their two ergodic MMEs are represented as asymptotic distributions of sets of periodic points of different unstable dimensions.