Complexity and directional entropy in two dimensions

TL;DR

This paper investigates the directional entropy of two-dimensional dynamical systems, revealing that under certain local complexity conditions, all directions have zero entropy or some are periodic, especially in nonexpansive directions.

Contribution

It establishes a dichotomy for directional entropy in $ ext{Z}^2$ systems based on local complexity assumptions, linking entropy to periodicity and nonexpansiveness.

Findings

All nonexpansive directions have zero directional entropy.

Under local complexity conditions, directions are either zero entropy or periodic.

The results connect local complexity with global dynamical properties.

Abstract

We study the directional entropy of the dynamical system associated to a $\Z^2$ configuration in a finite alphabet. We show that under local assumptions on the complexity, either every direction has zero topological entropy or some direction is periodic. In particular, we show that all nonexpansive directions in a $\Z^2$ system with the same local assumptions have zero directional entropy.