Directional uniformities, periodic points, and entropy

TL;DR

This paper explores the complex behavior of invariants like entropy and periodic points in multi-dimensional dynamical systems, emphasizing directional uniformities and their connections to number theory, including a new relationship between average entropy and periodic point growth.

Contribution

It introduces a new relationship between average entropy and periodic point growth, and discusses uniformity of dynamical invariants in algebraic $ ext{Z}^d$-actions, connecting to number theory.

Findings

Directional entropy can be defined even when natural entropy vanishes.

Uniformities in growth properties relate to number theory problems.

A new relationship between average entropy and periodic point growth is established.

Abstract

Dynamical systems generated by $d\ge2$ commuting homeomorphisms (topological $\mathbb{Z}^d$-actions) contain within them structures on many scales, and in particular contain many actions of $\mathbb{Z}^k$ for $1\le k\le d$. Familiar dynamical invariants for homeomorphisms, like entropy and periodic point data, become more complex and permit multiple definitions. We briefly survey some of these and other related invariants in the setting of algebraic $\mathbb{Z}^d$-actions, showing how, even in settings where the natural entropy as a $\mathbb{Z}^d$-action vanishes, a powerful theory of directional entropy and periodic points can be built. An underlying theme is uniformity in dynamical invariants as the direction changes, and the connection between this theory and problems in number theory; we explore this for several invariants. We also highlight Fried's notion of average entropy and its connection to uniformities in growth properties, and prove a new relationship between this entropy and periodic point growth in this setting.