A directional uniformity of periodic point distribution and mixing

TL;DR

This paper studies how periodic points and mixing behavior distribute uniformly in all directions for certain algebraic dynamical systems, showing that the convergence rate is uniform across directions when automorphisms have finite entropy.

Contribution

It proves that for mixing -actions generated by commuting automorphisms with finite entropy, the rate of periodic point distribution and mixing is uniformly consistent across all directions.

Findings

Directional mixing occurs at a uniform rate regardless of direction.

Periodic point measures converge uniformly to Haar measure.

Finite entropy automorphisms ensure uniformity in distribution and mixing rates.

Abstract

For mixing~$\mathbb Z^d$-actions generated by commuting automorphisms of a compact abelian group, we investigate the directional uniformity of the rate of periodic point distribution and mixing. When each of these automorphisms has finite entropy, it is shown that directional mixing and directional convergence of the uniform measure supported on periodic points to Haar measure occurs at a uniform rate independent of the direction.