Uniformly distributed periodic orbits of endomorphisms on $n$-tori

TL;DR

This paper proves that ergodic endomorphisms on n-tori have sequences of periodic orbits that are uniformly distributed, and characterizes ergodicity via approximation of Haar measure by periodic measures.

Contribution

It establishes a link between ergodicity and uniform distribution of periodic orbits for endomorphisms on n-tori, providing a new characterization of ergodic systems.

Findings

Existence of uniformly distributed periodic orbits in ergodic endomorphisms

Haar measure can be approximated by periodic measures if and only if the system is ergodic

Characterization of ergodicity through periodic orbit distribution

Abstract

We prove that any ergodic endomorphism on torus admits a sequence of periodic orbits uniformly distributed in the metric sense. As a corollary, an endomorphism on torus is ergodic if and only if the Haar measure can be approximated by periodic measures.