Standard decomposition of expansive ergodically supported dynamics

TL;DR

This paper introduces weak quasigroups in expansive ergodic systems and shows how their properties relate to entropy, ergodic period, and a decomposition of the dynamics into invariant subquasigroups.

Contribution

It establishes a connection between topological entropy, ergodic period, and the existence of almost everywhere defined quasigroup operations in expansive systems.

Findings

Existence of quasigroup operations almost everywhere in certain dynamical systems.

Decomposition of dynamics into invariant weak topological subquasigroups.

Criteria linking entropy and ergodic period to quasigroup structures.

Abstract

In this work we introduce the notion of weak quasigroups, that are quasigroup operations defined almost everywhere on some set. Then we prove that the topological entropy and the ergodic period of an invertible expansive ergodically supported dynamical system $(X,T)$ with the shadowing property establishes a sufficient criterion for the existence of quasigroup operations defined almost everywhere outside of universally null sets and for which $T$ is an automorphism. Furthermore, we find a decomposition of the dynamics of $T$ in terms of $T$-invariant weak topological subquasigroups.