Mixing endomorphisms on toroidal groups and their countable products

TL;DR

This paper demonstrates that all non-trivial continuous endomorphisms of the circle and certain torus groups are topologically mixing, and extends these results to infinite products of abelian polish semigroups, including countable toroidal groups.

Contribution

It establishes the topological mixing property for a broad class of endomorphisms on toroidal groups and their infinite products, extending previous understanding of dynamical behaviors in these structures.

Findings

All non-trivial continuous endomorphisms of the circle are topologically mixing.

A large class of endomorphisms of n-dimensional tori are topologically mixing.

Countable infinite products of abelian polish semigroups have topologically mixing endomorphisms.

Abstract

We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically mixing. Lastly, we prove that any continuous endomorphism on an abelian polish semigroup (with an identity) can be extended in a natural way to a topologically mixing endomorphism on the countable infinite product of said semigroup. This shows that every countable infinite product of an abelian polish semigroup has a topologically mixing endomorphism and, in particular, the countable infinite toroidal group has infinitely many topologically mixing endomorphisms.