Mixing endomorphisms on toroidal groups and their countable products

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.