Topologically mixing extensions of endomorphisms on Polish groups

Abstract

In this paper we study the dynamics of continuous endomorphisms on Polish groups. We offer necessary and sufficient conditions for a continuous endomorphism on a Polish group to be weakly mixing. We prove that any continuous endomorphism of an abelian Polish group can be extended in a natural way to a topologically mixing endomorphism on the countable infinite product of said group.

Keywords

• weak mixing
• polish group
• hypercyclicity criterion.