Weak mixing and mixing of a single transformation of a topological (semi)group

Abstract

We investigate some aspects of the iterative dynamics of a single continuous homomorphism \(T : X {\rightarrow} X\) of a Hausdorff topological (semi)group X. We show that if X is a Hausdorff topological group and \(T : X {\rightarrow} X\) is a continuous homomorphism such that either T is syndetically transitive, or T is non-wandering with a dense set of points having orbits converging to the identity element, then T is topologically weak mixing. We also show that for some familiar topological (semi)groups X, there is an (invertible) element \(a \in X\) such that \(T : X \rightarrow X\) given by \(T(x) = axa^{-1}\) is topologically mixing. As a corollary we get a zero-one law for generic dynamics on certain spaces such as the Cantor space, the Hilbert cube and \({\mathbb{R}}^k\).