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 → X of a Hausdorff topological (semi)group X. We show that if X is a Hausdorff topological group and T: X → 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 ∈ X such that T: X → 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 Rk. © Birkhäuser Verlag, Basel, 2009.