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 R k . (2000). 54H20. Mathematics Subject Classification

Point transitivity, -transitivity and multi-minimality

“…It is shown in [14] that frequently hypercyclic operators are weakly mixing. Later, several authors show that {positive upper Banach density sets}-point transitive operators are weakly mixing (see [25] or [28]). The authors in [4] studied in detail how fast the integers of the sets N (x, U ) could increase to ensure that the operator is weakly mixing.…”

Point transitivity, $Δ$-transitivity and multi-minimality

“…It is shown in [14] that frequently hypercyclic operators are weakly mixing. Latter, several authors show that {positive upper Banach density sets}point transitive operators are weakly mixing (see [25] or [28]). The authors in [4] studied in detail how fast the integers of the sets N(x,U ) could increase to ensure that the operator is weakly mixing.…”