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

作者：Moothathu T K Subrahmonian^*

来源：Aequationes Mathematicae, 2009, 78(1-2): 147-155.

DOI：10.1007/s00010-009-2958-x

• Topological transitivity
• weak mixing
• mixing
• syndetic set
• topological group
• topological semigroup
• generic dynamics

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 is an element of 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.

• 出版日期2009-10