On weak mixing of cascades

Abstract

The topological properties of "weak mixing" and "strong mixing" for topological transformation groups have been introduced and investigated and many analogous results have been derived. Let us adopt the following definitions and notations. We will restrict our attention to topological transformation groups (X, T) with compact metrizable phase space X. We call a topological transformation group ergodic if the only closed proper invariant subset is nowhere dense or, equivalently, there exists a point in X the orbit-closure of which is the whole space X. Such a point is called a transitive point. We call a topological transformation group weakly mixing if (X, T)x (X, T) is ergodic and strongly mixing if for any two open sets U and V, there is a compact subset K of T such that t ~ T ~ K implies Ut n V # ;J. Let ~'(X) be the set of all bounded complex-valued functions on X which are continuous on a co-meager set. (We identify functions which coincide on a co-meager set.) Let us define a cascade to be a topological transformation group with the discrete additive group of integers as phase group. In this situation, it is also denoted by (X, ~b), with ~b a homeomorphism of X. If Y is an invariant subset of X, then (Y, ~b) is also a cascade and we say that (Y, ~b) is a subcascade of (X, ~b). Let us consider the following properties of a topological transformation group (X, T):