Introduction

Topological dynamics studies the action of a group G acting as homeomorphisms of a topological space X, usually taken to be compact. In ergodic theory, a measure structure (B, μ) is added and one studies properties modulo μ-null sets. In generic dynamics we study properties modulo the topologically negligible sets — the sets of first category or meager sets. Compactness is not such a property, so the natural spaces for our investigations will be Polish spaces — the complete separable metric spaces. Continuous actions of G on such Polish spaces always have compactifications, so we could stay in the class of compact spaces — but this restricts the framework unnecessarily.

Our survey will deal mainly with three topics which we introduce now by describing their analogues in ergodic theory and topological dynamics. The first concerns ergodicity, weak mixing and the basic results of P. Haimos and J. von Neumann that characterize weak mixing as the absence of pure point spectrum, or as the ergodicity of cartesian products. As we will see, there is a very nice reduction of general systems to ergodic ones in generic dynamics and there are also interesting characterizations of weak mixing in the sense of multipliers.