Measurable Distal and Topological Distal Systems

In this paper we prove that any ergodic measurably distal system can be realized as a minimal topologically distal system with an invariant Borel measure of full support. The proof depends upon a theorem stating that every measurable function from a measurable system with its base space being a compact metric space to a connected compact group is cohomologous to a continuous function. A topological distal dynamical system is a dynamical system (X; T) (where X is a compact metric space and T: X ! X a continuous invertible transformation) such that for every a 6 = b 2 X, inf i2Z d(T i a; T i b) > 0: A minimal dynamical system is a dynamical system with no closed invariant proper subsets. One of the basic facts about distal systems is that they decompose into the disjoint union of minimal systems. In 3], H. Furstenberg proved a structure theorem for minimal topologically distal systems. If a dynamical system is topologically distal then any isometric topological extension of it is topologically distal, where an isometric topological extension is an extension of the original system (in the category of topological dynamic systems) such that each ber is isometric to a homogeneous compact metric space (see the next section for more detailed deenitions). H. Furstenberg proved that every minimal topologically distal system is the transsnite inverse limit of a tower of isometric topological extensions. An analogous notion in the category of measurable systems is the notion of measurably distal. It was introduced by W. Parry in 6], and played a major role in H. Furstenbergs proof of Szemeredi's theoremm4]. The deenition of measurably distal in 4] is particularly suitable to our needs. A measurable system is measurably distal if it is the transsnite inverse limit of a tower of isometric measurable extensions. What we wish to prove is that every ergodic measurably distal system is isomorphic (as a measurable system) to a minimal topologically distal system equipped with a Borel measure with full support, and it is clear that the natural way to prove this is to consider rst the connection between isometric measurable extensions and isometric topological extensions.