On the connectedness of homomorphisms in topological dynamics

Abstract:

Let (X, T) be a minimal transformation group with compact Hausdorff phase space. We show that if $\phi :(X,T) \to (Y,T)$ is a distal homomorphism and has a structure similar to the structure Furstenberg derived for distal minimal sets, then for T belonging to a class of topological groups T, the homomorphism $X \to X/S(\phi )$ has connected fibers, where $S(\phi )$ is the relativized equicontinuous structure relation. The class T is defined by Sacker and Sell as consisting of all groups T with the property that there is a compact set $K \subseteq T$ such that T is generated by each open neighborhood of K. They show that for such T, a distal minimal set which is a finite-to-one extension of an almost periodic minimal set is itself an almost periodic minimal set. We provide an example that shows that the restriction on T cannot be dropped. As one of the preliminaries to the above we show that given $\phi :(X,T) \to (Y,T)$, the relation $Rc(\phi )$ induced by the components in the fibers relative to $\phi$, i.e., $(x,x’) \in Rc(\phi )$ if and only if x and $x’$ are in the same component of ${\phi ^{ - 1}}(\phi (x))$, is a closed invariant equivalence relation. We also consider the question of when a minimal set (X, T) is such that $Q(x)$ is finite for some x in X, where Q is the regionally proximal relation. This problem was motivated by Veech’s work on almost automorphic minimal sets, i.e., the case in which $Q(x)$ is a singleton for some x in X.