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 ϕ : ( X , T ) → ( Y , T ) \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 → X / S ( ϕ ) X \to X/S(\phi ) has connected fibers, where S ( ϕ ) 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 ⊆ T 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 ϕ : ( X , T ) → ( Y , T ) \phi :(X,T) \to (Y,T) , the relation R c ( ϕ ) Rc(\phi ) induced by the components in the fibers relative to ϕ \phi , i.e., ( x , x ′ ) ∈ R c ( ϕ ) (x,x’) \in Rc(\phi ) if and only if x and x ′ x’ are in the same component of ϕ − 1 ( ϕ ( x ) ) {\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 ) 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 ) Q(x) is a singleton for some x in X.