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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the connectedness of homomorphisms in topological dynamics

Authors: D. McMahon and T. S. Wu
Journal: Trans. Amer. Math. Soc. 217 (1976), 257-270
MSC: Primary 54H20
MathSciNet review: 0413067
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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.

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Keywords: Minimal, transformation group homomorphism, relativized equicontinuous structure relation, connected
Article copyright: © Copyright 1976 American Mathematical Society