Compact groups and fixed point sets

Abstract

Abstract. Some structure theorems for compact abelian groups are derived and used to show that every closed subset of an infinite compact metrizable group is the fixed point set of an autohomeomorphism. It is also shown that any metrizable product containing a positive-dimensional compact group as a factor has the property that every closed subset is the fixed point set of an autohomeomorphism.

On uniform flow

“…Definition 2.1. [2] A continuous right action ϕ of an usual real line R on a metric space (X, d) is said to be uniform flow if the following conditions hold :…”

“…In case f can be found to be a homeomorphism, we say that the space enjoys the complete invariance property with respect to a homeomorphism (CIPH) [3]. Spaces possessing CIP, preservation of CIP under topological notions and various techniques to determine CIP and CIPH over topological spaces can be found in [2,3,4,5,6,7,8,9,10].…”

“…In 1997, the notion of uniform flow is introduced by A. Chigogidge, K. H. Hofman and J. R. Martin in their paper entitled "Compact groups and fixed point sets" [2] and Proved that a compact metric space with a uniform flow has the CIPH and the product of two metric spaces has the CIPH if one of them is compact and has uniform flow.…”

Structure of Finite-Dimensional Protori

“…The Structure Theorem for Protori is formulated for an arbitrary protorus by applying the key new Lemma 5, intrinsically engineered for protori, to the Resolution Theorem for Compact Abelian Groups (Proposition 2.2, [1]), which states that a compact abelian group H is topologically isomorphic to r∆ ˆLpHqs{Γ for a totally disconnected Γ and a profinite subgroup ∆ of H such that H{∆ is a torus, where LpHq is the Lie algebra of H (Proposition 7.36, [2]). An immediate consequence of the Structure Theorem for Protori is the existence of a universal resolution for a protorus G (Corollary 6): r p ∆ X 8 ˆLpGqs{X 8 , where p ∆ X 8 is a periodic (Definition 1.13, [3]), locally compact topological divisible hull of a profinite ∆ of a given resolution of G, X 8 is a minimal quotient-divisible extension of an intervening Pontryagin dual of G, and the concept of minimal divisible locally compact cover of G is introduced (Corollary 7) and realized via p ∆ X 8 ˆLpGq.…”

“…LCA topology by Proposition 4. The morphism exp G is continuous and injective (Corollary 8.47,[2]) with exp G LpGq dense in G. Thus, Y ˚is dense in G ô Y ˚is dense in exp G LpGq ô exp ´1 G pY ˚q is dense in LpGq because the map ϕ : ∆ ˚ˆLpGq Ñ G given by ϕpα, rq " α `exp G r is a local isometry (Proposition 2.14,[1]). However, exp ´1 G pY ˚q is dense in LpGq ô Y " θ ˚.…”