1989/12/01 by Bruce Kitchens, Klaus Schmidt · 2 citations
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Cellular Automata and Applications #semigroups and automata theory #Mathematics #Abelian group #Automorphism #Metrization theorem #Locally compact space #Chain (unit) #Totally disconnected space #Pure mathematics #Group (periodic table) #Automorphisms of the symmetric and alternating groups #Combinatorics #Nilpotent #Markov chain #Invariant (physics) #Mathematical analysis #Separable space
paper · pdf · doi:10.1017/s0143385700005290
openalex publication_date 1989/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
Abstract We study finitely generated, abelian groups Γ of continuous automorphisms of a compact, metrizable group X and introduce the descending chain condition for such pairs ( X , Γ). If Γ acts expansively on X then ( X , Γ) satisfies the descending chain condition, and ( X , Γ) satisfies the descending chain condition if and only if it is algebraically and topologically isomorphic to a closed, shift-invariant subgroup of G Γ , where G is a compact Lie group. Furthermore every such subgroup of G Γ is a (higher dimensional) Markov shift whose alphabet is a compact Lie group. By using the descending chain condition we prove, for example, that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X . Furthermore, if X is a compact group and ( X , Γ) satisfies the descending chain condition, then every ergodic element of Γ has a dense set of periodic points. Finally we give an algebraic description of pairs ( X , Γ) satisfying the descending chain condition under the assumption that X is abelian.