Rank-one actions, their (C,F)-models and constructions with bounded\n parameters

2016/10/31 by Alexandre I. Danilenko, Danilenko, Alexandre I.
Computer Science · Mathematics · #37A #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1610.09851

openalex publication_date 2016/10/31 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let G be a discrete countable infinite group. We show that each topological\n(C,F)-action T of G on a locally compact non-compact Cantor set is a free\nminimal amenable action admitting a unique up to scaling non-zero invariant\nRadon measure (answer to a question by Kellerhals, Monod and R ordam). We\nfind necessary and sufficient conditions under which two such actions are\ntopologically conjugate in terms of the underlying (C,F)-parameters. If G\nis linearly ordered Abelian then the topological centralizer of T is trivial.\nIf G is monotileable and amenable, denote by cal A_G the set of all\nprobability preserving actions of G on the unit interval with Lebesgue\nmeasure and endow it with the natural topology. We show that the set of\n(C,F)-parameters of all (C,F)-actions of G furnished with a suitable\ntopology is a model for cal A_G in the sense of Forman, Rudolph and Weiss.\nIf T is a rank-one transformation with bounded sequences of cuts and spacer\nmaps then we found simple necessary and sufficient conditions on the related\n(C,F)-parameters under which (i) T is rigid, (ii) T is totally ergodic.\nIt is found an alternative proof of Ryzhikov's theorem that if T is totally\nergodic and non-rigid rank-one map with bounded parameters then T has MSJ. We\nalso give a more general version of the criterium (by Gao and Hill) for\nisomorphism and disjointness of two commensurate non-rigid totally ergodic\nrank-one maps with bounded parameters. It is shown that the rank-one\ntransformations with bounded parameters and no spacers over the last subtowers\nis a proper subclass of the rank-one transformations with bounded parameters.\n

Related

• Classification of rank-one actions via the cutting-and-stacking parameters
• Rank-one nonsingular actions of countable groups and their odometer factors
• Invariant ergodic measures and the classification of crossed product C^∗-algebras
• Amenable purely infinite actions on the non-compact Cantor set
• Finite ergodic index and asymmetry for infinite measure preserving\n actions