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

Abstract

Let G be a discrete countable infinite group. We show that each topological (C, F)-action T of G on a locally compact non-compact Cantor set is a free minimal amenable action admitting a unique up to scaling non-zero invariant Radon measure (answer to a question by Kellerhals, Monod and Rørdam). We find necessary and sufficient conditions under which two such actions are topologically conjugate in terms of the underlying (C, F)-parameters. If G is linearly ordered abelian, then the topological centralizer of T is trivial. If G is monotileable and amenable, denote by $${{\cal A}_G}$$ the set of all probability preserving actions of G on the unit interval with Lebesgue measure and endow it with the natural topology. We show that the set of (C, F)-parameters of all (C, F)-actions of G furnished with a suitable topology is a model for $${{\cal A}_G}$$ in the sense of Foreman, Rudolph and Weiss. If T is a rank-one transformation with bounded sequences of cuts and spacer maps, then we found simple necessary and sufficient conditions on the related (C, F)-parameters under which (i) T is rigid, (ii) T is totally ergodic. An alternative proof is found of Ryzhikov’s theorem that if T is totally ergodic and a non-rigid rank-one map with bounded parameters, then T has MSJ. We also give a more general version of the criterion (by Gao and Hill) for isomorphism and disjointness of two commensurate non-rigid totally ergodic rank-one maps with bounded parameters. It is shown that the rank-one transformations with bounded parameters and no spacers over the last subtowers is a proper subclass of the rank-one transformations with bounded parameters.