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

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## 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.

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