Bounded rank-1 transformations

Topological mixing properties of rank‐one subshifts

“…This mirrors the results of [15] in giving an explicit characterization of weak mixing in terms of the spacer parameters. In general, topological weak mixing neither implies, nor is implied by, weak mixing in the measure-theoretic sense; but in the case of canonically bounded rank-one subshifts, it turns out that they are topologically weakly mixing exactly when they are weakly mixing as a rank-one transformation.…”

“…Other papers that looked at mixing properties of rank-one transformations include [1,3,5,16]. Of particular interest are the papers which attempted to classify mixing properties; the first author and Hill [15] classified when a rank-one transformation is weakly mixing in the canonically bounded case, and Creutz and Silva [7,8] classified when a rank-one transformation is mixing based on the ergodicity of the sequence of spacer parameters.…”

Partially bounded transformations have trivial centralizers

“…We instead work with symbolic properties of the transformation as started in del Junco [6]. In fact we extend to infinite measure the methods of Gao and Hill [11] (see also ), who showed that canonically bounded finite measure-preserving rank-one transformations have trivial centralizers.…”

Disjointness between bounded rank-one transformations

“…In Section 6 we apply our results about non-isomorphism and disjointness to give simple algorithms to determine isomorphism and disjointness for canonically bounded rank-one transformations that are commensurate (Corollary 5.5). The notion of canonically bounded rank-one transformations was defined in [6] and was used in [7] to characterize non-rigidity for bounded rank-one transformations. Our results on isomorphism and disjointness for canonically bounded rank-one transformations extend what was already known for a class of Chacon-like transformations.…”

“…Section 6 contains a proof of a case of Ryzhikov's theorem that a bounded rank-one transformation has minimal self-joinings if and only if it is non-rigid and totally ergodic; our proof applies to strictly bounded rank-one transformations (those for which no spacers are ever inserted at the last opportunity). We include, using characterizations of non-rigidity and total ergodicity for strictly bounded rank-one transformations (stated in [7]), a simple algorithm for determining whether a strictly bounded rankone transformation has minimal self-joinings of all orders.…”