Disjointness between Bounded
Rank-One Transformations
Su Gao and
Aaron Hill
Abstract
In
this paper some sufficient conditions are given for when two bounded rank-one
transformations are non-isomorphic and when they are disjoint. We also obtain sufficient
conditions for a bounded rank-one transformation to have minimal self-joinings. For commensurate, canonically bounded rank-one
transformations, isomorphism and disjointness are
completely determined by simple conditions in terms of their cutting and spacer
parameters.
Table of Contents
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1. Introduction 2. Preliminaries 2.1 Finite sequences, finite functions, and finite words 2.2 Infinite and bi-infinite sequences 2.3 Symbolic rank-one systems and rank-one transformations 2.4 Combinatorics of rank-one words 3. Non-Isomorphism and Disjointness 4. Minimal Self-Joinings 5. Applications to Canonically Bounded Transformations 5.1 Canonical generating sequences 5.2 Replacement schemes and topological conjugacy 5.3 Isomorphism and disjointness of canonically bounded transformations 5.4 A case of Ryzhikov's theorem 6. Concluding remarks Acknowledgments References |