Topological mixing properties of rank‐one subshifts

Gao, Su; Ziegler, Caleb

doi:10.1112/tlm3.12016

Cited by 9 publications

(12 citation statements)

References 15 publications

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“…Mixing properties of rank-one transformations were also studied in, e.g., [1], [3], [4], [6], [7], and [13]. In [12] we also completely characterized the maximal equicontinuous factors of rank-one subshifts and showed that they can only be finite.…”

Section: Introduction and Definitionsmentioning

confidence: 91%

“…This led to the definition of rank-one subshifts, which was first studied by the first author and Hill in [11], where they gave a characterization for the topological isomorphism relation of rank-one subshifts based on the cutting and spacer parameters. In [12] the current authors studied the topological mixing properties of rank-one subshifts. Because the concept of rank-one subshifts came from rank-one transformations, the study of their topological dynamical properties is often motivated by their ergodic-theoretic counterparts which tend to have a long history.…”

Section: Introduction and Definitionsmentioning

confidence: 99%

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Topological factors of rank-one subshifts

Gao

Ziegler²

2020

Proc. Amer. Math. Soc. Ser. B

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Section: Introduction and Definitionsmentioning

confidence: 91%

Section: Introduction and Definitionsmentioning

confidence: 99%

Topological factors of rank-one subshifts

Gao

Ziegler²

2020

Proc. Amer. Math. Soc. Ser. B

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“…In this section, we prove that (X M , σ) is not rank-one. Gao and Ziegler in [12] have shown that an infinite odometer is not a factor of a rank-one shift. Gao suggested to the authors that one way to show that the Thue-Morse system is not rank-one is to show that it has an infinite odometer factor.…”

Section: T T -M R -Omentioning

confidence: 99%

“…Using a similar definition to expected occurrences defined in [11], [12] and [13], given an infinite or bi-infinite word that is built from a finite word v, if there is a unique way to decompose the word to a collection of disjoint occurrences of v's separated by 1's, then we can call an occurrence of v an expected occurrence if it is an element of such collection. Specifically, given any any infinite word V , if there exists a unique decomposition of V to the form…”

Section: T T -M R -Omentioning

confidence: 99%

On finite symbolic rank words and subshifts

Leng¹,

Silva²,

Wu³

2020

Preprint

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A. We study rank-two symbolic systems (as topological dynamical systems) and prove that the Thue-Morse sequence and quadratic Sturmian sequences are rank-two and define rank-two symbolic systems. IRank-one measure-preserving transformations have played an important role in ergodic theory since the pioneering work of Chacón [4]. In this paper, rather than considering the notion of rankone in measurable dynamics we are interested in studying rank-one and higher rank systems strictly from the point of view of topological dynamics, in particular as symbolic shifts. Of course, symbolic systems have been used extensively in ergodic theory. In [13], Kalikow discusses a symbolic model for rank-one measure-preserving transformations, and Ferenczi [8] in his survey on rank-one finite measure-preserving transformations mentions the symbolic definition for rank-one transformations. Later in [3], Bourgain used a class of symbolic rank-one transformations for which he proved the Moebius disjointness law, and rank-one symbolic shifts are also considered in [1,5,7]. It was in [11] that Gao and Hill started a systematic study of (non-degenerate) rank-one shifts as topological dynamical systems, and proved several properties for rank-one symbolic shifts. In this paper we study higher rank systems and prove that the system defined by the Thue-Morse sequence and systems defined by quadratic Sturmian sequences are rank-two.The terminology "rank-one" comes from "rank-one" cutting and stacking systems [4]. As shown by Kalikow [13], one can encode cutting and stacking systems as a shift on a symbolic system and he shows that the two systems are measurably isomorphic when the symbolic sequence is non-periodic. Gao and Hill introduced the notion of (symbolic) rank-one shifts as an extension to earlier ideas and developed various results associated to symbolic rank-one systems. We start by first defining rank-one words using the definition provided by Gao and Hill in [11].Definition 1.1. Let F be the set consisting of all finite words in the alphabet {0, 1} that begin and end with 0. Let V ∈ {0, 1} N . We say that V is built from v ∈ F if there exists a sequence {a i } i≥1 of natural numbers such that V = v1 a 1 v1 a 2 v • • • . LetWe say that the infinite word V is rank-one if A V is infinite.

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“…As irrational rotations are totally ergodic, the natural question is whether weak mixing imposes any sort of stronger lower bound on word complexity. Topological mixing properties were considered by Gao and Ziegler [GZ19] (see also Gao and Hill [GH16a], [GH16b]); here we address the measure-theoretic question. The corresponding question regarding measure-theoretically strongly mixing subshifts is also of interest, the reader is referred to [CPR22] for recent progress.…”

Section: Introductionmentioning

confidence: 99%

Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One Subshifts

Creutz¹

2022

Preprint

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We exhibit subshifts admitting weakly mixing (probability) measures, for arbitrary ǫ > 0, with word complexity p satisfying lim sup p(q) /q < 1.5 + ǫ. For arbitrary f (q) → ∞, said subshifts can be made to satisfy p(q) < q + f (q) infinitely often.We establish that every subshift associated to a totally ergodic rank-one transformation (on a probability space) satisfies lim sup p(q) − 1.5q = ∞ and that this is optimal for rank-ones.

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Topological mixing properties of rank‐one subshifts

Cited by 9 publications

References 15 publications

Topological factors of rank-one subshifts

Topological factors of rank-one subshifts

On finite symbolic rank words and subshifts

Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One Subshifts

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