On the dimension group of unimodular $${\mathcal {S}}$$-adic subshifts

Berthé, Valérie; Cecchi, Bernales, Paulina; Durand, Fabien; Leroy, Julien; Perrin, Dominique; Petite, Samuel

doi:10.1007/s00605-020-01488-3

Cited by 13 publications

(17 citation statements)

References 58 publications

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“…Theorem 7.5.1 is from Berthé et al (2020). Corollary 7.5.2 extends a statement initially proved for interval exchanges by Ferenczi and Zamboni (2008).…”

Section: Recognizability and Unimodular S-adic Shiftssupporting

confidence: 68%

Dimension Groups and Dynamical Systems

Durand¹,

Perrin²

2020

Preprint

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The text follows, at least in an initial version, closely the notes of Bernard Host Host (1995), trying to develop more explicitly some arguments. We have also chosen to follow a slightly different presentation, not assuming systematically that the dynamical systems are minimal. We wish to thank Brian Marcus and Mike Boyle for fruitful discussions during the preparation of this text. We are also grateful to Paulina Cecchi, Francesco Dolce, Pavel Heller, Christophe Reutenauer and Revekka Kyriakoglou for reading the manuscript and founding many errors. Special thanks are due to Soren Eilers for reading closely Chapter 10 and to Christian for his careful reading of most chapters.Proposition 2.1.4 A topological dynamical system is minimal if and only if the positive orbit of every point is dense in X. Example 2.5.1 The Fibonacci shift (see Example 2.4.1) is Sturmian (see Exercise 2.32).As one might expect in view of its minimal word complexity, any Sturmian shift is minimal, moreover it is uniquely ergodic.

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“…Theorem 7.5.1 is from Berthé et al (2020). Corollary 7.5.2 extends a statement initially proved for interval exchanges by Ferenczi and Zamboni (2008).…”

Section: Recognizability and Unimodular S-adic Shiftssupporting

confidence: 68%

Dimension Groups and Dynamical Systems

Durand¹,

Perrin²

2020

Preprint

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“…Remark 6.23. That a generic subshift in TT ′ is not balanced may be deduced from Theorem 6.6 together with Proposition 5.4 of [3]. Proposition 6.22 proves something stronger however, namely that a generic subshift in TT ′ is not balanced for any letter.…”

Section: The Space Tt ′ Of Infinite Totally Transitive Systemsmentioning

confidence: 96%

“…One can see this for example using Bratteli diagrams: by [19,Prop. 20], if (X, σ X ) is a subshift associated to a substitution then (X, σ X ) is conjugate to the Vershik map on some stationary Bratteli diagram 3 . In particular, if (X, σ X ) is a subshift defined by a primitive substitution, then there exists an r × r integral matrix A such that G σX is isomorphic to the direct limit of the stationary system…”

Section: 32mentioning

confidence: 99%

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On the structure of generic subshifts

Pavlov¹,

Schmieding²

2022

Preprint

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We investigate generic properties (i.e. properties corresponding to residual sets) in the space of subshifts with the Hausdorff metric. Our results deal with four spaces: the space S of all subshifts, the space S ′ of non-isolated subshifts, the closure T ′ of the infinite transitive subshifts, and the closure TT ′ of the infinite totally transitive subshifts.In the first two settings, we prove that generic subshifts are fairly degenerate; for instance, all points in a generic subshift are biasymptotic to periodic orbits. In contrast, generic subshifts in the latter two spaces possess more interesting dynamical behavior. Notably, generic subshifts in both T ′ and TT ′ are zero entropy, minimal, uniquely ergodic, and have word complexity which realizes any possible subexponential growth rate along a subsequence. In addition, a generic subshift in T ′ is a regular Toeplitz subshift which is strongly orbit equivalent to the universal odometer.

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“…As card(V ) ≥ 2, we see that m ≥ 2. Suppose that σ B 2 [n, ] (w 0 ) = w 1 w 2 • • • w r , where w 1 , w 2 , . .…”

Section: On Topological Rank Of Factors Of Cantor Minimal Systemsmentioning

confidence: 99%

On topological rank of factors of Cantor minimal systems

Golestani

Hosseini²

2021

Ergod. Th. Dynam. Sys.

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A Cantor minimal system is of finite topological rank if it has a Bratteli–Vershik representation whose number of vertices per level is uniformly bounded. We prove that if the topological rank of a minimal dynamical system on a Cantor set is finite, then all its minimal Cantor factors have finite topological rank as well. This gives an affirmative answer to a question posed by Donoso, Durand, Maass, and Petite in full generality. As a consequence, we obtain the dichotomy of Downarowicz and Maass for Cantor factors of finite-rank Cantor minimal systems: they are either odometers or subshifts.

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On the dimension group of unimodular $${\mathcal {S}}$$-adic subshifts

Cited by 13 publications

References 58 publications

Dimension Groups and Dynamical Systems

Dimension Groups and Dynamical Systems

On the structure of generic subshifts

On topological rank of factors of Cantor minimal systems

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