Random product of substitutions with the same incidence matrix

Arnoux, Pierre; Mizutani, Masahiro; Sellami, Tarek

doi:10.1016/j.tcs.2014.06.002

Cited by 6 publications

(3 citation statements)

References 9 publications

(8 reference statements)

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“…Theorem 7.4.12 is (Berthé et al, 2019b, Theorem 3.1). Proposition 7.4.3 is from Arnoux et al (2014).…”

Section: Recognizability and Unimodular S-adic Shiftsmentioning

confidence: 98%

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 7.4.12 is (Berthé et al, 2019b, Theorem 3.1). Proposition 7.4.3 is from Arnoux et al (2014).…”

Section: Recognizability and Unimodular S-adic Shiftsmentioning

confidence: 98%

Dimension Groups and Dynamical Systems

Durand¹,

Perrin²

2020

Preprint

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“…Following [AMS14], we say that an infinite word ω ∈ A N is a limit word of σ = (σ n ) n∈N if there is a sequence of infinite words (ω (n) ) n∈N with…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, although there exist results for the generation of discrete hyperplanes in connection with continued fraction algorithms [IO93, IO94, ABI02, BBJS13, BBJS15], more information on convergence and renormalization properties is needed in order to deduce spectral properties. In [AMS14], S-adic sequences are considered where the substitutions all have the same Pisot irreducible unimodular matrix; in our case, the matrices are allowed to be different at each step.…”

Section: Introductionmentioning

confidence: 99%

Geometry, dynamics, and arithmetic ofS-adic shifts

Berthé¹,

Steiner²,

Thuswaldner³

2019

Annales De l'Institut Fourier

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This paper studies geometric and spectral properties of S-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions. Pure discrete spectrum for S-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to the S-adic framework. They are applied to families of S-adic shifts generated by Arnoux-Rauzy as well as Brun substitutions. It is shown that almost all of these shifts have pure discrete spectrum. Using S-adic words related to Brun's continued fraction algorithm, we exhibit bounded remainder sets and natural codings for almost all translations on the two-dimensional torus. Due to the lack of self-similarity properties present for substitutive systems we have to develop new proofs to obtain our results in the S-adic setting.

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S-adic Sequences: A Bridge Between Dynamics, Arithmetic, and Geometry

Thuswaldner

2020

Lecture Notes in Mathematics

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Random product of substitutions with the same incidence matrix

Cited by 6 publications

References 9 publications

Dimension Groups and Dynamical Systems

Dimension Groups and Dynamical Systems

Geometry, dynamics, and arithmetic ofS-adic shifts

S-adic Sequences: A Bridge Between Dynamics, Arithmetic, and Geometry

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