Growth-type invariants for ℤ d subshifts of finite type and arithmetical classes of real numbers

Meyerovitch, Tom

doi:10.1007/s00222-010-0296-1

Cited by 21 publications

(32 citation statements)

References 16 publications

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“…(10) By Lemma 3.2, η ϕ,ξ is periodic and its period vector lies in F . Combining equations (5), (7), and (8) we see this vector is not an element of the set F \ V x1 .…”

Section: Shifts Of Subquadratic Growthmentioning

confidence: 92%

The automorphism group of a shift of subquadratic growth

Cyr

Kra

2015

Proc. Amer. Math. Soc.

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Abstract. For a subshift over a finite alphabet, a measure of the complexity of the system is obtained by counting the number of nonempty cylinder sets of length n. When this complexity grows exponentially, the automorphism group has been shown to be large for various classes of subshifts. In contrast, we show that subquadratic growth of the complexity implies that for a topologically transitive shift X, the automorphism group Aut(X) is small: if H is the subgroup of Aut(X) generated by the shift, then Aut(X)/H is periodic.

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“…(10) By Lemma 3.2, η ϕ,ξ is periodic and its period vector lies in F . Combining equations (5), (7), and (8) we see this vector is not an element of the set F \ V x1 .…”

Section: Shifts Of Subquadratic Growthmentioning

confidence: 92%

The automorphism group of a shift of subquadratic growth

Cyr

Kra

2015

Proc. Amer. Math. Soc.

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“…The main problem is to prove a reverse statement: given a measure satisfying the computational obstructions, we want to construct a cellular automaton which, starting from any simple initial measure, reaches this measure asymptotically. Similar computational obstructions appear in topological dynamics, when characterizing possible properties of subshifts of finite type or cellular automata: possible entropies [29], possible growth-type invariants [38], possible sub-actions [1,28]... However, the construction is quite different here, since starting from a random configuration, the construction requires an ability to self-organize the space.…”

Section: Limit Measures Of Cellular Automata: a Computational Approachmentioning

confidence: 88%

Probability and algorithmics: a focus on some recent developments

Cénac¹,

Marcovici²,

Rovetta³

et al. 2017

ESAIM: Procs

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Abstract.This article presents different recent theoretical results illustrating the interactions between probability and algorithmics. These contributions deal with various topics: cellular automata and calculability, variable length Markov chains and persistent random walks, perfect sampling via coupling from the past. All of them involve discrete dynamics on complex random structures.Résumé. Cet article présente différents résultats récents de nature théorique illustrant les interactions entre probabilités et algorithmique. Ces contributions traitent de sujets variés : automates cellulaires et calculabilité, chaînes de Markov à mémoire variable et marches aléatoires persistantes, échantillonage parfait par la méthode de couplage par le passé. Leur point commun est de faire intervenir des dynamiques discrètes sur des structures aléatoires complexes.

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“…Ferenczi and Park [11] investigated a new entropy-like invariant for the action of Z or Z d on a probability space. Meyerovitch [16] considered the entropy dimension for Z d -subshifts of finite type.…”

Section: Introductionmentioning

confidence: 99%

Topological entropy dimension for noncompact sets

Kuang

2012

Dynamical Systems

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Similar with the fractal dimension, we introduce the concept of topological entropy dimension to classify the sets with entropy zero. We prove that the entropy dimension of the space in this article is not greater than that defined by De Carvalho, where he introduced the entropy dimension for the system, and give some examples indicating that such inequality is optimal. Some basic propositions of entropy dimension are discussed and it turns out that the entropy dimension is invariant under conjugacy. The property of the countable stability and a power rule for the entropy dimension of any set are obtained. It is shown that any set shares the same entropy dimension with its image set.

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Growth-type invariants for ℤ d subshifts of finite type and arithmetical classes of real numbers

Cited by 21 publications

References 16 publications

The automorphism group of a shift of subquadratic growth

The automorphism group of a shift of subquadratic growth

Probability and algorithmics: a focus on some recent developments

Topological entropy dimension for noncompact sets

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