Bounded density shifts

Stanley, Brett

doi:10.1017/etds.2013.38

Cited by 11 publications

(17 citation statements)

References 7 publications

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“…Bounded density shifts. We first recall the definition of the bounded density subshifts of [25]. Definition 8.7.…”

Section: R Pavlovmentioning

confidence: 99%

“…Our other application is to the bounded density shifts (denoted by X k,h ) from [25], as well as a natural generalization, which we call signed bounded density shifts (denoted by X ± k,h ), which are defined in terms of parameters n ∈ N and increasing subadditive h : N → N. These shifts have corresponding sets F k,h (and F ± k,h ) of forbidden words and sets G k,h (and G ± k,h ) of words which do not begin or end with more than one-third of a forbidden word (see §8.7 for details). (In the following, e is Euler's number 2.718 .…”

Section: Introductionmentioning

confidence: 99%

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On subshifts with slow forbidden word growth

Pavlov

2021

Ergod. Th. Dynam. Sys.

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In this work, we treat subshifts, defined in terms of an alphabet $\mathcal {A}$ and (usually infinite) forbidden list $\mathcal {F}$ , where the number of n-letter words in $\mathcal {F}$ has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633–647] and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a more general result, which states that bounded supermultiplicativity and a sort of measure-theoretic specification property together imply uniqueness of the measure of maximum entropy and our Gibbs bounds. We also show that some well-known classes of subshifts can be treated by our results, including the symbolic codings of $x \mapsto \alpha + \beta x$ (the so-called $\alpha $ - $\beta $ shifts of Hofbauer [Maximal measures for simple piecewise monotonic transformations. Z. Wahrsch. verw. Geb.52(3) (1980), 289–300]) and the bounded density subshifts of Stanley [Bounded density shifts. Ergod. Th. & Dynam. Sys.33(6) (2013), 1891–1928].

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“…Bounded density shifts. We first recall the definition of the bounded density subshifts of [25]. Definition 8.7.…”

Section: R Pavlovmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On subshifts with slow forbidden word growth

Pavlov

2021

Ergod. Th. Dynam. Sys.

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“…Examples of hereditary shifts include

β

‐shifts ,

scriptB

‐free shifts , spacing shifts , multi‐choice shifts , and bounded density shifts . Many of these examples have a unique MME, but not every hereditary subshift has this property (see ).…”

Section: Double-struckg‐subshiftsmentioning

confidence: 99%

Extender sets and measures of maximal entropy for subshifts

García-Ramos

Pavlov

2019

J. London Math. Soc.

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For countable amenable finitely generated torsion-free G, we prove inequalities relating μ(v) and μ(w) for any measure of maximal entropy μ on a G-subshift and any words v, w where the extender set of v is contained in the extender set of w. Our main results are two generalizations of a theorem of Meyerovitch (Ergodic Theory Dynam. Systems 33 (2013) 934-953): the first applies to all such v, w when G = Z, and the second to v, w with the same shape for any G. As a consequence of our results we give new and simpler proofs of several facts about synchronized subshifts (including a result from Thomsen, Ergodic Theory Dynam. Systems 26 (2006) 1235-1256) and we answer a question of Climenhaga.Theorem 3.11. Let X be a Z-subshift with positive topological entropy, μ an MME of X, and w, v ∈ L(X). If E X (v) ⊆ E X (w), then μ(v) μ(w)e htop(X) (|w|−|v|) .

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“…and call it the bounded density subshift generated by f . Bounded density shifts were introduced by Stanley in [33]. Stanley proved also that to define Ψ f we can consider only canonical functions f : Z + → [0, ∞).…”

Section: Multiple Ip-recurrence Propertymentioning

confidence: 99%

Multi-recurrence and van der Waerden systems

Kwietniak

Oprocha

et al. 2016

Sci. China Math.

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ABSTRACT. We explore recurrence properties arising from dynamical approach to the van der Waerden Theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynamics. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity of topologically weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.

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Bounded density shifts

Cited by 11 publications

References 7 publications

On subshifts with slow forbidden word growth

On subshifts with slow forbidden word growth

Extender sets and measures of maximal entropy for subshifts

Multi-recurrence and van der Waerden systems

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