Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group

Barbieri, Sebastián; García-Ramos, Felipe; Li, Hanfeng

doi:10.48550/arxiv.1911.00785

Cited by 5 publications

(17 citation statements)

References 89 publications

(113 reference statements)

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“…More generally, an analogous version of the TMP can be defined for arbitrary group actions on compact metrizable spaces, and it naturally generalizes the well known pseudo-orbit tracing property, also called shadowing. See [5] for further background.…”

Section: The Topological Markov Propertymentioning

confidence: 99%

“…For instance, it is known that for a fixed group Γ there are countably many subshifts of finite type up to topological conjugacy, whereas for Γ = Z 2 there exist uncountably many non-conjugate subshifts with the topological Markov property [14, page 233]. Moreover, every subshift which has a trivial asymptotic relation satisfies the property [5,Proposition 5.3]. Another interesting family of examples is algebraic: every subshift whose alphabet is a finite group and is closed under the pointwise group operation satisfies the property [7, Proposition 5.1], while there are examples with that structure which are not of finite type if the acting group is solvable but not polycyclic-by-finite [36].…”

Section: The Topological Markov Propertymentioning

confidence: 99%

“…We say that X is a group shift if X forms a group under the operation induced by H operating pointwise on every g ∈ Γ. In [7, Proposition 5.1] it was shown that every group shift satisfies the topological Markov property (in fact, every algebraic action satisfies the non-symbolic variant of the topological Markov property, see [5,Proposition 4.1]). Also, it is clear that for any sofic approximation sequence Σ we have h Σ (Γ X) ≥ 0 as X contains a constant configuration, namely its identity configuration e which satisfies e(g) = 1 H for every g ∈ Γ.…”

Section: Absolutely Summable Interactionsmentioning

confidence: 99%

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The Lanford-Ruelle theorem for actions of sofic groups

Barbieri¹,

Meyerovitch²

2021

Preprint

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Let Γ be a sofic group, Σ be a sofic approximation sequence of Γ and X be a Γ-subshift with nonnegative sofic topological entropy with respect to Σ. Further assume that X is a shift of finite type, or more generally, that X satisfies the topological Markov property. We show that for any sufficiently regular potential f : X → R, any translation-invariant Borel probability measure on X which maximizes the measure-theoretical sofic pressure of f with respect to Σ, is a Gibbs state with respect to f . This extends a classical theorem of Lanford and Ruelle, as well as previous generalizations of Ollangier, Pinchon, Tempelman and others, to the case where the group is sofic.As applications of our main result we present a criterion for uniqueness of an equilibrium measure, as well as sufficient conditions for having that the equilibrium states do not depend upon the chosen sofic approximation sequence. We also prove that for any group-shift over a sofic group, the Haar measure is the unique measure of maximal sofic entropy for every sofic approximation sequence, as long as the homoclinic group is dense.On the expository side, we present a short proof of Chung's variational principle for sofic topological pressure.

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Section: The Topological Markov Propertymentioning

confidence: 99%

Section: The Topological Markov Propertymentioning

confidence: 99%

Section: Absolutely Summable Interactionsmentioning

confidence: 99%

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The Lanford-Ruelle theorem for actions of sofic groups

Barbieri¹,

Meyerovitch²

2021

Preprint

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“…There is also a cumulative development on the expansive actions by continuous automorphisms on compact metrizable abelian groups appealing tools from commutative algebras, operator algebras, etc. [10,5,12,3].…”

Section: Introductionmentioning

confidence: 99%

The non-coexistence of distality and expansivity for group actions on infinite compacta

Liang¹,

Shi²,

Xie³

et al. 2021

Preprint

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Let X be a compact metric space and G a finitely generated group. Suppose φ : G → Homeo(X) is a continuous action. We show that if φ is both distal and expansive, then X must be finite. A counterexample is constructed to show the necessity of finite generation condition on G. This is also a supplement to a result due to Auslander-Glasner-Weiss which says that every distal action by a finitely generated group on a zerodimensional compactum is equicontinuous.

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“…This is known even for subshifts, where a counterexample was provided for a strongly irreducible subshifts in [24] and it was showed in [30] that the weak specification property for subshifts is equivalent to being strogly irreducible. We add one of the topological Markov properties that were introduced very recently by S. Barbieri, F. García-Ramos, and H. Li in [1] and which generalizes the pseudo-orbit tracing property, also known as shadowing, which for subshifts corresponds to being of finite type (see [16]).…”

Section: Introductionmentioning

confidence: 99%

Garden of Eden and weakly periodic points for certain expansive actions of groups

Doucha¹

2021

Preprint

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We present several applications of the weak specification property and certain topological Markov properties, recently introduced by S. Barbieri, F. García-Ramos and H. Li, and implied by the pseudoorbit tracing property, for general expansive group actions on compact spaces.First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, i.e. every surjective automorphism of such dynamical system is preinjective. This together with an earlier result of H. Li (where the strong topological Markov property is not needed) of the Myhill property, which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces.Second, we generalize the recent result of D. B. Cohen that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

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Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group

Cited by 5 publications

References 89 publications

The Lanford-Ruelle theorem for actions of sofic groups

The Lanford-Ruelle theorem for actions of sofic groups

The non-coexistence of distality and expansivity for group actions on infinite compacta

Garden of Eden and weakly periodic points for certain expansive actions of groups

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