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

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

doi:10.1016/j.aim.2022.108196

Cited by 7 publications

(8 citation statements)

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“…We shall need another variant of the topological Markov property, different from those introduced in [1], although it may turn out that it is equivalent to one of them. We formulate the following definition only for expansive actions, since our work is limited to them; however, a straightforward modification allows us to formulate it generally.…”

Section: Weakly Periodic Pointsmentioning

confidence: 99%

“…This is known even for subshifts, where a counterexample was provided for strongly irreducible subshifts in [26] and it was showed in [34] that for subshifts, the weak specification property is equivalent to being strongly irreducible. We add one of the topological Markov properties that were introduced, for general expansive dynamical systems, very recently by Barbieri, García-Ramos, and 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 [18]).…”

Section: Introductionmentioning

confidence: 99%

“…Here we generalize Cohen's result again to far more general dynamical systems. For this, we introduce a new topological Markov property, called here the cover strong topological Markov property, in the spirit of [1], which is implied by the pseudo-orbit tracing property and implies the uniform strong topological Markov property -we do not know if it is equivalent to the latter. We show the following.…”

Section: Introductionmentioning

confidence: 99%

“…For examples of dynamical systems satisfying various topological Markov properties, the pseudo-orbit tracing property, and of systems satisfying them together with the weak specification property, we refer the reader to e.g. [1,14]. We remark that among the standard examples are, in addition to various subshifts, also various finitely presented algebraic actions.…”

Section: Introductionmentioning

confidence: 99%

“…1 ) Indeed, suppose the contrary and let N ⊆ Y be a γ -separated subset of (Y , d F ) of size strictly bigger than sep(Y , γ ,F \ E) • sep(Y , γ /2, E). Let M ⊆ N be a maximal γ /2-separated subset of N with respect to d E .…”

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confidence: 99%

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

Doucha

2022

Ergod. Th. Dynam. Sys.

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We present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li [Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math.397 (2022), 52], and implied by the pseudo-orbit 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, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property [Garden of Eden and specification. Ergod. Th. & Dynam. Sys.39 (2019), 3075–3088], 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 Cohen [The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math.308 (2017), 599–626] 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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Section: Weakly Periodic Pointsmentioning

confidence: 99%