Effect of quantified irreducibility on the computability of subshift entropy

Gangloff, Silvère; Menibus, Benjamin Hellouin de

doi:10.3934/dcds.2019083

Cited by 30 publications

(12 citation statements)

References 13 publications

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“…Proof. This follows from the proof of Theorem 3.7 in [15]. We now show how to "encode" decidable mixing subshifts onto computable maps of the Cantor set.…”

Section: Upper Bound On the Minimal Cardinality Of A Sub-cover Ofmentioning

confidence: 85%

Computability of topological entropy: From general systems to transformations on Cantor sets and the interval

Gangloff

Herrera

Rojas

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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The dynamics of symbolic systems, such as multidimensional subshifts of finite type or cellular automata, are known to be closely related to computability theory. In particular, the appropriate tools to describe and classify topological entropy for this kind of systems turned out to be algorithmic. Part of the great importance of these symbolic systems relies on the role they have played in understanding more general systems over non-symbolic spaces. The aim of this article is to investigate topological entropy from a computability point of view in this more general, not necessarily symbolic setting. In analogy to effective subshifts, we consider computable maps over effective compact sets in general metric spaces, and study the computability properties of their topological entropies. We show that even in this general setting, the entropy is always a Σ 2 -computable number. We then study how various dynamical and analytical constrains affect this upper bound, and prove that it can be lowered in different ways depending on the constraint considered. In particular, we obtain that all Σ 2 -computable numbers can already be realized within the class of surjective computable maps over {0, 1} N , but that this bound decreases to Π 1 (or upper)-computable numbers when restricted to expansive maps. On the other hand, if we change the geometry of the ambient space from the symbolic {0, 1} N to the unit interval [0, 1], then we find a quite different situation -we show that the possible entropies of computable systems over [0, 1] are exactly the Σ 1 (or lower)-computable numbers and that this characterization switches down to precisely the computable numbers when we restrict the class of system to the quadratic family.

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“…Proof. This follows from the proof of Theorem 3.7 in [15]. We now show how to "encode" decidable mixing subshifts onto computable maps of the Cantor set.…”

Section: Upper Bound On the Minimal Cardinality Of A Sub-cover Ofmentioning

confidence: 85%

Computability of topological entropy: From general systems to transformations on Cantor sets and the interval

Gangloff

Herrera

Rojas

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…[12] with [6] or [18]) by breaking our ability to embed computation. In particular, finding the border where the difficulty jump occurs gives a fine understanding of the effect of the restriction [7].…”

Section: Introductionmentioning

confidence: 99%

The aperiodic Domino problem in higher dimension

Callard,

de Menibus

2022

Preprint

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The classical Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. In this paper, we consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling? The input may correspond to a subshift of finite type, a sofic subshift or an effective subshift.[8] proved that this problem is co-recursively enumerable (Π 1 0 -complete) in dimension 2 for geometrical reasons. We show that it is much harder, namely analytic (Σ 1 1 -complete), in higher dimension: d ≥ 4 in the finite type case, d ≥ 3 for sofic and effective subshifts. The reduction uses a subshift embedding universal computation and two additional dimensions to control periodicity. This complexity jump is surprising for two reasons: first, it separates 2-and 3-dimensional subshifts, whereas most subshift properties are the same in dimension 2 and higher; second, it is unexpectedly large.

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“…Once again, dynamical constraints are another relevant parameter: under strong irreducibility constraints, such as being block gluing, the entropy becomes computable [PS15]. It is possible to extend the notion of block gluingness by adding a gap function f that yields the distance f (n) which allows for the concatenation of two rectangular blocks of size n of the language: depending on the asymptotic behavior of f (n), the set of entropies can be either any Π 1 -computable number or only some computable numbers [Gd19]. The exact frontier for this parametrization is only known for subshifts with decidable language [GS20].…”

Section: Introductionmentioning

confidence: 99%

Parametrization by Horizontal Constraints in the Study of Algorithmic Properties of $\mathbb{Z}^2$-Subshift of Finite Type

Solène¹,

Sablik²

2022

Preprint

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The non-emptiness, called the Domino Problem, and the characterization of the possible entropies of 2 -subshifts of finite type are standard problems of symbolic dynamics. In this article we study these questions with horizontal constraints fixed beforehand as a parameter. We determine for which horizontal constraints the Domino Problem is undecidable and when all right-recursively enumerable numbers can be obtained as entropy, with two approaches: either the additional local rules added to the horizontal constraints can be of any shape, or they can only be vertical rules.Note that a Rauzy graph can be made of one or several strongly connected components (SCC for short). We recall that it is constituted of one unique strongly connected components if and only if the SFT associated is transitive; that is, for any u, w ∈ L H there exists v ∈ L H so that uvw ∈ L H . If the Rauzy graph has several SCCs it can also contain transient vertices, that are vertices with no path from themselves to themselves. We refer to [LM95] for more details.

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Effect of quantified irreducibility on the computability of subshift entropy

Cited by 30 publications

References 13 publications

Computability of topological entropy: From general systems to transformations on Cantor sets and the interval

Computability of topological entropy: From general systems to transformations on Cantor sets and the interval

The aperiodic Domino problem in higher dimension

Parametrization by Horizontal Constraints in the Study of Algorithmic Properties of $\mathbb{Z}^2$-Subshift of Finite Type

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