On pointwise periodicity in tilings, cellular automata, and subshifts

Meyerovitch, Tom; Salo, Ville

doi:10.4171/ggd/497

Cited by 8 publications

(7 citation statements)

References 17 publications

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“…As any finite G-invariant subset of A G necessarily consists only of periodic configurations, we have (c) =⇒ (d). The reverse implication follows from the finiteness of closed subshifts containing only periodic configurations proved in [3,Theorem 5.8] and in [24,Theorem 1.4] (see also [4,Theorem 3.8] for the case G = Z 2 ). Note that since A G is compact and τ is continuous, Ω(τ ) is closed in A G .…”

Section: Nilpotency Over Finite Alphabetsmentioning

confidence: 87%

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Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

Ceccherini‐Silberstein¹,

Coornaert²,

Phung³

2020

Preprint

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Let G be a group and let V be an algebraic variety over an algebraically closed field K. Let A denote the set of K-points of V . We introduce algebraic sofic subshifts Σ ⊂ A G and study endomorphisms τ : Σ → Σ. We generalize several results for dynamical invariant sets and nilpotency of τ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that τ is nilpotent if and only if its limit set, i.e., the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and Σ is topologically mixing, we show that τ is nilpotent if and only if its limit set consists of periodic configurations and has finite trace.

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Section: Nilpotency Over Finite Alphabetsmentioning

confidence: 87%

“…The following result is well known, at least in the case of full shifts with finite alphabets (cf. [18, Proposition 2], [33, Proposition 1], [24]).…”

Section: Generalizationsmentioning

confidence: 99%

Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

Ceccherini‐Silberstein¹,

Coornaert²,

Phung³

2020

Preprint

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“…It is known that expansive systems that have only periodic points are finite (see [35] for a general result). The following example shows this is no longer true for subshifts over sofics.…”

Section: Systems With Trivial Dynamicsmentioning

confidence: 99%

Sofically presented dynamical systems

Kopra¹,

Salo²

2021

Preprint

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Systems obtained by quotienting a subshift of finite type (SFT) by another SFT are called finitely presented in the literature. Analogously, if a sofic shift is quotiented by a sofic equivalence relation, we call the resulting system sofically presented. Generalizing an observation of Fried, for all discrete countable monoids M , we show that M -subshift SFT systems are precisely the expansive dynamical M -systems, where S 1 S 2 denotes the class of systems obtained by quotienting subshifts in S1 by (relative) subshifts in S2. We show that for all finitely generated infinite monoids M ,and that Mañé's theorem about the dimension of expansive systems characterizes the virtually cyclic groups.In the case of one-dimensional actions, Mañé's theorem generalizes to sofically presented systems, which also have finite topological dimension. The basis of this is the construction of an explicit metric for a sofically presented system. We show that any finite connected simplicial complex is a connected component of a finitely presented system, and prove that conjugacy of one-dimensional sofically presented dynamical systems is undecidable. A key idea is the introduction of so-called automatic spaces. We also briefly study the automorphism groups and periodic points of these systems.We also perform two case studies. First, in the context of β-shifts, we define the β-kernelthe least subshift relation that identifies 1 with its orbit. We give a classification of the β-shift/βkernel pair as a function of β. Second, we revisit the classical study of toral automorphisms in our framework, and in particular for the toral automorphism ( 1 1 1 0 ), we explicitly compute the kernel of the standard presentation.

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“…Gromov's definition is slightly different from ours, but we will see later in this section that they are equivalent. For the definition in terms of limits of balls, see for instance [MS19]. See also [AGW07], where horoballs "tangent to a base point" are called "cones" (in the context of finitely generated groups, where the base point is the identity element of the group).…”

Section: Introductionmentioning

confidence: 99%

Iterated Minkowski sums, horoballs and north-south dynamics

Epperlein¹,

Meyerovitch²

2020

Preprint

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Given a finite generating set A for a group Γ, we study the map W → W A as a topological dynamical system -a continuous self-map of the compact metrizable space of subsets of Γ. If the set A generates Γ as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when Γ = Z d and A ⊆ Z d a finite positively generating set containing the natural invertible extension of the map W → W + A is always topologically conjugate to the unique "northsouth" dynamics on the Cantor set. In contrast to this, we show that various natural "geometric" properties of the finitely generated group (Γ, A) can be recovered from the dynamics of this map, in particular, the growth type and amenability of Γ. When Γ = Z d , we show that the volume of the convex hull of the generating set A is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group Γ, related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.

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On pointwise periodicity in tilings, cellular automata, and subshifts

Cited by 8 publications

References 17 publications

Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

Sofically presented dynamical systems

Iterated Minkowski sums, horoballs and north-south dynamics

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