We provide an example of a non-finitely generated group which admits a nonempty strongly aperiodic SFT. Furthermore, we completely characterize the groups with this property in terms of their finitely generated subgroups and the roots of their conjugacy classes.
We give strongly aperiodic subshifts of finite type on every hyperbolic surface group; more generally, for each pair of expansive primitive symbolic substitution systems with incommensurate growth rates, we construct strongly aperiodic subshifts of finite type on their orbit graphs.
A theorem of Gao, Jackson and Seward, originally conjectured to be false by Glasner and Uspenskij, asserts that every countable group admits a $2$-coloring. A direct consequence of this result is that every countable group has a strongly aperiodic subshift on the alphabet $\{0,1\}$. In this article, we use Lovász local lemma to first give a new simple proof of said theorem, and second to prove the existence of a $G$-effectively closed strongly aperiodic subshift for any finitely generated group $G$. We also study the problem of constructing subshifts which generalize a property of Sturmian seq…
We consider the structure of aperiodic points in $\mathbb Z^2$-subshifts, and in particular the positions at which they fail to be periodic. We prove that if a $\mathbb Z^2$-subshift contains points whose smallest period is arbitrarily large, then it contains an aperiodic point. This lets us characterise the computational difficulty of deciding if an $\mathbb Z^2$-subshift of finite type contains an aperiodic point. Another consequence is that $\mathbb Z^2$-subshifts with no aperiodic point have a very strong dynamical structure and are almost topologically conjugate to some $\mathbb Z$-subshi…
It is well known that if $G$ admits a f.g. subgroup $H$ with a weaklyaperiodic SFT (resp. an undecidable domino problem), then $G$itself has a weakly aperiodic SFT (resp. an undecidable domino problem).We prove that we can replace the property "$H$ is a subgroup of $G$"by "$H$ acts translation-like on $G$", provided $H$ is finitely presented.In particular:* If $G\_1$ and $G\_2$ are f.g. infinite groups, then $G\_1 \times G\_2$ has a weakly aperiodic SFT (and actually a undecidable domino problem). In particular the Grigorchuk group has an undecidable domino problem. * Every infinite f.g. $p$-g…
We show that if $(X,T)$ is an extension of an aperiodic subshift (a subsystem of $({1,2,...,l}^{\mathbb{Z}},\mathrm{shift})$ for some $l\in\mathbb{N}$) and has mean dimension $mdim(X,T)<\frac{D}{2}$ $(D\in \mathbb{N}$), then it embeds equivariantly in (([0,1]^{D})^{\mathbb{Z}},\mathrm{shift})$. The result is sharp. If $(X,T)$ is an extension of an aperiodic zero-dimensional system then it embeds equivariantly in $(([0,1]^{D+1})^{\mathbb{Z}},\mathrm{shift})$.
In this note we prove the following results: $\bullet$ If a finitely presented group $G$ admits a strongly aperiodic SFT, then $G$ has decidable word problem. More generally, for f.g. groups that are not recursively presented, there exists a computable obstruction for them to admit strongly aperiodic SFTs. $\bullet$ On the positive side, we build strongly aperiodic SFTs on some new classes of groups. We show in particular that some particular monster groups admits strongly aperiodic SFTs for trivial reasons. Then, for a large class of group $G$, we show how to build strongly aperiodic SFTs ove…
This paper aims to better understand the link better understand the links between aperiodicity in subshifts and pattern complexity. Our main contribution deals with substitutive subshifts, an equivalent to substitutive tilings in the context of symbolic dynamics. For a class of substitutive subshifts, we prove a quadratic lower bound on their pattern complexity. Together with an already known upper bound, this shows that this class of substitutive subshifts has a pattern complexity in $Θ(n^2)$. We also prove that the recent bound of Kari and Moutot, showing that any aperiodic subshift has patt…
Let $A$ be a finite set and $G$ a group. A closed subset $X$ of $A^G$ is called a subshift if the action of $G$ on $A^G$ preserves $X$. If $K$ is a closed subset of $A^G$ such that membership in $K$ is determined by looking at a fixed finite set of coordinates, and $X$ is the intersection of all translates of $K$ under the action of $G$, then $X$ is called a subshift of finite type (SFT). If an SFT is nonempty and contains no finite $G$-orbits, it is said to be weakly aperiodic. A virtually cyclic group has no weakly aperiodic SFT, and Carroll and Penland have conjectured that a group with no …
We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of a group. This new class of subshifts forms a conjugacy class that contains all sofic subshifts. Motivated by the question of whether there exists a group where the class of sofic subshifts coincides with that of effective subshifts, we show that the inclusion is strict for several groups, including recursively presented groups with undecidable word problem, …
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. arXiv:1805.08829 proved that this problem is co-recursively enumerable ($Π_0^1$-complete) in dimension 2 for geometrical reasons. We show that it is much harder, namely analytic ($Σ_1^1$-complete), in higher dimension: $d…
At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely $α$-repetitive, $α$-repulsive and $α$-finite ($α\geq 1$), have been introduced and studied. We establish the equivalence of $α$-repulsive and $α$-finite for general subshifts over finite alphabets. Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk's infinite $2$-group $G$. In…
It is possible to define mixing properties for subshifts according to the intensity which allows to concatenate two rectangular blocks. We study the interplay between this intensity and computational properties. In particular we prove that there exists linearly block gluing subshift of finite type which are aperiodic and that all right-recursively enumerable positive number can be realized as entropy of linearly block gluing Z 2-subshift of finite type. Like linearly block gluing imply transitivity, this last point answer a question asked in [HM10] about the characterization of the entropy of …
We present a technique to lift some tilings of the discrete hyperbolic plane -- tilings defined by a 1D substitution -- into a zero entropy subshift of finite type (SFT) on non-abelian amenable Baumslag-Solitar groups $BS(1,n)$ for $n\geq2$. For well chosen hyperbolic tilings, this SFT is also aperiodic and minimal. As an application we construct a strongly aperiodic SFT on $BS(1,n)$ with a hierarchical structure, which is an analogue of Robinson's construction on $\mathbb{Z}^2$ or Goodman-Strauss's on $\mathbb{H}_2$.
For an ascending HNN-extension $G*_ψ$ of a finitely generated abelian group $G$, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal{A}^{G*_ψ}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our results imply that a $\mathrm{BS}(1,N)$-SFT which contains a configuration with period $a^{N^\ell}$, $\ell\ge 0$, must contain a strongly periodic configuration with monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-c…
We explicitly construct a strongly aperiodic subshift of finite type for the discrete Heisenberg group. Our example builds on the classical aperiodic tilings of the plane due to Raphael Robinson. Extending those tilings to the Heisenberg group by exploiting the group's structure and posing additional local rules to prune out remaining periodic behavior we maintain a rich projective subdynamics on $\mathbb Z^2$ cosets. In addition the obtained subshift factors onto a strongly aperiodic, minimal sofic shift via a map that is invertible on a dense set of configurations.
We compute an explicit representation of the (topological) automorphism group or a particular Toeplitz subshift. The automorphism group is a (non-finitely generated) subgroup of rational numbers under addition and the shift map corresponds to the rational number 1. The group is the additive subgroup of the rational numbers generated by the powers of 5/2.
We show that every Grigorchuk group $G_ω$ embeds in (the commutator subgroup of) the topological full group of a minimal subshift. In particular, the topological full group of a Cantor minimal system can have subgroups of intermediate growth, a question raised by Grigorchuk; it can also have finitely generated infinite torsion subgroups, as well as residually finite subgroups that are not elementary amenable, answering questions of Cornulier. By estimating the word-complexity of this subshift, we deduce that every Grigorchuk group $G_ω$ can be embedded in a finitely generated simple group that…
We study Schrödinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the standard theory that shows that the spectrum and the spectral type are almost surely constant, and that identifies the almost sure absolutely continuous spectrum with the essential closure of the set of energies with vanishing Lyapunov exponent. Using results of Damanik-Lenz and Klassert-Lenz-Stollmann, we also show that the spectrum is a Cantor set of zero Lebesgue …
We study random walk on topological full groups of subshifts, and show the existence of infinite, finitely generated, simple groups with the Liouville property. Results by Matui and Juschenko-Monod have shown that the derived subgroups of topological full groups of minimal subshifts provide the first examples of finitely generated, simple amenable groups. We show that if the (not necessarily minimal) subshift has a complexity function that grows slowly enough (e.g. linearly), then every symmetric and finitely supported probability measure on the topological full group has trivial Poisson-Furst…
We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.
In this paper we study the quenched distributions of hitting times for a class of random dynamical systems. We prove that hitting times to dynamically defined cylinders converge to a Poisson point process under the law of random equivariant measures with super-polynomial decay of correlations. In particular, we apply our results to uniformly aperiodic random subshifts of finite type equipped with random invariant Gibbs measures. We emphasize that we make no assumptions about the mixing property of the marginal measure.
We study implications of expansiveness and pointwise periodicity for certain groups and semigroups of transformations. Among other things we prove that every pointwise periodic finitely generated group of cellular automata is necessarily finite. We also prove that a subshift over any finitely generated group that consists of finite orbits is finite, and related results for tilings of Euclidean space.
Automorphism groups are intrincate conjugacy invariants for subshifts, which can reveal important features of the dynamical structure of a shift action. One important case is the study of automorphism groups when the underlying subshift has a very rigid structure, e.g. substitutive subshifts or aperiodic constructions with large-scale self-similarity, such as the Robinson shift. In this work we study the automorphism group of bijective substitutive subshifts, and a potential generalization in the form of the group of extended symmetries, studied previously by Michael Baake, John Roberts and Re…
We consider minimal, aperiodic symbolic subshifts and show how to characterize the combinatorial property of bounded powers by means of a metric property. For this purpose we construct a family of graphs which all approximate the subshift space, and define a metric on each graph which extends to a metric on the subshift space. The characterization of bounded powers is then given by the Lipschitz equivalence of a suitably defined infimum metric with the corresponding supremum metric. We also introduce zeta-functions and relate their abscissa of convergence to various exponents of complexity of …
In this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non superlinear, we prove that the automorphism group is, modulo a finite cyclic group, generated by a unique root of the shift. In the subquadratic complexity case, we show that the automorphism group modulo the torsion is generated by the roots of the shift map and that the result of the non superlinear case is optimal. Namely, for any $\varepsilon > 0$ we construct examples of minimal Toeplitz subshifts with complexity…
Using a deterministic version of the self-similar (or hierarchical, or fixed-point ) method for constructing 2-dimensional subshifts of finite type (SFTs), we construct aperiodic 2D SFTs with a unique direction of non-expansiveness and prove that the emptiness problem of SFTs is undecidable even in this restricted case. As an additional application of our method, we characterize the sets of directions that can be the set of non-expansive directions of 2D SFTs.
We study internal Lie algebras in the category of subshifts on a fixed group -- or Lie algebraic subshifts for short. We show that if the acting group is virtually polycyclic and the underlying vector space has dense homoclinic points, such subshifts can be recoded to have a cellwise Lie bracket. On the other hand there exist Lie algebraic subshifts (on any finitely-generated non-torsion group) with cellwise vector space operations whose bracket cannot be recoded to be cellwise. We also show that one-dimensional full vector shifts with cellwise vector space operations can support infinitely ma…
In this note we show that for any subshift $X$ of finite $S$-rank every invariant measure $μ$ is determined by its values on finitely many cylinders. Under mild conditions these cylinders are given by the letters of the alphabet in question.
Tata Institute of Fundamental Research（塔塔基础研究所）；International Centre for Theoretical Sciences (ICTS-TIFR), Tata Institute of Fundamental Research（塔塔基础研究所国际理论科学中心）
We prove that if $G$ is a finitely generated group and $Z$ is a uniformly recurrent subgroup of $G$ then there exists a minimal system $(X,G)$ with $Z$ as its stability system. This answers a query of Glasner and Weiss \cite{GW} in the case of finitely generated groups. Using the same method (introduced by Alon, Grytczuk, Haluszczak and Riordan \cite{AGHR}) we will prove that finitely generated sofic groups have free Bernoulli-subshifts admitting an invariant probability measure.
In this article we study the automorphism group ${\rm Aut}(X,σ)$ of subshifts $(X,σ)$ of low word complexity. In particular, we prove that Aut$(X,σ)$ is virtually $\mathbb{Z}$ for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneous…
We introduce subshifts of quasi-finite type as a generalization of the well-known subshifts of finite type. This generalization is much less rigid and therefore contains the symbolic dynamics of many non-uniform systems, e.g., piecewise monotonic maps of the interval with positive entropy. Yet many properties remain: existence of finitely many ergodic invariant probabilities of maximum entropy; lots of periodic points; meromorphic extension of the Artin-Mazur zeta function.
In this paper, we mainly elucidate a close relationship between the topological entropy and mean dimension theory for actions of polynomial growth groups. We show that metric mean dimension and mean Hausdorff dimension of subshifts with respect to the lower rank subgroup are equal to its topological entropy multiplied by the growth rate of the subgroup. Meanwhile, we also prove the above result holds for the rate distortion dimension of subshifts with respect to the lower rank subgroup and measure entropy. Furthermore, some relevant examples are indicated.
We study the (Hölder-)continuous behavior of the spectra belonging to a family of linear bounded operators $(A_t)_{t\in T}$ indexed by a topological space $T$. For the cases of self-adjoint, unitary and normal operators, a characterization of the continuity of $Σ:T\to \mathcal{K}(\mathbb{R}), t\mapsto σ(A_t),$ is proven while the distance of the spectra is measured by the Hausorff metric. If $T$ is a metric space, the Hölder-continuous behavior of $Σ$ is characterized for self-adjoint and unitary operators. Here we observe interesting effects, namely the rate of convergence is bisect whenever …
We study the density of periodic configurations for shift spaces defined on (the Cayley graph of) a finitely generated group. We prove that in the case of a full shift on a residually finite group and in that of a group shift space on an abelian group, the periodic configurations are dense. In the one-dimensional case we prove the density for irreducible sofic shifts. In connection with this we study the surjunctivity of cellular automata and local selfmappings. Some related decision problems for shift spaces of finite type are also investigated.
We consider the problem of when a symbolic dynamical system supports a Borel probability measure that is invariant under every element of its automorphism group. It follows readily from a classical result of Parry that the full shift on finitely many symbols, and more generally any mixing subshift of finite type, supports such a measure. Frisch and Tamuz recently dubbed such measures characteristic, and further showed that every zero entropy subshift has a characteristic measure. While it remains open if every subshift over a finite alphabet has a characteristic measure, we define a new class …
For each $n, d \in \mathbb{N}$ and $0 < α< 1$, we define a random subset of $\mathcal{A}^{\{1, 2, \dots, n\}^d}$ by independently including each element with probability $α$ and excluding it with probability $1-α$, and consider the associated random subshift of finite type. Extending results of McGoff and of McGoff and Pavlov, we prove there exists $α_0 = α(d, |\mathcal{A}|) > 0$ such that for $α< α_0$ and with probability tending to $1$ as $n \to \infty$, this random subshift will contain only finitely many elements. In the case $d = 1$, we obtain the best possible such $α_0$, $1/|\mathcal{A}…
Dimension groups are complete invariants of strong orbit equivalence for minimal Cantor systems. This paper studies a natural family of minimal Cantor systems having a finitely generated dimension group, namely the primitive unimodular proper S-adic subshifts. They are generated by iterating sequences of substitutions. Proper substitutions are such that the images of letters start with a same letter, and similarly end with a same letter. This family includes various classes of subshifts such as Brun subshifts or dendric subshifts, that in turn include Arnoux-Rauzy subshifts and natural coding …
In the present paper, we use the generalized multifractal framework introduced by Olsen to study the Bowen entropy and packing entropy of historic sets with typical weights over aperiodic and irreducible shifts of finite type. Following those results and a transfer from almost everywhere to everywhere, we show that for each point $ω$ in a irreducible shift of finite type $Σ_A$, the Bowen entropy of the set consisting of all the points that are mean Li-Yorke pairs with $ω$ is $0$, and its packing entropy is full. This result is beyond the ergodic theory. Also, by the transfer from almost everyw…
Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the i-th and (i+1)-th Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index i. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.
To every dynamical system $(X,φ)$ over a totally disconnected compact space, we associate a left-orderable group $T(φ)$. It is defined as a group of homeomorphisms of the suspension of $(X,φ)$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple, and if it is a subshift then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that i…
We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed $\mathbb{Z}^d$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^2$. Let $H$ be a finitely generated group and $G = \mathbb{Z}^2 \rtimes H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,f)$ where $Y$ is a Cantor set, there exists a $G$-subshift of finite type $(X,σ)$ such that the $H$-subaction of $(X,σ)$ is an extens…
Gao, Jackson, and Seward (see arXiv:1201.0513) proved that every countably infinite group $Γ$ admits a nonempty free subshift $X \subseteq \{0,1\}^Γ$. Furthermore, a theorem of Seward and Tucker-Drob (see arXiv:1402.4184) implies that every countably infinite group $Γ$ admits a free subshift $X \subseteq \{0,1\}^Γ$ that supports an invariant probability measure. Aubrun, Barbieri, and Thomassé (see arXiv:1507.03369) used the Lovász Local Lemma to give a short alternative proof of the Gao--Jackson--Seward theorem. Recently, Elek (see arXiv:1702.01631) followed another approach involving the Lová…
