For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group, study its algebraic properties, and use them to distinguish many of the stabilized automorphism groups. We also show that for a full shift, the subgroup of the stabilized automorphism group generated by elements of finite order is simple, and that the stabilized automorphism group is an extension of a free abelian group of finite rank by this simple group.
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.
It has been recently proved that the automorphism group of a minimal subshift with non-superlinear word complexity is virtually $\mathbb{Z}$ [DDPM15, CK15]. In this article we extend this result to a broader class proving that the automorphism group of a minimal S-adic subshift of finite alphabet rank is virtually $\mathbb{Z}$. The proof is based on a fine combinatorial analysis of the asymptotic classes in this type of subshifts, which we prove are a finite number.
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…
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…
For a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the stabilized automorphism group $\text{Aut}^{(\infty)}(T)$ consists of all self-homeomorphisms of $X$ which commute with some power of $T$. Motivated by the study of these groups in the context of shifts of finite type, we introduce a certain entropy for groups called local $\mathcal{P}$ entropy. We show that when $(X,T)$ is a non-trivial mixing shift of finite type, the local $\mathcal{P}$ entropy of the group $\text{Aut}^{(\infty)}(T)$ is determined by the topological entropy of $(X,T)$. We use this to give a complete cl…
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 introduce the notions in the title for endomorphisms of subshifts, and using them we characterize various classes of "resolving endomorphisms of subshifts" in the broad sense including onesided and weak ones. Resolving endomorphisms of transitive topological Markov shifts and resolving automorphisms of topological Markov shifts are treated particularly in detail. Moreover, we understand what the limits of onesided resolving directions are for "expansiveness" (in the broad sense including onesided ones) of endomorphisms of subshifts, and we understand the relation between "resolvingness" and…
We prove that for any transitive subshift $X$ with word complexity function $c_n(X)$, if $\liminf \frac{\log (c_n(X)/n)}{\log \log \log n} = 0$, then the quotient group $\textrm{Aut}(X,σ) / \langle σ\rangle$ of the automorphism group of $X$ by the subgroup generated by the shift $σ$ is locally finite. We prove that significantly weaker upper bounds on $c_n(X)$ imply the same conclusion if the Gap Conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of $X$ of range $n$ in terms of word complexity, which may be of independent in…
This article studies the complexity of the word problem in groups of automorphisms of subshifts. We show in particular that for any Turing degree, there exists a subshift whose automorphism group contains a subgroup whose word problem has exactly this degree.
We present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of $\mathcal{B}$-free subshifts satisfying them, extending [10]. On the other hand we provide an example of a $\mathcal{B}$-free Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 in [13].
The notions of symbolic matrix system and $λ$-graph system for a subshift are generalizations of symbolic matrix and $λ$-graph (= finite symbolic matrix) for a sofic shift respectively ([Doc. Math. 4(1999), 285-340]). M. Nasu introduced the notion of textile system for a pair of graph homomorphisms to study automorphisms and endomorphisms of topological Markov shifts ([Mem. Amer. Math. Soc. 546,114(1995)]). In this paper, we formulate textile systems on $λ$-graph systems and study automorphisms on subshifts. We will prove that for a forward automorphism $ϕ$ of a subshift $(Λ,σ)$, the automorph…
For a subshift over a finite alphabet, a measure of the complexity of the system is obtained by counting the number of nonempty cylinder sets of length $n$. When this complexity grows exponentially, the automorphism group has been shown to be large for various classes of subshifts. In contrast, we show that subquadratic growth of the complexity implies that for a topologically transitive shift $X$, the automorphism group $\Aut(X)$ is small: if $H$ is the subgroup of $\Aut(X)$ generated by the shift, then $\Aut(X)/H$ is periodic.
The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the direct sum of countably many copies of $\mathbb{Z}$. In contrast, the group of automorphisms of a symbolic system of zero entropy seems to be highly constrained. Our main result is that the automorphism group of any minimal subshift of stretched exponential growth with exponent $<1/2$, is amenable (as a countable discrete group). For shifts of polynomial grow…
Suppose $(X,σ)$ is a subshift, $P_X(n)$ is the word complexity function of $X$, and ${\rm Aut}(X)$ is the group of automorphisms of $X$. We show that if $P_X(n)=o(n^2/\log^2 n)$, then ${\rm Aut}(X)$ is amenable (as a countable, discrete group). We further show that if $P_X(n)=o(n^2)$, then ${\rm Aut}(X)$ can never contain a nonabelian free semigroup (and, in particular, can never contain a nonabelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a semigroup.
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 $\mathscr{B}$-free subshift given by $\mathscr{B}=\{2^ib_i\}_{i\in\mathbb{N}}$, where $\{b_i\}_{i\in\mathbb{N}}$ is a set of pairwise coprime odd numbers greater than one, it is shown that its automorphism group consists solely of powers of the shift.
Given a dynamical system $(X,G)$, the centralizer $C(G)$ denotes the group of all homeomorphisms of $X$ which commute with the action of $G$. This group is sometimes called the automorphism group of the dynamical system $(X,G)$. In this note, we generalize the construction of Bulatek and Kwiatkowski (1992) to $\mathbb Z^d$-Toepltiz systems by identifying a class of $\mathbb Z^d$-Toeplitz systems that have trivial centralizers. We show that this class of $\mathbb Z^d$-Toeplitz with trivial centralizers contains systems with positive topological entropy.
We will introduce a notion of normal subshifts. A subshift $(Λ,σ)$ is said to be normal if it satisfies a certain synchronizing property called $λ$-synchronizing and is infinite as a set. We have lots of purely infinite simple $C^*$-algebras from normal subshifts including irreducible infinite sofic shifts, Dyck shifts, $β$-shifts, and so on. Eventual conjugacy of one-sided normal subshifts and topological conjugacy of two-sided normal subshifts are characterized in terms of the associated $C^*$-algebras and the associated stabilized $C^*$-algebras with its diagonals and gauge actions, respect…
We generalize Wagoner's representation of the automorphism group of a two-sided subshifts of finite type as the fundamental group of a certain CW-complex to groupoids having a certain refinement structure. This significantly streamlines the original proof and allows us to extend this construction to, e.g., the automorphism group of subshifts of finite type over arbitrary finitely generated groups and the automorphism group of G-SFTs.
Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable ${\mathcal S}$-adic subshifts. This is done establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like all…
For a finite digraph $D$, we define the corresponding subshift of finite type $(X_D, σ_D)$ to be the dynamical system where $X_D$ is the set of all bi-infinite walks through $D$ and $σ_D$ is the shift operator. Two digraphs $D_1$ and $D_2$ are called shift equivalent if there is $m\geq 1$ such that $(X_{D_1}, σ_{D_1}^n)$ and $(X_{D_2}, σ_{D_2}^n)$ are topologically conjugate for all $n\geq m$. They are called strong shift equivalent if this holds for $m=1$. In this paper we introduce a new category which generalises the category of subshifts of finite type and topological conjugacy. Our catego…
For a homeomorphism $T \colon X \to X$ of a Cantor set $X$, the mapping class group $\mathcal{M}(T)$ is the group of isotopy classes of orientation-preserving self-homeomorphisms of the suspension $Σ_{T}X$. The group $\mathcal{M}(T)$ can be interpreted as the symmetry group of the system $(X,T)$ with respect to the flow equivalence relation. We study $\mathcal{M}(T)$, focusing on the case when $(X,T)$ is a minimal subshift. We show that when $(X,T)$ is a subshift associated to a substitution, the group $\mathcal{M}(T)$ is an extension of $\mathbb{Z}$ by a finite group; for a large class of sub…
Let $G$ be a residually finite group and let $A$ be a finite set. We prove that if $X \subset A^G$ is a strongly irreducible subshift of finite type containing a periodic configuration then periodic configurations are dense in $X$. The density of periodic configurations implies in particular that every injective endomorphism of $X$ is surjective and that the group of automorphisms of $X$ is residually finite. We also introduce a class of subshifts $X \subset A^\Z$, including all strongly irreducible subshifts and all irreducible sofic subshifts, in which periodic configurations are dense.
A $λ$-graph bisystem $\mathcal{L}$ consists of two labeled Bratteli diagrams $(\mathcal{L}^-,\mathcal{L}^+)$, that presents a two-sided subshift $Λ_\mathcal{L}$. We will construct a compact totally disconnected metric space with a shift homeomorphism consisting of two-dimensional configurations from a $λ$-graph bisystem. The configuration space has a certain étale AF-equivalence relation written $R_{\mathcal{L}}$ with a natural shift homeomorphism $σ_{\mathcal{L}}$ coming from the shift homeomorphism $σ_{Λ_{\mathcal{L}}}$ on the subshift $Λ_{\mathcal{L}}$. The equivalence relation $R_{\mathcal…
The self-similar structure of the attracting subshift of a primitive substitution is carried over to the limit set of the repelling tree in the boundary of Outer Space of the corresponding irreducible outer automorphism of a free group. Thus, this repelling tree is self-similar (in the sense of graph directed constructions). Its Hausdorff dimension is computed. This reveals the fractal nature of the attracting tree in the boundary of Outer Space of an irreducible outer automorphism of a free group.
Let $X$ be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in $X$. We show that there exist arbitrarily small finite overlapping union of shifted cylinders which intersect every orbit under the shift map. We also show that for any proper subshift $Y$ of $X$ there exists a finite overlapping unions of shifted cylinders such that its survivor set contains $Y$ (in particular, it can have entropy arbitrarily close to the entropy of $X$). Both results may be seen as somewhat counter-intuitive. Finally, we apply these results to …
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 S_1 by (relative) subshifts in S_2. We show that for all finitely generated infinite monoids M, M-SFT \s…
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…
Let S be an ergodic measure-preserving automorphism on a non-atomic probability space, and let T be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Holder ceiling function. We show that if the measure-theoretic entropy of S is strictly less than the topological entropy of T, then there exists an embedding from the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen and Ratner, we also ob…
For any infinite transitive sofic shift $X$ we construct a reversible cellular automaton (i.e. an automorphism of the shift $X$) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. This shows in addition that every infinite transitive sofic shift has a reversible CA which is sensitive with respect to all directions. As another application we prove a finitary Ryan's theorem: the automorphism group aut$(X)$ contains a two-element subset whose centralizer consists only of shift maps. We also show that in the class of $S$-gap …
We work with attracting subshifts generated by substitutions which are also irreducible parageometric automorphisms of free groups. For such a dynamical system, we construct a tree substitution to approximate the repelling real tree of the automorphism. We produce images of this tree inside the Rauzy fractal when the substitution is irreducible Pisot. We describe the contour of this tree and compute an interval exchange transformation of the circle covering the original substitution.
We introduce a notion of $λ$-graph bisystem. It consists of a pair $({\frak L}^-, {\frak L}^+)$ of two labeled Bratteli diagrams ${\frak L}^-, {\frak L}^+$ over alphabets $Σ^-, Σ^+$, respectively, and satisfy certain compatibility condition of their labeling on edges. Its matrix presentation is called a symbolic matrix bisystem. We first show that any $λ$-graph bisystem presents subshifts and conversely any subshift is presented by a $λ$-graph bisystem, called the canonical $λ$-graph bisystem for the subshift. We introduce a notion of properly strong shift equivalence on symbolic matrix bisyst…
We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group $\Z$) and the one-sided case (spatially acting monoid $\N$, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sof…
Given a locally finite graph $Γ$, an amenable subgroup $G$ of graph automorphisms acting freely and almost transitively on its vertices, and a $G$-invariant activity function $λ$, consider the free energy $f_G(Γ,λ)$ of the hardcore model defined on the set of independent sets in $Γ$ weighted by $λ$. Under the assumption that $G$ is finitely generated and its word problem can be solved in exponential time, we define suitable ensembles of hardcore models and prove the following: if $\|λ\|_\infty < λ_c(Δ)$, there exists a randomized $ε$-additive approximation scheme for $f_G(Γ,λ)$ that runs in ti…
