Sofic Subshifts Research Articles

Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

Abstract 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 ${\Sigma \subset A^G}$ and study endomorphisms $\tau \colon \Sigma \to \Sigma $ . We generalize several results for dynamical invariant sets and nilpotency of $\tau $ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $\tau $ is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and $\Sigma $ is topologically mixing, we show that $\tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.

High pointwise emergence via saturated sets

In this article, we prove that the set of points whose pointwise emergence have same degree in with repect to polynomial has strong dynamical complexity in sense of entropy, residuality and density, and distributional chaos for a β-shift, a C 1 diffeomorphism having a nontrival homoclinic class, or a diffeomorphism preserving a hyperbolic ergodic measure. In this process, we construct nonempty compact connected set with fixed box-counting dimension in the space of invariant measures. Then combining with the relation between pointwise emergence of saturated set and box-counting dimension, we get the goal from two frameworks about dynamical complexity of saturated set. One framework was introduced by C. Pfister and W. Sullivan and can be applicable to transitive Anosov diffeomorphisms, mixing expanding maps, β-shifts, transitive subshifts of finite type, mixing sofic subshifts, and mixing S-gap shifts. The other framework is applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. We also get some combined observations with irregular set.

On dynamical finiteness properties of algebraic group shifts

Let G be a group and let V be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts Σ ⊂ VG which generalize both the class of algebraic sofic subshifts of VG and the class of closed group subshifts over finite group alphabets. When G is a polycyclic-by-finite group, we show that VG satisfies the descending chain condition and that the notion of algebraic group subshifts, the notion of algebraic group sofic subshifts, and that of algebraic group subshifts of finite type are all equivalent. Thus, we obtain extensions of well-known results of Kitchens and Schmidt to cover the case of many non-compact group alphabets.

Slopes of Multidimensional Subshifts

In this paper we study the directions of periodicity of multidimensional subshifts of finite type (SFTs) and of multidimensional effectively closed and sofic subshifts. A configuration of a subshift has a slope of periodicity if it is periodic in exactly one direction, the slope representing that direction. In this paper, we prove that ${{\Sigma }^{0}_{1}}$ sets of non-commensurable $\mathbb {Z}^{2}$ vectors are exactly the sets of slopes of 2D SFTs and that ${{\Sigma }^{0}_{2}}$ sets of non-commensurable vectors are exactly the sets of slopes of 3D SFTs, and exactly the sets of slopes of 2D and 3D sofic and effectively closed subshifts.

Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

Every self-similar group acts on the space X^\omega of infinite words over some alphabet X . We study the Schreier graphs \Gamma_w for w\in X^\omega of the action of self-similar groups generated by bounded automata on the space X^\omega . Using sofic subshifts we determine the number of ends for every Schreier graph \Gamma_w . Almost all Schreier graphs \Gamma_w with respect to the uniform measure on X^\omega have one or two ends, and we characterize bounded automata whose Schreier graphs have two ends almost surely. The connection with (local) cut-points of limit spaces of self-similar groups is established.

Fractal dimensions of subfractals induced by sofic subshifts

In this paper, we will consider subfractals, defined as a subset of the attractor of hyperbolic iterated function systems, which satisfy the open set condition. The subfractals will consist of points associated with infinite strings from a sofic subshift on the symbolic space. We find that the zeros of the lower and upper topological pressure functions are lower and upper bounds, respectively, for the Hausdorff, packing, lower and upper box dimensions of the subfractal.

A notion of effectiveness for subshifts on finitely generated groups

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, amenable groups and groups with more than two ends. We also provide an extended model of Turing machine which uses the group itself as a tape and characterizes our extended notion of effectiveness. As applications of these machines we prove that the origin constrained domino problem is undecidable for any group of the form G×Z subject to a technical condition on G and we present a simulation theorem which is valid in any finitely generated group.

Languages of lossless seeds

Several algorithms for similarity search employ seeding techniques to quickly discard very dissimilar regions. In this paper, we study theoretical properties of lossless seeds, i.e., spaced seeds having full sensitivity. We prove that lossless seeds coincide with languages of certain sofic subshifts, hence they can be recognized by finite automata. Moreover, we show that these subshifts are fully given by the number of allowed errors k and the seed margin l. We also show that for a fixed k, optimal seeds must asymptotically satisfy l ~ m^(k/(k+1)).

Subshifts as models for MSO logic

We study the Monadic Second Order (MSO) Hierarchy over colorings of the discrete plane, and draw links between classes of formula and classes of subshifts. We give a characterization of existential MSO in terms of projections of tilings, and of universal sentences in terms of combinations of “pattern counting” subshifts. Conversely, we characterize logic fragments corresponding to various classes of subshifts (subshifts of finite type, sofic subshifts, all subshifts). Finally, we show by a separation result how the situation here is different from the case of tiling pictures studied earlier by Giammarresi et al.

On the density of periodic configurations in strongly irreducible subshifts

Let G be a residually finite group and let A be a finite set. We prove that if X ⊏ AG 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 , including all strongly irreducible subshifts and all irreducible sofic subshifts, in which periodic configurations are dense.

Projective subdynamics and universal shifts

We study the projective subdynamics of two-dimensional shifts of finite type, which is the set of one-dimensional configurations that appear as columns in them. We prove that a large class of one-dimensional shifts can be obtained as such, namely the effective subshifts which contain positive-entropy sofic subshifts. The proof involves some simple notions of simulation that may be of interest for other constructions. As an example, it allows us to prove the undecidability of all non-trivial properties of projective subdynamics.

Graph-directed systems and self-similar measures on limit spaces of self-similar groups

Let G be a group and ϕ : H → G be a contracting homomorphism from a subgroup H < G of finite index. V. Nekrashevych (2005) [25] associated with the pair ( G , ϕ ) the limit dynamical system ( J G , s ) and the limit G -space X G together with the covering ⋃ g ∈ G T ⋅ g by the tile T . We develop the theory of self-similar measures m on these limit spaces. It is shown that ( J G , s , m ) is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile T has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles T ∩ ( T ⋅ g ) for g ∈ G . We present applications to the invariant measures for the rational functions on the Riemann sphere and to the evaluation of the Lebesgue measure of integral self-affine tiles.

Möbius number systems with sofic subshifts

A real Möbius iterative system is an action of a free semigroup of finite words acting via Möbius transformations on the extended real line. Its convergence space consists of all infinite words, such that the images of the Cauchy measure by the prefixes of the word converge to a point measure. A Möbius number system consists of a Möbius iterative system and a subshift included in the convergence space, such that any point measure can be obtained as the limit of some word of the subshift. We give some sufficient conditions on sofic subshifts to form Möbius number systems. We apply our theory to several number systems based on continued fractions.

A new algebraic invariant for weak equivalence of sofic subshifts

Research programme AutoMathA of ESF; Pessoa bilateral project Egide/Grices 11113YM Automata, profinite semigroups and symbolic dynamics; FCT, grant SFRH/BD/24200/2005; POCI 2010; FSE

Subshift attractors of cellular automata

A subshift attractor is a two-sided subshift which is an attractor of a cellular automaton. We prove that each subshift attractor is chain-mixing, contains a configuration which is both F-periodic and σ-periodic and the complement of its language is recursively enumerable. We prove that a subshift of finite type is an attractor of a cellular automaton iff it is mixing. We identify a class of chain-mixing sofic subshifts which are not subshift attractors. We construct a cellular automaton whose maximal attractor is a non-sofic mixing subshift, answering a question raised in Maass (1995 Ergod. Theory Dyn. Syst. 15 663–84). We show that a cellular automaton is surjective on its small quasi-attractor which is the non-empty intersection of all subshift attractors of all Fqσp, where q > 0 and .

One Head Machines from a symbolic approach

We consider the Turing Machine as a dynamical system and we study a particular partition projection of it. In this way, we define a language (a subshift) associated to each machine. The classical definition of Turing Machines over a one-dimensional tape is generalized to allow for a tape in the form of a Cayley Graph. We study the complexity of the language of a machine in terms of realtime recognition by putting it in relation with the structure of its tape. In this way, we find a large set of realtime subshifts some of which are proved not to be deterministic in realtime. Sofic subshifts of this class correspond to machines that cannot make arbitrarily large tours. We prove that these machines always have an ultimately periodic behavior when starting with a periodic initial configuration, and this result is proved for any Cayley Graph.

Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts

For a pseudovariety V of ordered semigroups, let S ( V ) be the class of sofic subshifts whose syntactic semigroup lies in V . It is proved that if V contains Sl − then S ( V ∗ D ) is closed under taking shift equivalent subshifts, and conversely, if S ( V ) is closed under taking conjugate subshifts then V contains LSl − and S ( V ) = S ( V ∗ D ) . Almost finite type subshifts are characterized as the irreducible elements of S ( LInv ) , which gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.

CONJUGACY INVARIANTS OF SUBSHIFTS: AN APPROACH FROM PROFINITE SEMIGROUP THEORY

It is given a structural conjugacy invariant in the set of pseudowords whose finite factors are factors of a given subshift. Some profinite semigroup tools are developed for this purpose. With these tools a shift equivalence invariant of sofic subshifts is obtained, improving an invariant introduced by Béal, Fiorenzi and Perrin using different techniques. This new invariant is used to prove that some almost finite type subshifts with the same zeta function are not shift equivalent.

Finite type approximations of Gibbs measures on sofic subshifts

Consider a Hölder continuous potential ϕ defined on the full shift , where A is a finite alphabet. Let be a specified sofic subshift. It is well known that there is a unique Gibbs measure μϕ on X associated with ϕ. In addition, there is a natural nested sequence of subshifts of finite type (Xm) converging to the sofic subshift X. To this sequence we can associate a sequence of Gibbs measures . In this paper, we prove that these measures converge weakly at exponential speed to μϕ (in the classical distance metrizing weak topology). We also establish a mixing property that implies that μϕ is Bernoulli. Finally, we prove that the measure-theoretic entropy of converges to the one of μϕ exponentially fast. We indicate how to extend our results to more general subshifts and potentials.

Realtime subshifts

We generalize the class of sofic subshifts, which correspond to regular languages, to subshifts accepted by either nondeterministic or deterministic Turing machines in real time. We show that every substitutive system can be accepted by a deterministic Turing machine in real time.