On the selection of subaction and measure for perturbed potentials

We characterize precisely the possible sets of periods and least periods for the periodic points of a shift of finite type (SFT). We prove that a set is the set of least periods of some mixing SFT if and only if it is either {1} or cofinite, and the set of periods of some mixing SFT if and only if it is cofinite and closed under multiplication by arbitrary natural numbers. We then use these results to derive similar characterizations for the class of irreducible SFTs and the class of all SFTs. Specifically, a set is the set of (least) periods for some irreducible SFT if and only if it can be written as a natural number times the set of (least) periods for some mixing SFT, and a set is the set of (least) periods for an SFT if and only if it can be written as the finite union of the sets of (least) periods for some irreducible SFTs. Finally, we prove that the possible sets of (least) periods of mixing sofic shifts are exactly the same as for mixing SFTs, and that the same is not true for the class of nonmixing sofic shifts.

Krieger’s embedding theorem provides necessary and sufficient conditions for an arbitrary subshift to embed in a given topologically mixing $\mathbb {Z}$ -subshift of finite type. For certain families of $\mathbb {Z}^d$ -subshifts of finite type, Lightwood characterized the aperiodic subsystems. In the current paper, we prove a new embedding theorem for a class of subshifts of finite type over any countable abelian group. Our theorem provides necessary and sufficient conditions for an arbitrary subshift X to embed inside a given subshift of finite type Y that satisfies a certain natural condition. For the particular case of $\mathbb {Z}$ -subshifts, our new theorem coincides with Krieger’s theorem. Our result gives the first complete characterization of the subsystems of the multidimensional full shift $Y= \{0,1\}^{\mathbb {Z}^d}$ . The natural condition on the target subshift Y, introduced explicitly for the first time in the current paper, is called the map extension property. It was introduced implicitly by Mike Boyle in the early 1980s for $\mathbb {Z}$ -subshifts and is closely related to the notion of an absolute retract, introduced by Borsuk in the 1930s. A $\mathbb {Z}$ -subshift has the map extension property if and only if it is a topologically mixing subshift of finite type. We show that various natural examples of $\mathbb {Z}^d$ subshifts of finite type satisfy the map extension property, and hence our embedding theorem applies for them. These include any subshift of finite type with a safe symbol and the k-colorings of $\mathbb {Z}^d$ with $k \ge 2d+1$ . We also establish a new theorem regarding lower entropy factors of multidimensional subshifts that extends Boyle’s lower entropy factor theorem from the one-dimensional case.

In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of free extension shifts on a group G, which takes a shift on a subgroup H of G, and naturally extends it to a shift on all of G.

\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in "rational polygons" on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just 'Subshifts of Finite Type' or their dense subsets. We further show that 'Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate that they approximate any shift dynamics.

Let (Σ, σ) be a Z d -subshift of finite type. Under a strong irreducibility condition (strong specification), we show that Aut( Σ) contains any finite group. For Z d -subshift of finite type without strong specification, examples show that topological mixing is not sufficient to give any finite group in the automorphism group in general: in particular, End (Σ) may be an abelian semigroup. For an example of a topologically mixing Z 2 -subshift of finite type, the endomorphism semigroup and automorphism group are computed explicitly. This subshift has periodic-point permutations that do not extend to automorphisms.

Let be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that K can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of K as well as the set of elements in K with unique codings using the machinery of Mauldin and Williams (1988 Trans. Am. Math. Soc. 309 811–29). We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of K with multiple codings. Secondly, in the setting of β-expansions, when the set of all the unique codings is not a subshift of finite type, we can calculate in some cases the Hausdorff dimension of the univoque set. Motivated by this application, we prove that the set of all the unique codings is a subshift of finite type if and only if it is a sofic shift. This equivalent condition was not mentioned by de Vries and Komornik (2009 Adv. Math. 221 390–427, theorem 1.8). Thirdly, for the doubling map with asymmetrical holes, we give a sufficient condition such that the survivor set can be identified with a subshift of finite type. The third application partially answers a problem posed by Alcaraz Barrera (2014 PhD Thesis University of Manchester).

In this chapter we introduce some further topics which are closely tied to subshifts of finite type. In the first section we look at sofic systems. The continuous symbolic image of a subshift of finite type need not be a subshift of finite type. It may have an unbounded memory. Sofic systems are the symbolic systems that arise as continuous images of subshifts of finite type. There are three equivalent characterizations of these systems. The characterizations are explained and then we investigate some of the basic dynamical properties of sofic systems. The second section contains a discussion of Markov measures. These are the measures which have a finite memory. We define the measures, compute their measure-theoretic entropy, characterize the measures using conditional entropy and then prove that for a fixed subshift of finite type there is a unique Markov measure whose entropy is greater than the entropy of any other measure on the subshift of finite type. The third section investigates symbolic systems that have a group structure. These are the Markov subgroups. We show that any symbolic system with a group structure is a subshift of finite type. Then we classify them up to topological conjugacy. The fourth section contains a very brief introduction to cellular automata. The point is to see how they fit into the framework we have developed. The final section discusses channel codes as illustrated in Example 1.2.8. We describe a class of codes and develope an algorithm to construct them.KeywordsCellular AutomatonFinite TypeTopological EntropyMaximal MeasureBorel Probability MeasureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let G,H be two countable amenable groups. We introduce the notion of group charts, which gives us a tool to embed an arbitrary H -subshift into a G -subshift. Using an entropy addition formula derived from this formalism we prove that whenever H is finitely presented and admits a subshift of finite type (SFT) on which H acts freely, then the set of real numbers attained as topological entropies of H -SFTs is contained in the set of topological entropies of G -SFTs modulo an arbitrarily small additive constant for any finitely generated group G which admits a translation-like action of H . In particular, we show that the set of topological entropies of G -SFTs on any such group which has decidable word problem and admits a translation-like action of \mathbb{Z}^2 coincides with the set of non-negative upper semi-computable real numbers. We use this result to give a complete characterization of the entropies of SFTs in several classes of groups.

We look at constructions of aperiodic subshifts of finite type (SFTs) on fundamental groups of graph of groups. In particular, we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by Whyte and on two constructions of strongly aperiodic SFTs on $\mathbb {F}_n\times \mathbb {Z}$ and $BS(m,n)$ of our own. Our two constructions rely on a path-folding technique that lifts an SFT on $\mathbb {Z}^2$ inside an SFT on $\mathbb {F}_n\times \mathbb {Z}$ or an SFT on the hyperbolic plane inside an SFT on $BS(m,n)$ . In the case of $\mathbb {F}_n\times \mathbb {Z}$ , the path folding technique also preserves minimality, so that we get minimal strongly aperiodic SFTs on unimodular GBS groups.

For any d ≥ 1, random ℤd shifts of finite type (SFTs) were defined in previous work of the authors. For a parameter α ∈ [0, 1], an alphabet $$\mathcal{A}$$ , and a scale n ∈ ℕ, one obtains a distribution of random ℤd SFTs by randomly and independently forbidding each pattern of shape {1,..., n}d with probability 1 − α from the full shift on $$\mathcal{A}$$ . We prove twomain results concerning random ℤd SFTs. First, we establish sufficient conditions on α, $$\mathcal{A}$$ , and a ℤd subshift Y so that a random ℤd SFT factors onto Y with probability tending to one as n tends to infinity. Second, we provide sufficient conditions on α, $$\mathcal{A}$$ and a ℤd subshift X so that X embeds into a random ℤd SFT with probability tending to one as n tends to infinity.

Iterated function systems and the code space

We investigate the stability of maximizing measures for a penalty function of a two-dimensional subshift of finite type, building on the work of Gonschorowski et al. [Support stability of maximizing measures for shifts of finite type, Ergodic Theory Dyn. Syst. 41(3) (2021), pp. 869–880]. In the one-dimensional case, such measures remain stable under Lipschitz perturbations for any subshift of finite type. However, instability arises for a penalty function of the Robinson tiling, which is a two-dimensional subshift of finite type with no periodic points and zero entropy. This raises the question of whether stability persists in two-dimensional subshifts of finite type with positive topological entropy. In this paper, we address this question by studying a nearest-neighbor subshift of finite type satisfying the single-site fillability property. Our main theorem establishes that, in contrast to previous results, a penalty function of such a subshift of finite type remains stable under Lipschitz perturbations.

We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension $d$ can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension $d+1$.

We investigate what happens when we try to work with continuing block codes (i.e., left or right continuing factor maps) between shift spaces that may not be shifts of finite type. For example, we demonstrate that continuing block codes on strictly sofic shifts do not behave as well as those on shifts of finite type; a continuing block code on a sofic shift need not have a uniformly bounded retract, unlike one on a shift of finite type. A right eresolving code on a sofic shift can display any behavior arbitrary block codes can have. We also show that a right continuing factor of a shift of finite type is always a shift of finite type.

For any fixed alphabet A, the maximum topological entropy of a Z^d subshift with alphabet A is obviously log |A|. We study the class of nearest neighbor Z^d shifts of finite type which have topological entropy very close to this maximum, and show that they have many useful properties. Specifically, we prove that for any d, there exists beta_d such that for any nearest neighbor Z^d shift of finite type X with alphabet A for which log |A| - h(X) < beta_d, X has a unique measure of maximal entropy. Our values of beta_d decay polynomially (like O(d^(-17))), and we prove that the sequence must decay at least polynomially (like d^(-0.25+o(1))). We also show some other desirable properties for such X, for instance that the topological entropy of X is computable and that the unique m.m.e. is isomorphic to a Bernoulli measure. Though there are other sufficient conditions in the literature which guarantee a unique measure of maximal entropy for Z^d shifts of finite type, this is (to our knowledge) the first such condition which makes no reference to the specific adjacency rules of individual letters of the alphabet.