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For any mixing SFT X we construct a reversible shift-commuting continuous map (automorphism) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. As an application we prove a finitary Ryan’s theorem: the automorphism group $${{\,\mathrm{Aut}\,}}(X)$$ Aut ( X ) contains a two-element subset S whose centralizer consists only of shift maps. We also give an example which shows that a stronger finitary variant of Ryan’s theorem does not hold even for the binary full shift.

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.

We study perturbations of topological pressures, Gibbs measures and measure-theoretic entropies of these measures concerning perturbed potentials defined on topologically transitive subshift of finite type. The subshift with respect to non-perturbed system is not assumed to be topologically transitive. Therefore, the subshift of the perturbed systems and the subshift of the unperturbed system are different. We reduce this situation to a perturbation problem of certain irreducible nonnegative matrices generated by Ruelle operators. Moreover, under suitable conditions of potentials, we characterize the limit points of those thermodynamics and give a necessary and sufficient condition for convergence of Gibbs measures and the measure-theoretic entropy of this measure when the subshift of the non-perturbed system has or transitive components with the maximal pressure. Finally we apply our results to a problem of collapse of perturbed graph iterated function systems.

The dynamics of a class of one-dimensional polynomial maps are studied, and interesting dynamics are observed under certain conditions: the existence of periodic points with even periods except for one fixed point; the coexistence of two attractors, an attracting fixed point and a hidden attractor; the existence of a double period-doubling bifurcation, which is different from the classical period-doubling bifurcation of the Logistic map; the existence of Li-Yorke chaos. Furthermore, based on this one-dimensional map, the corresponding generalized Henon map is investigated, and some interesting dynamics are found for certain parameter values: the coexistence of an attracting fixed point and a hidden attractor; the existence of Smale horseshoe for a subshift of finite type and also Li-Yorke chaos.

The dynamical structure of the rational map ax+1/x on the projective line $$\mathbb{P}^1(\mathbb{Q}_2)$$ over the field ℚ2 of 2-adic numbers, is fully described.

In this paper, we introduce a noisy framework for SFTs, allowing some amount of forbidden patterns to appear. Using the Besicovitch distance, which permits a global comparison of configurations, we then study the closeness of noisy measures to non-noisy ones as the amount of noise goes to 0. Our first main result is the full classification of the (in)stability in the one-dimensional case. Our second main result is a stability property under Bernoulli noise for higher-dimensional periodic SFTs, which we finally extend to an aperiodic example through a variant of the Robinson tiling., Comment: The v2 fixes a few typos, as well as a small numerical mistake in the last theorem. 35 pages, 7 figures

In this text I study the asymptotics of the complexity function of minimal multidimensional subshifts of finite type through their entropy dimension, a topological invariant that has been introduced in order to study zero entropy dynamical systems. Following a recent trend in symbolic dynamics I approach this using concepts from computability theory. In particular it is known [12] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the \begin{document}$ \Delta_2 $\end{document}-computable numbers in \begin{document}$ [0, d] $\end{document}. The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all \begin{document}$ d \ge 2 $\end{document} the possible values for entropy dimensions of \begin{document}$ d $\end{document}-dimensional SFT are the \begin{document}$ \Delta_2 $\end{document}-computable numbers in \begin{document}$ [0, d-1] $\end{document}. In the present text I prove formally this result for \begin{document}$ d = 3 $\end{document}. Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case \begin{document}$ d = 2 $\end{document} requires some substantial changes to be made in order to adapt the construction that are not discussed here.

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are &beta, shifts, namely transformations of the form T &beta, &alpha, x ↦ &beta, x + &alpha, mod 1 acting on [ &minus, / ( &beta, &minus, 1 ) , ( 1 &minus, ) / ( &beta, 1 ) ] , where ( &beta, ) &isin, &Delta, is fixed and where &Delta, ≔ { ( &beta, R 2 : &beta, &isin, ( 1 , 2 ) and 0 &le, &le, 2 &minus, &beta, } . Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045&ndash, 2055, 2019), that the set of ( &beta, ) such that T &beta, has the subshift of finite type property is dense in the parameter space &Delta, Here, they proposed the following question. Given a fixed &beta, ( 1 , 2 ) which is the n-th root of a Perron number, does there exists a dense set of &alpha, in the fiber { &beta, } &times, ( 0 , 2 &minus, ) , so that T &beta, has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the sofic property (that is a factor of a subshift of finite type). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269&ndash, 278, 1980) from the case when &alpha, = 0 to the case when &alpha, ) . That is, we examine the structure of the set of eventually periodic points of T &beta, when &beta, is a Pisot number and when &beta, is the n-th root of a Pisot number.

In this paper, we describe the relationship between the Perron root and eigenvectors of an irreducible subshift of finite type with the correlation between the forbidden words in the subshift. In particular, we derive an expression for the Perron eigenvectors of the associated adjacency matrix. As an application, we obtain the Perron eigenvectors for irreducible $(0,1)$ matrices which are adjacency matrices for directed graphs. Moreover, we derive an alternate definition of the Parry measure in ergodic theory on an irreducible subshift of finite type., Python code added

We introduce the notion of $(\Gamma,E)$-determinacy for $\Gamma$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where $S=2=\{0,1\}$ or $S=\omega$. We show that for all shift actions by countable groups $G$, and any "reasonable" pointclass $\Gamma$, that $(\Gamma,E_G)$-determinacy implies $\Gamma$-determinacy. We also prove a corresponding result when $E$ is a subshift of finite type of the shift map on $2^\mathbb{Z}$., Comment: 13 pages

A restricted permutation of a locally finite directed graph $G=(V,E)$ is a vertex permutation $\pi: V\to V$ for which $(v,\pi(v))\in E$, for any vertex $v\in V$. The set of such permutations, denoted by $\Omega(G)$, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. We focus on the particular case presented by Schmidt and Strasser (2016) of restricted $\mathbb{Z}^d$ permutations, in which $\Omega(G)$ is a subshift of finite type. We show a correspondence between restricted permutations and perfect matchings (also known as dimer coverings). We use this correspondence in order to investigate and compute the topological entropy in a class of cases of restricted $\mathbb{Z}^d$-permutations. We discuss the global and local admissibility of patterns, in the context of restricted $\mathbb{Z}^d$-permutations. Finally, we review the related models of injective and surjective restricted functions., Comment: To be published in Discrete and Continuous Dynamical Systems Journal

Consider $m \in \mathbb{N}$ and $\beta \in (1, m + 1]$. Assume that $a\in \mathbb{R}$ can be represented in base $\beta$ using a development in series $a = \sum^{\infty}_{n = 1}x(n)\beta^{-n}$ where the sequence $x = (x(n))_{n \in \mathbb{N}}$ take values in the alphabet $\mathcal{A}_m := \{0, \ldots, m\}$. The above expression is called the $\beta$-expansion of $a$ and it is not necessarily unique. We are interested in sequences $x = (x(n))_{n \in \mathbb{N}} \in \mathcal{A}_m^\mathbb{N}$ which are associated to all possible values $a$ which have a unique expansion. We denote the set of such $x$ (with some more technical restrictions) by $X_{m,\beta} \subset\mathcal{A}_m^\mathbb{N}$. The space $X_{m, \beta}$ is called the symmetric $\beta$-shift associated to the pair $(m, \beta)$. It is invariant by the shift map but in general it is not a subshift of finite type. Given a H\"older continuous potential $A:X_{m, \beta} \to\mathbb{R}$, we consider the Ruelle operator $\mathcal{L}_A$ and we show the existence of a positive eigenfunction $\psi_A$ and an eigenmeasure $\rho_A$ for some appropriated values of $m$ and $\beta$. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies $h(\mu_{tA})_{t>1}$ converges, when $t \to\infty$, to the maximal value among the set of all possible values of entropy of all $A$-maximizing probabilities.

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts \(S_\beta \). We show that any reversible CA \(F:S_\beta \rightarrow S_\beta \) has an almost equicontinuous direction whenever \(S_\beta \) is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations \(X\times Y\) of two nontrivial subshifts X and Y. We also make some preliminary observations on direct topological factorizations of beta-shifts that are subshifts of finite type.

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., Comment: 28 pages, 8 figures; revised version accepted to Illinois Journal of Mathematics

We give a necessary and sufficient condition on a $d$-dimensional affine subspace of $\mathbb{R}^n$ to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of {\em coincidence} and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns., Comment: 24 pages, 12 figures

Let$f:X\rightarrow X$be a continuous dynamical system on a compact metric space$X$and let$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$be an$m$-dimensional continuous potential. The (generalized) rotation set$\text{Rot}(\unicode[STIX]{x1D6F7})$is defined as the set of all$\unicode[STIX]{x1D707}$-integrals of$\unicode[STIX]{x1D6F7}$, where$\unicode[STIX]{x1D707}$runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy$\unicode[STIX]{x210B}(w)$to each$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where$f$is a subshift of finite type. We prove that$\text{Rot}(\unicode[STIX]{x1D6F7})$is computable and that$\unicode[STIX]{x210B}(w)$is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general,$\unicode[STIX]{x210B}$is not continuous on the boundary of the rotation set when considered as a function of$\unicode[STIX]{x1D6F7}$and$w$. In particular,$\unicode[STIX]{x210B}$is, in general, not computable at the boundary of$\text{Rot}(\unicode[STIX]{x1D6F7})$.

We study equilibrium measures (K\"aenm\"aki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure. In particular, we do this by showing that the K\"aenm\"aki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type., Comment: 20 pages. A few minor clarifications made and some typos corrected. To appear in the Journal of Fractal Geometry

In this paper, we construct a special class of subshifts of finite type. By studying the spectral radius of the transfer matrix associated with the subshift of finite type, we obtain an estimation of its topological entropy. Interestingly, we find that the topological entropy of this class of subshifts of finite type converges monotonically to log(n + 1) (a constant only depends on the structure of the transfer matrices) as the increasing of the order of the transfer matrices.

We study the set of p-adic Gibbs measures of the q-state Potts model on the Cayley tree of order three. We prove the vastness of the set of the periodic p-adic Gibbs measures for such model by showing the chaotic behavior of the corresponding Potts–Bethe mapping over $$\mathbb {Q}_p$$ for the prime numbers $$p\equiv 1 \ (\mathrm {mod} \ 3)$$ . In fact, for $$0< |\theta -1|_p< |q|_p^2 < 1$$ where $$\theta =\exp _p(J)$$ and J is a coupling constant, there exists a subsystem that is isometrically conjugate to the full shift on three symbols. Meanwhile, for $$0< |q|_p^2 \le |\theta -1|_p< |q|_p < 1$$ , there exists a subsystem that is isometrically conjugate to a subshift of finite type on r symbols where $$r \ge 4$$ . However, these subshifts on r symbols are all topologically conjugate to the full shift on three symbols. The p-adic Gibbs measures of the same model for the prime numbers $$p=2,3$$ and the corresponding Potts–Bethe mapping are also discussed. On the other hand, for $$0< |\theta -1|_p< |q|_p < 1,$$ we remark that the Potts–Bethe mapping is not chaotic when $$p=3$$ and $$p\equiv 2 \ (\mathrm {mod} \ 3)$$ and we could not conclude the vastness of the set of the periodic p-adic Gibbs measures. In a forthcoming paper with the same title, we will treat the case $$0< |q|_p \le |\theta -1|_p < 1$$ for all prime numbers p.

Ergodic optimization aims to single out dynamically invariant Borel probability measures which maximize the integral of a given 'performance' function. For a continuous self-map of a compact metric space and a dense set of continuous functions, we show the existence of uncountably many ergodic maximizing measures. We also show that, for a topologically mixing subshift of finite type and a dense set of continuous functions there exist uncountably many ergodic maximizing measures with full support and positive entropy.

We prove a conjecture of Viana which states that Lyapunov exponents vary continuously when restricted to $GL(2,\mathbb{R})$-valued cocycles over a subshift of finite type which admit invariant holonomies that depend continuously on the cocycle., 34 pages, 1 figure

In this paper we consider the structure of ω-limit sets in subshifts of Baire space. We consider both subshifts of finite type and subshifts of bounded type and we demonstrate that many classical structure theorems for ω-limit sets fail in this context. Nevertheless, we obtain characterizations of ω-limit sets in subshifts of finite types and of attracting ω-limit sets in subshifts of bounded type.

In this paper, we derive a sharp condition on the equivalence of topological transitivity among an interval autonomous dynamical system, its induced set-valued system and induced normal fuzzified system. We also prove that their sensitivity (resp., total transitivity) are equivalent. For a general non-autonomous dynamical system, we show the equivalence of topological mixing (resp., mild mixing, cofinite sensitivity, multi-sensitivity and syndetic sensitivity) among the non-autonomous system and its two induced systems. In contrast, we construct a non-autonomous system that is weakly mixing but neither of its two induced systems is weakly mixing. We extend the topological equi-conjugacy between two non-autonomous systems to their two induced systems. Finally, we verify some basic properties of topological entropy among a non-autonomous system and its two induced systems, and establish some sufficient conditions for the topological equi-conjugacy between the fuzzification of a non-autonomous system and a subshift of finite type.

We construct an explicit example of a family of non-uniformly hyperbolic diffeomorphisms, at the boundary of a set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this heteroclinic tangency is periodic, and there is a Cantor set of choices of the second one. For a non-countable subset of these choices, the second leaf is not periodic and the diffeomorphism is Kupka–Smale: every periodic point is hyperbolic and the intersections of stable and unstable leaves of periodic points are transverse. The bifurcating system is Holder-conjugated to a subshift of finite type; thus every Holder potential admits a unique equilibrium state associated with it.

Let X be a compact metric space and f: X → X be a continuous map. In [14], it was shown that if a dynamical system (X, f) has strictly coupled-expanding property, then the Hyperspace dynamical system ( K ( X ) , f ¯ ) , induced by (X, f), has a subsystem which is topologically semi-conjugated to a full shift (Σk, σ). In this paper, we show that under some conditions more weaker than those of [14] , ( K ( X ) , f ¯ ) has a subsystem which not only is topologically semi-conjugated to a subshift of finite type (ΣA, σA), but also is bigger than the subsystem builded in [14] . Furthermore, we expand above results to the fuzzy dynamical system, extended by (X, f).

A subshift on a group G is a closed, G-invariant subset of A^G, for some finite set A. It is said to be a subshift of finite type (SFT) if it is defined by a finite collection of 'forbidden patterns', to be strongly aperiodic if all point stabilizers are trivial, and weakly aperiodic if all point stabilizers are infinite index in G. We show that groups with at least 2 ends have a strongly aperiodic SFT, and that having such an SFT is a QI invariant for finitely presented torsion free groups. We show that a finitely presented torsion free group with no weakly aperiodic SFT must be QI-rigid. The domino problem on G asks whether the SFT specified by a given set of forbidden patterns is empty. We show that decidability of the domino problem is a QI invariant., Comment: 23 pages, 6 figures. The proof of the main theorem has been simplified and some new corollaries deduced

We prove that Mobius orthogonality does not hold for subshifts of finite type with positive topological entropy. This, in particular, shows that all C1+alpha surface diffeomorphisms with positive e ...

We study equidistribution of certain subsets of periodic orbits for subshifts of finite type. Our results solely rely on the growth of these subsets. As a consequence, effective equidistribution results are obtained for both hyperbolic diffeomorphisms and expanding maps on compact manifolds., 9 pages

We describe the multifractal nature of random weak Gibbs measures on some classes of attractors associated with \begin{document}$C^1$\end{document} random dynamics semi-conjugate to a random subshift of finite type. This includes the validity of the multifractal formalism, the calculation of Hausdorff and packing dimensions of the so-called level sets of divergent points, and a \begin{document}$0$\end{document} - \begin{document}$∞$\end{document} law for the Hausdorff and packing measures of the level sets of the local dimension.