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We study higher order expansions both in the Berry-Ess\'een estimate (Edgeworth expansions) and in the local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We establish general results under technical assumptions, discuss the verification of these assumptions and illustrate our results by different examples (subshifts of finite type, Young towers, Sinai billiards, random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting., Comment: 66 pages, 2 figures

In this paper we show that a “typical” random product of conservative surface diffeomorphism has positive Lyapunov exponents. We prove that for any compact oriented surface S, any $$r\ge 1$$ , and any $$d\ge 2$$ , there exists a $$C^1$$ -open and $$C^1$$ -dense subset of $${\text {Diff}}^r_{{\text {vol}}}(S)^d$$ such that if $$(f_1, \ldots , f_d)$$ belongs to this subset, the random product generated by them has positive Lyapunov exponents. Our proof also allows us to deal with more general skew products, for example skew products with a volume preserving Anosov diffeomorphism on the basis, or with a subshift of finite type on the basis preserving a measure with product structure. In these cases we prove the $$C^1$$ -density and $$C^r$$ -openness of the existence of positive Lyapunov exponents.

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., Comment: 4 nicely drawn figures. Last update improves the results of Sec 3.2 and fixes a few typos

For sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth Expansions for the Central Limit Theorem. We apply our results to show that Diophantine iid sequences, finite state Markov chains, strongly ergodic Markov chains and Birkhoff sums of smooth expanding maps & subshifts of finite type satisfy these strong large deviation results., 37 pages, 1 figure, original version split into two parts: discrete time (part I) and continuous time (Part II)

We propose a general framework for proving that a compact, infinite-dimensional map has an invariant set on which the dynamics is semiconjugated to a subshift of finite type. The method is then applied to certain Poincare map of the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter ν = 0.1212 . We give a computer-assisted proof of the existence of symbolic dynamics and countable infinity of periodic orbits with arbitrary large periods.

Let G be an infinite countable group and A be a finite set. If ∑ ⊆ AG is a strongly irreducible subshift of finite type, we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation ${\cal G}$ on ∑ and show that the reduced C*-algebra $C_r^*\left({\cal G} \right)$ of ${\cal G}$ is a unital simple approximately finite (AF)-dimensional C*-algebra. The shift action of G on ∑ induces a canonical automorphism action of G on the C*-algebra $C_r^*\left({\cal G} \right)$ . We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C*-algebras, and show that, if G is an amenable group, then the noncommutative topological entropy of the canonical automorphism action of G on $C_r^*\left({\cal G} \right)$ is equal to the topology entropy of the shift action of G on ∑. We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C*-dynamical system ( $C_r^*\left({\cal G} \right)$ , G).

We show that typical [in the sense of Bonatti and Viana (Ergod Theory Dyn Syst 24(5):1295–1330, 2004) and Avila and Viana (Port Math 64:311–376, 2007)] Holder and fiber-bunched $$\text {GL}_d(\mathbb {R})$$-valued cocycles over subshifts of finite type are uniformly quasi-multiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding (necessarily unique) equilibrium state for such cocycles, and apply this result to repellers. Moreover, we show that the pointwise Lyapunov spectrum is closed and convex, and establish partial multifractal analysis on the level sets of pointwise Lyapunov exponents for such cocycles.

We prove a new result in the area of hitting time statistics. Currently, there is a lot of papers showing that the first entry times into cylinders or balls are often faster than the Birkhoff's Ergodic Theorem would suggest. We provide an opposite counterpart to these results by proving that the hitting times into shrinking balls are also often much larger than these theorems would suggest, by showing that for many dynamical systems $$ \displaystyle \limsup_{r\to 0} \tau_{B(y,r)}(x)\mu(B(y,r))=+\infty, $$ for an appropriately large, at least of full measure, set of points $y$ and $x$. We first do this for all transitive open distance expanding maps and Gibbs/equilibrium states of H\"older continuous potentials; in particular for all irreducible subshifts of finite type with a finite alphabet. Then we prove such result for all finitely irreducible subshifts of finite type with a countable alphabet and Gibbs/equilibrium states for H\"older continuous summable potentials. Next, we show that the \emph{limsup} result holds for all graph directed Markov systems (far going natural generalizations of iterated function systems) and projections of aforementioned Gibbs states on their limit sets. By utilizing the first return map techniques, we then prove the \emph{limsup} result for all tame topological Collect--Eckmann multimodal maps of an interval, all tame topological Collect--Eckmann rational functions of the Riemann sphere, and all dynamically semi--regular transcendental meromorphic functions from $\mathbb{C}$ to $\widehat{\mathbb{C}}$., Comment: arXiv admin note: text overlap with arXiv:1609.03610

This paper establishes a fundamental difference between $\mathbb{Z}$ subshifts of finite type and $\mathbb{Z}^{2}$ subshifts of finite type in the context of ergodic optimization. Specifically, we consider a subshift of finite type $X$ as a subset of a full shift $F$. We then introduce a natural penalty function $f$, defined on $F$, which is 0 if the local configuration near the origin is legal and $-1$ otherwise. We show that in the case of $\mathbb{Z}$ subshifts, for all sufficiently small perturbations, $g$, of $f$, the $g$-maximizing invariant probability measures are supported on $X$ (that is, the set $X$ is stably maximized by $f$). However, in the two-dimensional case, we show that the well-known Robinson tiling fails to have this property: there exist arbitrarily small perturbations, $g$, of $f$ for which the $g$-maximizing invariant probability measures are supported on $F\setminus X$.

In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These results have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known results and proves new results. Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius operators and Tauberian theorems as used in classical problems of prime number theory.

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.

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.

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 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$\text{Rot}(\unicode[STIX]{x1D6F7})$.

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 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.

We study a totally discontinuous interval map defined in [ 0 , 1 ] which is associated to a deformation of the shift map on two symbols 0 − 1 . We define a sequence of transition matrices which characterizes the effect of the interval map on a family of partitions of the interval [ 0 , 1 ] . Recursive algorithms that build the sequence of matrices and their left and right eigenvectors are deduced. Moreover, we compute the Artin zeta function for the interval map.

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 $

International audience; We examine the problem of self-stabilisation, as introduced by Dijkstra in the 1970's, in the context of cellular automata stabilising on k-colourings, that is, on infinite grids which are coloured with k distinct colours in such a way that adjacent cells have different colours. Suppose that for whatever reason (e.g., noise, previous usage, tampering by an adversary), the colours of a finite number of cells in a valid k-colouring are modified, thus introducing errors. Is it possible to reset the system into a valid k-colouring with only the help of a local rule? In other words, is there a cellular automaton which, starting from any finite perturbation of a valid k-colouring, would always reach a valid k-colouring in finitely many steps? We discuss the different cases depending on the number of colours, and propose some deterministic and probabilistic rules which solve the problem for k = 3. We also explain why the case k = 3 is more delicate. Finally, we propose some insights on the more general setting of this problem, passing from k-colourings to other tilings (subshifts of finite type).

In this paper we study the dimension spectrum of general conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We perform a comprehensive study of the dimension spectrum addressing questions regarding its size and topological structure. As a corollary we prove that the dimension spectrum of infinite conformal iterated function systems is compact and perfect. On the way we revisit the role of the parameter $$\theta $$ in graph directed Markov systems and we show that new phenomena arise. We also establish topological pressure estimates for subsystems in the abstract setting of symbolic dynamics with countable alphabets. These estimates play a crucial role in our proofs regarding the dimension spectrum, and they allow us to study Hausdorff dimension asymptotics for subsystems. Finally, we narrow our focus to the dimension spectrum of conformal iterated function systems and we prove, among other things, that the iterated function system resulting from the complex continued fractions algorithm has full dimension spectrum. As a result, we provide a positive answer to the Texan conjecture for complex continued fractions.

We show that effective 2l-multiple correlations imply quantitative l-multiple pointwise ergodic theorems. The result has a wide class of applications which include subgroup actions on homogeneous spaces, ergodic nilmanifold automorphisms, subshifts of finite type and Young towers.

Given a topological dynamical system (X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of (X, T) is the system (S(X), F T ), where F T is defined by F T (φ) = T ◦ φ for any φ ∈ S(X). We show that (1) If (Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ; (2) If (S(Σ), F σ ) is transitive then it is Devaney chaos, where (Σ, σ) is a subshift of finite type; (3) If (Σ, T) has shadowing property, then (S U (Σ), F T ) has shadowing property, where Σ is any closed subset of a Cantor set and T a selfmap of Σ; (4) If (X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T: X → X is continuous, then (S U (X), F T ) is sensitive; (5) If Σ is a closed subset of a Cantor set with infinite points and T: Σ → Σ is positively expansive then the entropy ent U (F T ) of the functional envelope of (Σ, T) is infinity.

A necessary and sufficient condition is given for the existence of an embedding of an irreducible subshift of finite type into the Fibonacci–Dyck shift.

In this paper we obtain exponential large-deviation bounds in the Shannon–McMillan–Breiman convergence formula for entropy in the case of weak Gibbs measures and topologically mixing subshifts of finite type. We also prove almost sure estimates for the error term in the convergence to entropy given by the Shannon–McMillan–Breiman formula for both uniformly and non-uniformly expanding shifts. Finally, we establish a topological characterization of large-deviation bounds for Gibbs measures and deduce some of their topological and geometrical aspects: the local entropy is zero and the topological pressure of positive measure sets is total. Some applications include large-deviation estimates for Lyapunov exponents, pointwise dimension and slow return times.

In this paper, topological conjugacy for two-sided non-hyperbolic and non-autonomous discrete dynamical systems is studied. It is shown that if the system has covering relations with weak Lyapunov condition determined by a transition matrix, there exists a sequence of compact invariant sets restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by the transition matrix. Moreover, if the systems have covering relations with exponential dichotomy and small Lipschitz perturbations, then there is a constructive verification proof of the weak Lyapunov condition, and so topological dynamics of these systems are fully understood by symbolic representations. In addition, the tolerance of Lipschitz perturbation can be characterised by the dichotomy tuple . Here, the weak Lyapunov condition is adapted from [12,24,15] and the exponential dichotomy is from [2].

We treat step skew products over transitive subshifts of finite type with interval fibers. The fiber maps are diffeomorphisms on the interval; we assume that the end points of the interval are fixed under the fiber maps. Our paper thus extends work by V. Kleptsyn and D. Volk who treated step skew products where the fiber maps map the interval strictly inside itself. We clarify the dynamics for an open and dense subset of such skew products. In particular we prove existence of a finite collection of disjoint attracting invariant graphs. These graphs are contained in disjoint areas in the phase space called trapping strips. Trapping strips are either disjoint from the end points of the interval (internal trapping strips) or they are bounded by an end point (border trapping strips). The attracting graphs in these different trapping strips have different properties.

In this paper, some necessary and sufficient conditions are given for maps in Hausdorff spaces to be topologically conjugate or semi-conjugate to subshifts of finite type. These results not only extend some related existing results in metric spaces to Hausdorff spaces but also relax their conditions. In addition, it is shown that the strict A-coupled-expansion for a map in a Hausdorff space is a sufficient condition for it to be semi-conjugate to the subshift σA in Σ+m(A). Based on these results, two new criteria of chaos are established, in which the maps are shown to be chaotic either in the sense of both Devaney and Li-Yorke or in the sense of Li-Yorke.

This paper considers the general question of when a topological action of a countable group can be factored into a direct product of a nontrivial actions. In the early 1980's D. Lind considered such questions for $\mathbb{Z}$-shifts of finite type. We study in particular direct factorizations of subshifts of finite type over $\mathbb{Z}^d$ and other groups, and $\mathbb{Z}$-subshifts which are not of finite type. The main results concern direct factors of the multidimensional full $n$-shift, the multidimensional $3$-colored chessboard and the Dyck shift over a prime alphabet. A direct factorization of an expansive $\mathbb{G}$-action must be finite, but a example is provided of a non-expansive $\mathbb{Z}$-action for which there is no finite direct prime factorization. The question about existence of direct prime factorization of expansive actions remains open, even for $\mathbb{G}=\mathbb{Z}$., Comment: 21 pages, some changes and remarks added in response to suggestions by the referee. To appear in ETDS

We prove that if $G_{P}$ is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group $P$ of pattern size $d$, $d\geq 2$, and if $G_{P}$ has maximal Hausdorff dimension (equal to $1-1/2^{d-1}$), then $G_{P}$ is not topologically finitely generated. We describe precisely all essential pattern groups $P$ that yield finitely constrained groups with maximal Hausdorff dimension. For a given size $d$, $d\geq 2$, there are exactly $2^{d-1}$ such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree of depth $d$.

In this paper we study the directions of periodicity of three-dimensional subshifts of finite type (SFTs) and in particular their slopes. A configuration of a subshift has a slope of periodicity if it is periodic in exactly one direction, the slope being the angle of the periodicity vectors. In this paper, we prove that any $\Sigma^0_2$ set may be realized as a a set of slopes of an SFT., Comment: 12 pages, published in CSR 2018

We provide a topological classification of locally constant functions over subshifts of finite type via their zero-temperature measures. Our approach is to analyze the relationship between the distribution of the zero-temperature measures and the boundary of higher dimensional generalized rotation sets. We also discuss the regularity of the localized entropy function on the boundary of the generalized rotation sets., To appear in Transactions of A.M.S

International audience; From a classical point of view, the domino problem is the question of the existence of an algorithm which can decide whether a finite set of square tiles with colored edges can tile the plane, subject to the restriction that adjacent tiles share the same color along their adjacent edges. This question has already been settled in the negative by Berger in 1966; however, these tilings can be reinterpreted in dynamical terms using the formalism of subshifts of finite type, and hence the same question can be formulated for arbitrary finitely generated groups. In this chapter we present the state of the art concerning the domino problem in this extended framework. We also discuss different notions of effectiveness in subshifts defined over groups, that is, the ways in which these dynamical objects can be described through Turing machines.

We define a hierarchy of systems with topological completely positive entropy in the context of continuous countable amenable group actions on compact metric spaces. For each countable ordinal we construct a dynamical system on the corresponding level of the aforementioned hierarchy and provide subshifts of finite type for the first three levels. We give necessary and sufficient conditions for entropy pairs by means of the asymptotic relation on systems with the pseudo-orbit tracing property, and thus create a bridge between a result by Pavlov and a result by Meyerovitch. As a corollary, we answer negatively an open question by Pavlov regarding necessary conditions for completely positive entropy., 26 pages and 11 gorgeous figures (new version fixes various typos and improves readability of figures in grayscale)

We study the difficulty of computing topological entropy of subshifts subjected to mixing restrictions. This problem is well-studied for multidimensional subshifts of finite type : there exists a threshold in the irreducibility rate where the difficulty jumps from computable to uncomputable, but its location is an open problem. In this paper, we establish the location of this threshold for a more general class, subshifts with decidable languages, in any dimension., Comment: 28 pages, 5 figures, 1 table, submitted to Discrete & Continuous Dynamical Systems. There was an error in the previous preprint version. This version presents a more involved construction

International audience; We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan's property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane.

We generalize the definition of topological entropy due to Adler, Konheim, and McAndrew \cite{AKM} to set-valued functions from a closed subset $A$ of the interval to closed subsets of the interval. We view these set-valued functions, via their graphs, as closed subsets of $[0,1]^{2}$. We show that many of the topological entropy properties of continuous functions of a compact topological space to itself hold in our new setting, but not all. We also compute the topological entropy of some examples, relate the entropy to other dynamical and topological properties of the examples, and we give an example of a closed subset $G$ of $[0,1]^2$ that has $0$ entropy but $G\cup \{ (p,q) \},$ where $(p,q) \in [0,1]^2\setminus G,$ has infinite entropy., Comment: 37 pages, 7 figures

In this paper, we study stability of non-autonomous discrete dynamical systems. For a two-sided non-autonomous systems with covering relations determined by a transition matrix A, we show that any small C0 perturbed system has a sequence of compact invariant sets restricted to which the system is topologically semi-conjugate to σA, the two-sided subshift of finite type induced by A. Together with Lyapunov condition of good rate, the semi-conjugacy will become conjugacy. Moreover, if the Lyapunov condition is strict and has perfect rate, then any small C1 perturbed systems is topological conjugate to σA. We also study topological chaos of one-sided systems and systems with limit functions. Lack of hyperbolicity and the time dependence of the rate prevent us from applying classical hyperbolic results or earlier works for autonomous systems: cone condition and Lyapunov function. Two examples are provided to demonstrate the existence of a non-trivial, non-hyperbolic invariant and essential of controlling rate.