We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions. We show the variational principle for topological pressure. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. As an application, we extend the theory of factors of (generalized) Gibbs measures on subshifts on finite alphabets to that on certain subshifts over countable alphabets.
We provide a quantification of the uniqueness of Gibbs measure for topologically mixing countable Markov shifts with locally Hölder continuous potentials. Corollaries for speed of convergence for approximation by finite subsystems are also given.
We deal with countable alphabet locally compact random subshifts of finite type (the latter merely meaning that the symbol space is generated by an incidence matrix) under the absence of Big Images Property and under the absence of uniform positivity of the transfer operator. We first establish the existence of random conformal measures along with good bounds for the iterates of the Perron-Frobenius operator. Then, using the technique of positive cones and proving a version of Bowen's type contraction (see \cite{Bow75}), we also establish a fairly complete thermodynamical formalism. This means…
In this work we characterize shift spaces over infinite countable alphabets that can be endowed with an inverse semigroup operation. We give sufficient conditions under which zero-dimensional inverse semigroups can be recoded as shift spaces whose correspondent inverse semigroup operation is a 1-block operation, that is, it arises from a group operation on the alphabet. Motivated by this, we go on to study block operations on shift spaces and, in the end, we prove our main theorem, which states that Markovian shift spaces, which can be endowed with a 1-block inverse semigroup operation, are co…
This note revisits the problem of finding necessary and sufficient conditions for a subshift to have a continuous g-function. Results obtained by Krieger (IMS Lecture Notes-Monograph Series, 48, 306--316, 2006) on finite alphabet subshifts are generalized to countable state subshifts.
We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is giv…
We consider a hard core (HC) model with a countable set $\mathbb{Z}$ of spin values on the Cayley tree. This model is defined by a countable set of parameters $λ_{i}>0, i \in \mathbb{Z}\setminus\{0\}$. For all possible values of parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model.
We prove the existence of calibrated uniformly continuous subactions for coercive potentials with bounded variation defined on topologically transitive Markov shifts with countable alphabet through the construction of the Peierls barrier in this context. Also, we characterize the existence of bounded calibrated subactions in the same context.
For a finitely irreducible countable Markov shift and a potential with summable variations, we provide a condition on the associated pressure function which ensures that Bowen's Gibbs state, the equilibrium state, and the minimizer of the level-2 large deviations rate function are all unique and they coincide. From this, we deduce that all periodic points weighted with the potential equidistribute with respect to the Gibbs-equilibrium state as the periods tend to infinity. Applications are given to the Gauss map, and the Bowen-Series map associated with a finitely generated free Fuchsian group…
In the present paper we provide a new construction of measure, called $p$-adic quasi Gibbs measure, for countable state of $p$-adic Potts model on the Cayley tree. Such a construction depends on a parameter $\frak{p}$ and wights. In particular case, i.e. if $\frak{p}=\exp_p$, the defined measure coincides with $p$-adic Gibbs measure. In this paper, under some condition on weights we establish the existence of $p$-adic quasi Gibbs measures associated with the model. Note that this condition does not depend on values of the prime $p$. An analogues fact is not valid when the number of spins is fi…
Let $Σ_{A}(\mathbb{N})$ be a topologically mixing countable Markov shift with the BIP property over the alphabet $\mathbb{N}$ and $f: Σ_{A}(\mathbb{N}) \rightarrow \mathbb{R}$ a potential satisfying the Walters condition with finite Gurevich pressure. Under suitable hypotheses, we prove the existence of a Large Deviation Principle for the family $(μ_β)_{β> 0}$ where each $μ_β$ is the Gibbs measure associated to the potential $βf$. Our main theorem generalizes from finite to countable alphabets and also to a larger class of potentials a previous result of A. Baraviera, A. O. Lopes and P. Thieul…
We introduce a definition of pressure for almost-additive sequences of continuous functions defined over (non-compact) countable Markov shifts. The variational principle is proved. Under certain assumptions we prove the existence of Gibbs and equilibrium measures. Applications are given to the study of maximal Lypaunov exponents of product of matrices.
Dense periodic optimization for countable Markov shift via Aubry points
Institute of Mathematics, Statistics and Scientific Computing, University of Campinas（坎皮纳斯大学数学、统计与科学计算研究所）；Center of Exact and Technological Sciences, Federal University of Recôncavo da Bahia（巴伊亚河谷联邦大学精确与科学技术中心）
The study of escape rates for a ball in a dynamical systems has been much studied. Understanding the asymptotic behavior of the escape rate as the radius of the ball tends to zero is an especially subtle problem. In the case of hyperbolic conformal systems this has been addressed by various authors. In this paper we consider a far more general realm of conformal maps where the analysis is correspondingly more complicated. We prove the existence of escape rates and calculate them in the context of countable alphabets, either finite or infinite, uniformly contracting conformal graph directed Mar…
We characterize the finiteness of Gibbs measures for geodesic flows on negatively curved manifolds by several criteria, analogous to those proposed by Sarig for symbolic dynamical systems over an infinite alphabet. As an application, we recover Dal'bo-Otal-Peign{é} criterion of finiteness for the Bowen-Margulis measure on geometrically finite hyperbolic manifolds, as well as Peign{é}' examples of gemetrically infinite manifolds having a finite Bowen-Margulis measure. These criteria should be useful in the future to find more examples with finite Gibbs measures.
We formulate and prove a very general relative version of the Dobrushin-Lanford-Ruelle theorem which gives conditions on constraints of configuration spaces over a finite alphabet such that for every absolutely summable relative interaction, every translation-invariant relative Gibbs measure is a relative equilibrium measure and vice versa. Neither implication is true without some assumption on the space of configurations. We note that the usual finite type condition can be relaxed to a much more general class of constraints. By "relative" we mean that both the interaction and the set of allow…
Let $σ:\boldsymbolΣ\to\boldsymbolΣ$ be the left shift acting on $ \boldsymbolΣ $, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of $σ$-invariant Borel probabilities that maximize the integral of a given locally Hölder continuous potential $ A : \boldsymbolΣ \to \mathbb R $. Under certain conditions, we are able to show not only that $A$-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to …
This paper is devoted to study ergodic optimisation problems for almost-additive sequences of functions (rather than a fixed potential) defined over countable Markov shifts (that is a non-compact space). Under certain assumptions we prove that any accumulation point of a family of Gibbs equilibrium measures is a maximising measure. Applications are given in the study of the joint spectral radius and to multifractal analysis of Lyapunov exponent of non-conformal maps.
For a large class of irreducible shift spaces $X\subset\tA^{\Z^d}$, with $\tA$ a finite alphabet, and for absolutely summable potentials $Φ$, we prove that equilibrium measures for $Φ$ are weak Gibbs measures. In particular, for $d=1$, the result holds for irreducible sofic shifts.
We establish the large deviation principle for a topological Markov shift over infinite alphabet which satisfies strong combinatorial assumptions called ``finite irreducibility'' or ``finite primitiveness''. More precisely, we assume the existence of a Gibbs state for a potential $ϕ$ in the sense of Bowen, and prove the level-2 Large Deviation Principles for the distribution of empirical means under the Gibbs state, as well as that of weighted periodic points and iterated pre-images. The rate function is written with the pressure and the free energy associated with the potential $ϕ$.
We consider the multifractal analysis of the pointwise dimension for Gibbs measures on countable Markov shifts. Our paper analyses the set of non-analytic points or phase transitions of the multifractal spectrum. By Sarig's thermodynamic formalism for countable Markov shifts and Iommi's expression of the multifractal spectrum, we apply analyticity arguments on the pressure function for the countable shift. Finally, we apply our results about the phase transitions of the multifractal spectrum to the Gauss map.
We prove that if $Σ_{\mathbf A}(\mathbb N)$ is an irreducible Markov shift space over $\mathbb N$ and $f:Σ_{\mathbf A}(\mathbb N) \rightarrow \mathbb R$ is coercive with bounded variation then there exists a maximizing probability measure for f, whose support lies on a Markov subshift over a finite alphabet. Furthermore, the support of any maximizing measure is contained in this same compact subshift. To the best of our knowledge, this is the first proof beyond the finitely primitive case on the general irreducible non-compact setting. It's also noteworthy that our technique works for the full…
Starting from the full--shift on a finite alphabet $A$, mingling some symbols of $A$, we obtain a new full shift on a smaller alphabet $B$. This amalgamation defines a factor map from $(A^{\mathbb N},T_A)$ to $(B^{\mathbb N},T_B)$, where $T_A$ and $T_B$ are the respective shift maps. According to the thermodynamic formalism, to each regular function (`potential') $ψ:A^{\mathbb N}\to{\mathbb R}$, we can associate a unique Gibbs measure $μ_ψ$. In this article, we prove that, for a large class of potentials, the pushforward measure $μ_ψ\circπ^{-1}$ is still Gibbsian for a potential $ϕ:B^{\mathbb …
This paper is devoted to study how do thermodynamic formalism quantities varies for time changes of suspension flows defined over countable Markov shifts. We prove that in general no quantity is preserved. We also make a topological description of the space of suspension flows according to certain thermodynamic quantities. For example, we show that the set of suspension flows defined over the full shift on a countable alphabet having finite entropy is open. Of independent interest might be a set of analytic tools we use to construct examples with prescribed thermodynamic behaviour.
We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures---a version of de Finetti's Theorem. Generalizing the other half,…
For a dynamical system, we study the set of points $\cal W$ whose orbit approximates any chosen point at certain specified rates. Our basic setting is that of left shift acting on topological Markov chains endowed with a local weak Gibbs measure. Our rates of recurrence are so fast that the corresponding set $\cal W$ has measure zero, but we obtain a generalized Bowen formula for Carathéodory dimension. For the case of Markov transformations with countable partition and big image (BI) property a Bowen-type formula is obtained for the Hausdorff dimension of those exceptional sets. In particular…
Consider a Hölder continuous potential $ϕ$ defined on the full shift $A^\nn$, where $A$ is a finite alphabet. Let $X\subset A^\nn$ be a specified sofic subshift. It is well-known that there is a unique Gibbs measure $μ_ϕ$ on $X$ associated to $ϕ$. Besides, there is a natural nested sequence of subshifts of finite type $(X_m)$ converging to the sofic subshift $X$. To this sequence we can associate a sequence of Gibbs measures $(μ_ϕ^m)$. In this paper, we prove that these measures weakly converge at exponential speed to $μ_ϕ$ (in the classical distance metrizing weak topology). We also establish…
We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact. We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras …
We define a notion of (one-sided) shift spaces over infinite alphabets. Unlike many previous approaches to shift spaces over countable alphabets, our shift spaces are compact Hausdorff spaces. We examine shift morphisms between these shift spaces, and identify three distinct classes that generalize the shifts of finite type. We show that when our shift spaces satisfy a property that we call "row-finite", then shift morphisms on them may be identified with sliding block codes. As applications, we show that if two (possibly infinite) directed graphs have edge shifts that are conjugate, then the …
We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible two-sided topological Markov shifts with countable alphabe…
A theorem of Gao, Jackson and Seward, originally conjectured to be false by Glasner and Uspenskij, asserts that every countable group admits a $2$-coloring. A direct consequence of this result is that every countable group has a strongly aperiodic subshift on the alphabet $\{0,1\}$. In this article, we use Lovász local lemma to first give a new simple proof of said theorem, and second to prove the existence of a $G$-effectively closed strongly aperiodic subshift for any finitely generated group $G$. We also study the problem of constructing subshifts which generalize a property of Sturmian seq…
We demonstrate the equivalence of two definitions of a Gibbs measure on a subshift over a countable group, namely a conformal measure and a Gibbs measure in the sense of the Dobrushin-Lanford-Ruelle (DLR) equations. We formulate a more general version of the classical DLR equations with respect to a measurable cocycle, which reduce to the classical equations when the cocycle is induced by an interaction or a potential, and show that a measure satisfying these equations must be conformal. To ensure the consistency of these results with earlier work, we review methods of constructing an interact…
In this note we propose an alternative definition for sliding block codes between shift spaces. This definition coincides with the usual definition in the case that the shift space is defined on a finite alphabet, but it encompass a larger class of maps when the alphabet is infinite. In any case, the proposed definition keeps the idea that a sliding block code is a map with a local rule. Using this new definition we prove that the Curtis-Hedlund-Lyndon Theorem always holds for shift spaces over countable alphabets.
Consider a compact metric space $(M, d_M)$ and $X = M^{\mathbb{N}}$. We prove a Ruelle's Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in [Bull. Braz. Math. Soc. 45 (2014), pp. 53-72] which are defined from a continuous function $A : M \times M \to \mathbb{R}$ that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle $S^1$. Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associa…
We identify a class of hyperbolic transcendental entire maps and we prove that some of its elements generate a class of potentials for which exhibit a conformal and invariant probability Gibbs measure. The methods and techniques from the thermodynamic formalism can be extended to this class of potentials. To complement this study we highlight that the dynamics of such a map on some subset of the Julia set is conjugated to the shift map over a code space with countable alphabet and the euclidean metric on the complex plane induces a metric on the symbolic space which is not compatible with the …
Given a countable sofic group $Γ$, a finite alphabet $A$, a subshift $X \subseteq A^Γ$, and a potential $ϕ: X \to \mathbb{R}$, we give sufficient conditions on $X$ and $ϕ$ for expressing, in the uniqueness regime, the sofic entropy of the associated Gibbs measure $μ$ as the limit of the Shannon entropies of some suitable finite systems approximating $Γ\curvearrowright (X,μ)$. Next, we prove that if $μ$ satisfies strong spatial mixing, then the sofic pressure admits a formula in terms of the integral of a random information function with respect to any $Γ$-invariant Borel probability measure wi…
Countable Markov shifts, denoted by $Σ_A$ for a 0-1 infinite matrix $A$, are central objects in symbolic dynamics and ergodic theory. R. Exel and M. Laca introduced the corresponding operator algebras, a generalization of the Cuntz-Krieger algebras for infinite countable alphabet, and the set $X_A$, a kind of Generalized Markov Shift (GMS) that coincides with $Σ_A$ in the locally compact case. The set $Σ_A$ is dense in $X_A$, and its complement, a set of finite allowed words, is dense in $X_A$ when non-empty. We develop the thermodynamic formalism for $X_A$, introducing the notion of conformal…
Consider a subshift over a finite alphabet, $X\subset Λ^{\mathbb Z}$ (or $X\subsetΛ^{\mathbb N_0}$). With each finite block $B\inΛ^k$ appearing in $X$ we associate the empirical measure ascribing to every block $C\inΛ^l$ the frequency of occurrences of $C$ in $B$. By comparing the values ascribed to blocks $C$ we define a metric on the combined space of blocks $B$ and probability measures $μ$ on $X$, whose restriction to the space of measures is compatible with the weak-$\star$ topology. Next, in this combined metric space we fix an open set $\mathcal U$ containing all ergodic measures, and we…
We show that a shift space on a finite alphabet with a non-uniform specification property can be modeled by a strongly positive recurrent countable-state Markov shift to which every equilibrium state lifts. In addition to uniqueness of the equilibrium state, this gives strong statistical properties including the Bernoulli property, exponential decay of correlations, central limit theorem, and analyticity of pressure, which are new even for uniform specification. We give applications to shifts of quasi-finite type, synchronised and coded shifts, and factors of beta-shifts and S-gap shifts.
In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures. One of the main results of the article is to show that a probability is DLR-Gibbs (associated to a continuous translation invariant specification), if and only if, is an eigenprobability for the transpose of the Ruelle operator. Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the pr…
We show that the image of a subshift $X$ under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, resp. a sofic subshift, if and only if so is $X$. Similarly, let $G$ be a countable monoid and let $A$, $B$ be Artinian modules over a ring. We prove that for every closed subshift submodule $Σ\subset A^G$ and every injective $G$-equivariant uniformly continuous module homomorphism $τ\colon Σ\to B^G$, a subshift $Δ\subset Σ$ is of finite type, resp. sofic, if and only if so is the image $τ(Δ)$. Generaliza…
In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space). We compare and analyze these "in principle" distinct notions: DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of the Ruelle operator (also called conformal measures). Among other things we extended the classical notion of a Gibbsian specification on $\mathbb{N}$ in such way that the similarity of many results in Statistical Mechanics and Dynamical System becomes apparent. One of our main result claims…
Consider a topologically transitive countable Markov shift and, let $f$ be a summable potential with bounded variation and finite Gurevic pressure. We prove that there exists an equilibrium state $μ_{tf}$ for each $t > 1$ and that there exists accumulation points for the family $(μ_{tf})_{t>1}$ as $t \to \infty$. We also prove that the Kolmogorov-Sinai entropy is continuous at $\infty$ with respect to the parameter $t$, that is $\lim_{t \to \infty} h(μ_{tf})=h(μ_{\infty})$, where $μ_{\infty}$ is an accumulation point of the family $(μ_{tf})_{t>1}$. These results do not depend on the existence …
We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $r$ expansions, and their various generalisations: generalised Lüroth series expansions and $β$-expansions. In fact, we consider subshifts over a countable alphabet generated by all possible expansions of numbers in $[0,1)$. Then normal numbers correspond to generic points of shift-invariant measures. It turns out that for these subshifts the set of generic points for a shift-invariant probability measur…
This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. We give a unified model for all continuous maps on a metric space, by representing a map through a general subshift over usually an uncountable alphabet. It is shown that at most the second order representation is enough for a continuous map. In particular, it is shown that the dynamics of one-dimensional continuous maps to a great extent can be transformed to the study of subshift structure of a general symbolic dynamics system. By introducing distillations, partial …
