Jerome Buzzi, Mike M. Boyle, Ricardo Gomez
Almost isomorphism is an equivalence relation on countable state Markov shifts which provides a strong version of Borel conjugacy; still, for mixing SPR shifts, entropy is a complete invariant of almost isomorphism. In this paper, we establish a class of potentials on countable state Markov shifts whose thermodynamic formalism is respected by almost isomorphism.
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Countable state Markov shifts are a natural generalization of the well-known subshifts of finite type. They are the subject of current research both for their own sake and as models for smooth dynamical systems. In this paper, we investigate their almost isomorphism and entropy conjugacy and obtain a complete classification for the especially important class of strongly positive recurrent Markov shifts. This gives a complete classification up to entropy conjugacy of the natural extensions of smooth entropy expanding maps, including all smooth interval maps with non-zero topological entropy.
Isomorphism of pointed minimal systems is not classifiable by countable structures
本文证明极小系统及带点极小系统的拓扑共轭关系无法通过可数结构进行 Borel 归约分类，揭示了其分类复杂性的本质。
We study ergodic properties of certain piecewise smooth two-dimensional systems by constructing countable Markov partitions. Using thermodynamic formalism we prove exponential decay of correleations.
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 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.
We study hyperbolic attractors of some dynamical systems with apriori given countable Markov partitions. Assuming that contraction is stronger than expansion we construct new Markov rectangles such that their crossections by unstable manifolds are Cantor sets of positive Lebesgue measure. Using new Markov partitions we develop thermodynamical formalism and prove exponential decay of correlations and related properties for certain Hölder functions. The results are based on the methods developed by Sarig.
Countable Markov shifts with exponential mixing
本文针对可数马尔可夫移位上的势，给出其强正递归性与相关性指数衰减等价的一组条件，并通过量化移位在边界处的行为完成证明，揭示了两者的等价关系。
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.
We consider topological Markov chains (also called Markov shifts) on countable graphs. We show that a transient graph can be extended to a recurrent graph of equal entropy which is either positive recurrent of null recurrent, and we give an example of each type. We extend the notion of local entropy to topological Markov chains and prove that a transitive Markov chain admits a measure of maximal entropy (or maximal measure) whenever its local entropy is less than its (global) entropy.
Pressure at infinity on countable Markov shifts
本文研究了可数马尔可夫移位上无穷远处的压力，建立了压力的上半连续性结果，并探讨了平衡态和最大化测度的存在条件。
For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and the measure of maximal entropy in terms of the difference of their entropies. This extends an earlier result for subshifts of finite type, due to Kadyrov. We provide a similar bound for equilibrium measures of strongly positively recurrent potentials, in terms of the pressure difference. For measures with nearly maximal entropy, we have new, and sharp, bounds. The strong positive recurrence condition is necessary.
Cover time for countable Markov shifts
研究了可数全移位中盖尔伯格测度下的轨道覆盖速率，证明其与最小球测度相关，并对度量变化敏感。
We study the equilibrium behaviour of a two-sided topological Markov shift with a countable number of states. We assume the potential associated with this shift is Walters with finite first variation and that the shift is topologically transitive. We show the equilibrium measure of the system is Bernoulli up to a period. In the process we generalize several theorems on countable Markov shifts. We prove a variational principle and the uniqueness of equilibrium measures. A key step is to show that functions with Walters property on a two-sided shift are cohomologous to one-sided functions with t…
Dense periodic optimization for countable Markov shift via Aubry points
本文研究了可数马尔可夫移位中密集周期优化问题，通过Aubry点概念证明了局部Hölder连续势函数的最大测度存在性。
Gurevic pressure and equidistribution for amenable extensions of countable state Markov shifts
本文证明了可均斜积扩张的加权等分布定理，并建立了Gurevic压力与阿贝尔化系统的一致性，为等分布结果提供了关键证明。
Let $Y$ be a topological Markov chain with finite leading and follower sets. Special flow over $Y$ whose height function depends on the time zero of elements of $Y$ is constructed. Then a formula for computing the entropy of this flow will be given. As an application, we give a lower estimate for the entropy of a class of geodesic flows on the modular surface. We also give sufficient conditions to guarantee the existence of a measure with maximal entropy.
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…
We show that strongly positively recurrent Markov shifts (in particular shifts of finite type) are classified up to Borel conjugacy by their entropy, period and their numbers of periodic points.
In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of …
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…
In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain regularity assumptions on the potential the Birkhoff spectrum is real analytic. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.
This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions …
We study the pointwise perturbations of countable Markov maps with infinitely many inverse branches and establish the following continuity theorem: Let $T_k$ and $T$ be expanding countable Markov maps such that the inverse branches of $T_k$ converge pointwise to the inverse branches of $T$ as $k \to \infty$. Then under suitable regularity assumptions on the maps $T_k$ and $T$ the following limit exists: $$\lim_{k \to \infty} \dim_\mathrm{H} \{x : θ_k'(x) \neq 0\} = 1,$$ where $θ_k$ is the topological conjugacy between $T_k$ and $T$ and $\dim_\mathrm{H}$ stands for the Hausdorff dimension. This…
Level-2 large deviation principle for countable Markov shifts without Gibbs states
本文在无Bowen吉布斯态假设下，通过诱导充分条件保证周期构型Borel概率测度列的指数紧性，建立了单侧可数全移位的2级大偏差原理，并将其应用于Rényi映射连分数展开，得到对应大偏差原理及二次无理数的加权等分布结果。
Let $T$ be an expanding Markov map with a countable number of inverse branches and a repeller $Λ$ contained within the unit interval. Given $α\in \R_+$ we consider the set of points $x \in Λ$ for which $T^n(x)$ hits a shrinking ball of radius $e^{-nα}$ around $y$ for infinitely many iterates $n$. Let $s(α)$ denote the infimal value of $s$ for which the pressure of the potential $-s\log|T'|$ is below $s α$. Building on previous work of Hill, Velani and Urbański we show that for all points $y$ contained within the limit set of the associated iterated function system the Hausdorff dimension of th…
The Lyapunov spectrum as the Newton-Raphson method for countable Markov interval maps
本文研究具有无限多分支的MRL区间映射，证明其Lyapunov谱可表示为拓扑压的Legendre变换，并与应用于该拓扑压的Newton-Raphson方法直接相关，推广了已有结论。
This work is concerned with competitive Lotka-Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective of the paper is to reduce the computational complexity by using the two-time-scale systems. Because existence and uniqueness as well as continuity of solutions for Lotka-Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the genera…
In this paper we generalize the recently introduced concept of fair measure (M. Misiurewicz and A. Rodrigues, Counting preimages. Ergod. Th. & Dynam. Sys. 38 (2018), no. 5, 1837 -- 1856). We study transitive countable state Markov shift maps and extend our results to a particular class of interval maps, Markov and mixing interval maps. Finally, we move beyond the interval and look for fair measures for graph maps.
We show that the (Gurevich) topological entropy for the countable Markov shift associated with an infinite transition matrix $A$ coincides with the non-commutative topological entropy for the Exel--Laca algebra associated with $A$, under certain conditions on $A$. An important example satisfying the conditions is the renewal shift, which is not locally finite. We also pose interesting questions for future research on non-commutative topological entropy for non-locally finite transition matrices.
In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.
It is well known that the space of invariant probability measures for transitive sub-shifts of finite type is a Poulsen simplex. In this article we prove that in the non-compact setting, for a large family of transitive countable Markov shifts, the space of invariant sub-probability measures is a Poulsen simplex and that its extreme points are the ergodic invariant probability measures together with the zero measure. In particular we obtain that the space of invariant probability measures is a Poulsen simplex minus a vertex and the corresponding convex combinations. Our results apply to finite…
We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in Part I. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semi-continuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov--Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergod…
Using elementary methods, we prove that for a countable Markov chain $P$ of ergodic degree $d > 0$ the rate of convergence towards the stationary distribution is subgeometric of order $n^{-d}$, provided the initial distribution satisfies certain conditions of asymptotic decay. An example, modelling a renewal process and providing a markovian approximation scheme in dynamical system theory, is worked out in detail, illustrating the relationships between convergence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Ma…
We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért--Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin ent…
Quasi-invariant measures concentrating on countable structures
本文研究了可数结构上准不变测度的存在性，发现其与结构的代数性密切相关，提出了高度代数性的定义，并展示了相关测度的性质。
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.
A Large deviations principle at zero temperature for stationary Markov equilibrium states on countable Markov shifts
本文针对可数 Markov 移位上满足条件的局部常值势，证明平稳 Markov 平衡态族在零温极限下的第一层级大偏差原理，并推广至双侧情形。
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 …
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 $ϕ$.
Discrete group extensions of countable Markov shifts and pressure gap
本文将关于顺从性和增长间隙的若干经典结果推广到强正递归的可数马尔可夫移位，利用扭曲Patterson--Sullivan测度的方法，并给出扭曲测度的背景及一个例子的估计。
We show that any finite-entropy, countable-valued finitary factor of an i.i.d process can also be expressed as a finitary factor of a finite-valued i.i.d process whose entropy is arbitrarily close to the target process. As an application, we give an affirmative answer to a question of van den Berg and Steif about the critical Ising model on $\mathbb{Z}^d$. En route, we prove several results about finitary isomorphisms and finitary factors. Our results are developed in a new framework for processes invariant to a permutation group of a countable set satisfying specific properties. This new fram…
Random attractors on countable state spaces
本文通过随机动力系统研究可数状态空间上离散时间 Markov 链的同步行为，在温和条件下证明了随机吸引子的存在唯一性及正向与拉回吸引的等价性，并利用绝缘集与同步集刻画了其大小。
We introduce the notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words. Firstly, the scaling function allows a direct access to important thermodynamical quantities, which are usually given only implicitly by certain identities involving the classically defined pressure. In this context we generalise Savchenko's definition of entropy for special flows to a corresponding notion of topological pressure and show that this new notion coincides with the induced pressure for a large class of Hö…
A new proof of finitary isomorphism for Markov chains
本文为Rudolph关于可数状态混合马尔可夫链与IID过程有限同构的结论提供了一个简短直接的新证明，并将适用范围扩展到有限熵与无限熵过程。
Non-compact spaces of invariant measures
本文研究传递可数马尔可夫位移不变概率测度空间的紧化，证明其仿射同胚于Poulsen单纯形，并将遍历测度空间同胚于$\ell_2$的结果推广到非紧情形，同时给出对偶变分原理的应用。
L. A. Bunimovich and B. Z. Webb developed a theory for transforming a finite weighted graph while preserving its spectrum, referred as isospectral reduction theory. In this work we extend this theory to a class of operators on Banach spaces that include Markov type operators. We apply this theory to infinite countable weighted graphs admitting a finite structural set to calculate the stationary measures of a family of countable Markov chains.
In this article, we study countable sofic shifts of Cantor-Bendixson rank at most 2. We prove that their conjugacy problem is complete for GI, the complexity class of graph isomorphism, and that the existence problems of block maps, factor maps and embeddings are NP-complete.
$\overline{d}$-continuity for countable state shifts
本文研究了可数状态移位中$\overline{d}$-连续性的扩展，证明了Hölder连续势函数与平衡态之间的映射连续性，并探讨了强正回复势函数的规范性连续性。
Measures of maximal entropy for Markovian dynamics on the Gehman dendrite
研究盖曼枝状图上特定传递动力系统，通过引入马尔可夫映射，用可数马尔可夫图编码动力学，利用相关准则控制最大熵测度存在性，给出任意给定系统的两个马尔可夫扰动，一个有唯一最大熵测度，一个没有。
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 alp…
Convergence of pushforward measures for certain countably piecewise linear Markov maps
本文受2022年Miklós Schweitzer竞赛问题启发，研究可数分段线性马尔可夫映射，通过引入$\ell$-马尔可夫划分和符号动力学方法，证明了绝对连续测度前推测度的收敛定理，并用于证明遍历性质及解决原竞赛问题。
Existence of the zero temperature limit of equilibrium states on topologically transitive countable Markov shifts
本文证明了拓扑传递可数马尔可夫移位上满足有限 Gurevich 压力和可和条件的马尔可夫势的平衡态在零温极限下的存在性，并给出了更一般条件下的例子。
Strong Orbit Equivalence in Cantor dynamics and simple locally finite groups
本文研究极小同胚相关的可数局部有限群，证明其同构关系为Borel $S_\infty$-作用的泛关系，并为强轨道等价的刻画提供动力学途径。
Recurrence to rare events for all points in countable Markov shifts
研究了可数马尔可夫转移中罕见事件的复发行为，揭示了不同点的返回时间统计分布特性及极限定律分类。
Locally conformal expanding Actions: Markov Partition and Thermodynamic of the induced skew product
本文研究了由有限个 $C^{1+α}$ 共形局部微分同胚生成的拓扑混合局部共形半群作用，提出了满足有限像和有限环性质的可数 Markov 分割，并基于诱导映射等价于可数型子移位的塔构造，证明了可提升测度的存在性及诱导斜积的热力学形式。
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…
Maximizing measures for countable alphabet shifts via blur shift spaces
本文研究了可数字母移位上上半连续势函数的最大化测度存在性条件，引入模糊移位概念并扩展至更一般的移位空间，提供凸刻画和紧致性。
Multifractal analysis for Markov interval maps with countably many branches
本文研究可数分支马尔可夫区间映射的Birkhoff平均多重分形分解，推广至拓扑混合可数马尔可夫移位编码，并揭示导数下确界不趋于无穷时（尤其有限拓扑熵情形）的新行为。
We consider the natural definition of DLR measure in the setting of $σ$-finite measures on countable Markov shifts. We prove that the set of DLR measures contains the set of conformal measures associated with Walters potentials. In the BIP case, or when the potential normalizes the Ruelle's operator, we prove that the notions of DLR and conformal coincide. On the standard renewal shift, we study the problem of describing the cases when the set of the eigenmeasures jumps from finite to infinite measures when we consider high and low temperatures, respectively. For this particular shift, we prov…
We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of the Hjorth's conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube hom…
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