V. Baladi, B. Vallee
We extend Dolgopyat's bounds on iterated transfer operators to suspensions of interval maps with infinitely many intervals of monotonicity.
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A construction is given of Markov partitions for some rational maps, which persist over regions of parameter space, not confined to single hyperbolic components. The set on which the Markov partition exists, and its boundary, are analysed.
Markov partitions for non-transitive expansive flows
该研究针对三维非传递膨胀流，构造了对应的马尔可夫划分，丰富了动力系统领域相关研究内容。
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.
We give an explicit geometric description of the $\times2,\times3$ system, and use his to study a uniform family of Markov partitions related to those of Wilson and Abramov. The behaviour of these partitions is stable across expansive cones and transitions in this behaviour detects the non-expansive lines.
Markov partitions persisting in a neighbourhood of hyperbolic components of rational maps were constructed under the condition that closures of Fatou components are disjoint in \cite{R1}. Given such a partition, we characterize all nearby hyperbolic components in terms of the symbolic dynamics. This means we can count them, and also obtain topological information. We also determine extra conditions under which all nearby type IV hyperbolic components are given by matings. These are probably the first known results of this type.
This monographic short book is intended to a brief introduction in the classic topic of the theory of Hiperbolic Dynamical Systems, for Spanish speaking students of an undergraduate course in Mathematics. We revisit the classic definition of Markov Partitions for Anosov diffeomorphisms on compact manifolds. We focus on the theorem of Sinai (Func. Anal. and its Applic. 2, 1968) stsating the existence of such partitions with arbitrarily small diameter. To prove the theorem of Sinai we restate the constructive method in Bowen, Lecture Notes in Math. 470, 1975. Finally we also relate the results, …
The decimal expansion real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by symbolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely hyperbolic automorphisms of the two…
In the paper we give an introduction to Anosov diffeomorphisms, ways to represent their chaotic properties and some historical remarks on this subject. A complete classification of hyperbolic linear automorphisms of 2-torus is presented. We introduce a notion of pre-Markov partition for such automorphisms and give their classification and an algorithm for their construction.
For 2-d hyperbolic systems with singularities, statistical properties are rather difficult to establish because of the fragmentation of the phase space by singular curves. In this paper, we construct a Markov partition of the phase space with countable states for a general class of hyperbolic systems with singularities. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentia…
Markov partitions designed by Sinai(1968) and Bowen(1970) proved to be an efficient tool for descibing statistical properties of uniformly hyperbolic systems. For hyperbolic systems with singularities, in particular, for hyperbolic billiards the construction of a Markov partition by Bunimovich and Sinai(1980) was a delicate and hard task. Therefore later more and more flexible and simple variants of Markov partitions appeared: Markov sieves (Bunimovich-Chernov-Sinai, 1990), Markov towers (Young, 1998), standard pairs (Dolgopyat). This remarkable evolution of Sinai's original idea is surveyed i…
In the present paper we investigate two special types of non-normal numbers. On the one hand we call a number extremely non-normal if the set of accumulation points of its frequency of blocks vector is the full set of shift invariant probability vectors. On the other hand we call a number having maximal oscillation frequency if for any fixed block the set of accumulation points of its frequency vector is the full set of possible probability vectors. The goal is to investigate the Baire category of these numbers for infinite Markov partitions.
Nonstationary Markov Partitions and Multidimensional Continued Fraction Algorithms
本文研究了非平稳马尔可夫划分与多维连续分数算法的关联，通过皮索型条件证明了非平稳动力系统的双曲性，并构建了基于符号动力系统的非平稳边移位模型。
Let T be a C^2-expanding self-map of a compact, connected, smooth, Riemannian manifold M. We correct a minor gap in the proof of a theorem from the literature: the set of points whose forward orbits are nondense has full Hausdorff dimension. Our correction allows us to strengthen the theorem. Combining the correction with Schmidt games, we generalize the theorem in dimension one: given a point x in M, the set of points whose forward orbit closures miss x is a winning set.
Primitive Geometric Markov Partitions for pseudo-Anosov Homeomorphisms
本文提出了一种基于简单组合准则的有效算法，用于构造伪Anosov同胚的原始几何马尔可夫划分，并证明了其规范性和有限性。
Smoothness of Markov Partitions for Expanding Toral Endomorphisms
研究二维扩张流形内映射的光滑性条件，证明其存在光滑马尔可夫划分当且仅当对应整数矩阵的某次幂可对角化且具有整数特征值，同时探讨不同光滑性行为及马尔可夫划分边界 Hausdorff 维数估计。
It is well-known that hyperbolic flows admit Markov partitions of arbitrarily small size. However, the constructions of Markov partitions for general hyperbolic flows are very abstract and not easy to understand. To establish a more detailed understanding of Markov partitions, in this paper we consider the geodesic flow on Riemann surfaces of constant negative curvature. We provide a rigorous construction of Markov partitions for this hyperbolic flow with explicit forms of rectangles and local cross sections. The local product structure is also calculated in detail.
Experimental Results on Potential Markov Partitions for Wang Shifts
本文针对三种 Wang 瓷砖原型集（阶数 24、16 和 11）提出潜在马尔可夫划分的实验生成方法，并应用 Labbé 理论分析，以编码相应非周期平铺。
We build a Markov partition of the Feigenbaum dynamical plane, and a system of "external rays", which are invariant under rescaling. Applications to geometry of the Feigenbaum map are given.
A classification of pseudo-Anosov homeomorphisms via geometric Markov partitions
本文利用带竖直方向的几何 Markov 配分为 pseudo-Anosov 同胚赋予几何类型，证明其为拓扑共轭的完全不变量，并给出可实现性判据及判定两类共轭的算法。
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。
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。
Geometric Markov partitions for pseudo-Anosov homeomorphisms with prescribed combinatorics
本文研究伪阿诺索夫同胚的几何马尔可夫分割构造与细化，提出从单点出发的算法化构造，并引入二元细化、s和U边界细化及角点细化，为伪阿诺索夫同胚的拓扑共轭分类提供组合工具。
A classification of pseudo-Anosov homeomorphisms I: the geometric type is a complete conjugacy invariant
本文通过几何类型证明伪Anosov同胚的拓扑共轭性等价于其几何Markov分割类型相同。
We study partially hyperbolic sets $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the center-unstable direction $E^{cu}$ is non-uniformly expanding on some local unstable disk. We prove that the (stretched) exponential decay of recurrence times for an induced scheme can be deduced under the assumption of (stretched) exponential decay of the time that typical points need to achieve some uniform expanding in the center-unstable direction. This extends a result by Alves and Pinheiro to the (stretched) exponential case. As an application of our…
An Algorithmic Classification of Generalized Pseudo-Anosov Homeomorphisms via Geometric Markov Partitions
本文利用几何 Markov 分区的几何类型作为完全不变量，给出实现判据与判定算法，实现广义 pseudo-Anosov 同胚拓扑共轭类的算法分类。
Symbolic dynamics, shadowing and representation of real numbers with some countably piecewise linear Markov maps
受2022年施韦策竞赛问题启发，研究可数分段线性马尔可夫映射，引入ℓ-马尔可夫分割并关联符号动力学与马尔可夫转移，证明竞赛系统的伪轨跟踪性质，研究有理数轨道及最终周期性，提供示例系统。
We call a Markov partition of a two dimensional hyperbolic toral automorphism a Berg partition if it contains just two rectangles. We describe all Berg partitions for a given hyperbolic toral automorphism. In particular there are exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same connectivity matrix (k, l, m, n).
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。
A construction of the hat tilings by a Markov partition
本文提出了一种简单构造帽子铺砌的方法，通过在特殊着色图像上叠加三角网格来读取瓷砖方向，证明了该方法能生成有效帽子铺砌，并且所有有效帽子铺砌均可由此方法生成。
We define a partition $\mathcal{P}_0$ and a $\mathbb{Z}^2$-rotation ($\mathbb{Z}^2$-action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel-Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition $\mathcal{P}_\mathcal{U}$ and a $\mathbb{Z}^2$-rotation on $\mathbb{T}^2$ whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that $\mathcal{P}_\mathcal{U}$ is a Markov partition for the $\mathbb{Z}^2$-rotation on $\math…
Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov processes with random switching between flows
本文研究了一类在流之间随机切换的分段确定性马尔可夫过程，提出了仅依赖流性质和跳跃后位置转移律的可验证条件，以保证其在有界Lipschitz距离下的指数遍历性，并为适用于生物模型的特例建立了简单判据。
We study a Markov process with two components: the first component evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of the second component. The second component is discrete and its jump rates may depend on the position of the whole process. Under regularity assumptions on the jump rates and Wasserstein contraction conditions for the underlying dynamics, we provide a concrete criterion for the convergence to equilibrium in terms of Wasserstein distance. The proof is based on a coupling argument and a weak form of …
Markovian reduction and exponential mixing in total variation for random dynamical systems
本文研究了有限维紧致相空间下由随机噪声驱动的离散时间动力系统的大时间行为，通过马尔可夫约化和Markov过程混合理论，证明了轨迹在全变差度量下收敛于唯一测度，并将该结果推广至平稳噪声驱动的系统。
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