标题： 非平稳马尔可夫分划和多维连分数算法 显示英文标题

标题： Nonstationary Markov Partitions and Multidimensional Continued Fraction Algorithms

摘要： 众所周知，根据Sinaĭ和Bowen的结果，双曲环面自同态具有马尔可夫划分。我们的目标是将这一概念推广到非平稳情况，即为非平稳的环面自同态序列分配马尔可夫划分。特别强调的是由强收敛的多维连分数算法产生的环面自同态序列。算法的收敛性用Pisot型条件来表达，该条件为非平稳动力系统提供了双曲性。对于多维连分数映射，我们首先考虑其自然扩展，其轨道由行列式为$\pm 1$的双无限矩阵序列给出。双曲性性质使我们可以将该自然扩展的几乎所有轨道解释为Anosov映射族，即具有明确稳定和不稳定流形的双无限环面自同态序列。我们证明这个Anosov映射族 admits 一个显式的双无限非平稳马尔可夫划分序列。为了获得马尔可夫划分的原子，必须在Anosov映射族上叠加一种以替换和$\mathcal{S}$-进动力系统形式表达的组合结构。特别是，马尔可夫划分的原子是通过悬挂$\mathcal{S}$-进Rauzy分形定义的$\mathcal{S}$-进动力系统的几何实现。然后这些马尔可夫划分提供了一个非平稳边移位的符号模型，用于Anosov映射族。作为指导示例，允许显式实现结果，我们使用与各种版本的Brun连分数算法相关的二维和三维环面上的Anosov映射族。 显示英文摘要

摘要： It is well known from results of Sina\u{\i} and Bowen that a hyperbolic toral automorphism admits a Markov partition. Our aim is to generalize this concept to the nonstationary case, i.e., we associate Markov partitions to nonstationary sequences of toral automorphisms. Special emphasis is placed on sequences of toral automorphisms produced by strongly convergent multidimensional continued fraction algorithms. The convergence of the algorithms is expressed in terms of a Pisot type condition which yields hyperbolicity for the nonstationary dynamics. For a multidimensional continued fraction map, we first consider its natural extension, whose orbits are given by bi-infinite sequences of matrices with determinant $\pm 1$. The hyperbolicity property allows us to interpret almost every orbit of this natural extension as an Anosov mapping family, i.e., as a bi-infinite sequence of toral automorphisms with well-defined stable and unstable manifolds. We prove that this Anosov mapping family admits a bi-infinite sequence of explicit nonstationary Markov partitions. To obtain the atoms of the Markov partitions, a combinatorial structure, expressed in terms of substitutions and $\mathcal{S}$-adic dynamical systems, has to be superimposed on the Anosov mapping family. In particular, the atoms of the Markov partitions are geometric realizations of $\mathcal{S}$-adic dynamical systems, defined by suspensions of $\mathcal{S}$-adic Rauzy fractals. These Markov partitions then provide a symbolic model as a nonstationary edge shift for the Anosov mapping family. As a guiding example, allowing explicit realization results, we use Anosov mapping families on 2- and 3-dimensional tori associated to various versions of the Brun continued fraction algorithm.