论文标题
简单系统的动力学特性和持续的分数算法
Dynamical properties of simplicial systems and continued fraction algorithms
论文作者
论文摘要
我们提出了一种新的观点，该观点是关于受Rauzy诱导启发的多维持续分数算法的提议。这种算法的通用行为在这里被描述为我们称为Simplicial System的图表上的随机步行。这些系统为随机步行提供了一家示例，其中有限维矢量记录的内存。 我们在这些图表上介绍了一个通用标准，该标准与许多其他动力学属性一起诱导了怪异性。特别是，在计算Brun，Selmer和Arnoux-rauzy-Poincare算法的表示之后，它为这些经典示例提供了统一的怪异性证明，以及新的结果，例如在规范悬架上的最大熵度量的独特性。 这些物体还为某些分形套件（例如Rauzy垫片）带来了新的视角。我们显示了在这种形式主义中描述的分形的一般上的上限以及其最大熵度量的构造。这特别意味着所有维度的rauzy垫圈的尺寸严格小于其环境空间，并且在维度和渐近结果上的范围更加清晰。
We propose a new point of view on multidimensional continued fraction algorithms inspired by Rauzy induction. The generic behaviour of such an algorithm is described here as a random walk on a graph that we call simplicial system. These systems provide a family of examples for random walks with memory recorded by a finite dimensional vector. We introduce a general criterion on these graphs that induces ergodicity together with a bundle of many other dynamical properties. In particular, after computing the representation of Brun, Selmer and Arnoux-Rauzy-Poincare algorithm in this formalism, it provides a unified proof of ergodicity for these classical examples as well as new results such as uniqueness of the measure of maximal entropy on a canonical suspension. These objects also bring a new perspective to some fractal sets such as Rauzy gaskets. We show general explicit upper bound on Hausdorff dimensions of fractals described in this formalism as well as a construction of their measure of maximal entropy. This implies in particular that the Rauzy gasket in all dimensions has Hausdorff dimension strictly smaller than its ambient space, as well as sharper bounds on the dimension and an asymptotic result.
