论文标题
分段双曲图的最大熵度量的独特性和指数混合
Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps
论文作者
论文摘要
对于一类二维分段双曲线图，我们通过计算地图迭代的相位空间的最大，开放，连接的组件来提出拓扑熵的组合定义。我们证明，该数量主导了系统所有不变概率度量的理论熵，然后构建一个不变的度量，其熵等于所提出的拓扑熵。我们证明，我们的度量是对最大熵的独特度量，即它是千古的，为每个开放集提供了积极的措施，并且具有针对Hölder连续功能的相关性衰减。结果，我们还证明了周期性轨道生长速率的下限。本文中使用的主要工具是建造各向异性的Banach分布空间，相关加权转移操作员在其上具有光谱间隙。然后，我们通过将该操作员的左右最大特征向量的产物构建最大熵的度量。
For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We prove that this quantity dominates the measure theoretic entropies of all invariant probability measures of the system, and then construct an invariant measure whose entropy equals the proposed topological entropy. We prove that our measure is the unique measure of maximal entropy, that it is ergodic, gives positive measure to every open set, and has exponential decay of correlations against Hölder continuous functions. As a consequence, we also prove a lower bound on the rate of growth of periodic orbits. The main tool used in the paper is the construction of anisotropic Banach spaces of distributions on which the relevant weighted transfer operator has a spectral gap. We then construct our measure of maximal entropy by taking a product of left and right maximal eigenvectors of this operator.
