论文标题
统一扩展的耦合地图：自一致的转移操作员和混乱的传播
Uniformly Expanding Coupled Maps: Self-Consistent Transfer Operators and Propagation of Chaos
论文作者
论文摘要
在本文中，当$ n $有限但大时，我们研究$ n $均匀扩展的耦合地图的系统。我们介绍了近似于动力学下措施演变的自洽转移操作员，并对$ n $明确量化此近似值。使用此结果，我们证明，当$ n \ rightarrow \ infty $均匀地扩展耦合地图可以满足混乱的繁殖，并表征了有限尺寸系统的绝对连续不变的度量。主要的工作假设是，扩展并不太小，尽管两者都可以是一个顺序，但相互作用的强度也不太大。与以前的方法相反，我们不需要耦合图和相互作用是相同的。允许我们描述系统的技术进步是：引入一个框架来研究沿着某些非变变叶子的条件度量的演变，其中所有估计值对维度的依赖是显式的；以及与满足指数浓度不平等的产品接近的不变类别的表征。
In this paper we study systems of $N$ uniformly expanding coupled maps when $N$ is finite but large. We introduce self-consistent transfer operators that approximate the evolution of measures under the dynamics, and quantify this approximation explicitly with respect to $N$. Using this result, we prove that uniformly expanding coupled maps satisfy propagation of chaos when $N\rightarrow \infty$, and characterize the absolutely continuous invariant measures for the finite dimensional system. The main working assumption is that the expansion is not too small and the strength of the interactions is not too large, although both can be of order one. In contrast with previous approaches, we do not require the coupled maps and the interactions to be identical. The technical advances that allow us to describe the system are: the introduction of a framework to study the evolution of conditional measures along some non-invariant foliations where the dependence of all estimates on the dimension is explicit; and the characterization of an invariant class of measures close to products that satisfy exponential concentration inequalities.
