决定性动力系统中的随机现象:动力系统统计理论简介  

Stochastic phenomena in deterministic dynamical systems: Introduction to statistical properties of dynamical systems

作　　者：胡虎翼 

机构地区：[1]南方科技大学数学系,深圳518055 [2]Department of Mathematics, Michigan State University

出　　处：《中国科学：数学》2017年第12期1665-1680,共16页Scientia Sinica：Mathematica

摘　　要：本文简要介绍动力系统的统计理论研究中所得到的一些概率定律.考虑动力系统(M,f)相空间M上的函数φ,随着系统演化,函数族{φ?fn}可以看作一个过程.虽然这个过程是确定性的,但是当系统的运动具有一定的紊动性,且函数φ具有一定的正则性时,这个过程具有随机过程的很多性质,例如,服从大数定律、中心极限定理、重对数律、大偏差原理、局部极限定理和几乎确定不变性原理等.In this paper, we introduce some probability results in the study of statistical properties of dynamical systems. With the evolution of a dynamical system（M, f）, a function φ defined on the phase space M generates a process {φ ？ f^n}. Even though it is given by a deterministic system, the process can exhibit many stochastic properties whenever the system is of certain chaotic behavior and the function φ is of some regularity. Such properties include the law of large number, the central limit theorem, the law of iterated logarithm, the principle of large deviations, local limit theorems, almost sure invariant principle, etc.

关 键 词：动力系统统计性质 双曲性 物理测度 大数定律 中心极限定理 几乎确定不变原理 

分 类 号：O19[理学—数学]