Amenable群作用动力系统的熵与大偏差

发布时间：2018-01-27 09:29

本文关键词： amenable群变分原理 Bowen熵 SMB定理局部熵通有点集大偏差　出处：《南京师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本文主要研究的是amenable群作用动力系统的Bowen拓扑熵和大偏差公式.我们对紧致度量空间上的amenable群作用动力系统引入了 Bowen拓扑熵的定义,通过几何测度论的方法,我们建立了 amenable群作用下Bowen拓扑熵的变分原理:紧致子集上的Bowen拓扑熵等于该集合上满测度的Borel概率测度的下局部测度熵的上确界.同时我们还证明了关于局部测度熵的熵公式.利用这些结果,我们对amenable群作用动力系统证明了:1.全空间上的Bowen拓扑熵等于系统的拓扑熵;2.不变Borel概率测度的测度熵小于等于它的任何全测度子集上的Bowen拓扑熵;3.遍历测度的测度熵等于该测度在其通有点集上的Bowen拓扑熵.这将Bowen 1973年的经典结果推广到了amenable群作用的框架下.此外,我们还研究了某些与遍历平均相关的集合的测度的指数增长率,得到了 amenable群作用情形下的几个大偏差定理.
[Abstract]:In this paper, we mainly study the Bowen topological entropy and large deviation formula of amenable group interacting dynamical system. We introduce the amenable group action dynamical system on compact metric space. Definition of Bowen topological entropy. By means of geometric measure theory. The variational principle of Bowen topological entropy under the action of amenable group is established. The Bowen topological entropy on a compact subset is equal to the upper bound of the lower local measure entropy of the Borel probability measure of the full measure on the set. At the same time, we also prove the entropy formula of the local measure entropy. Fruit. We prove that the Bowen topological entropy on the whole space is equal to the topological entropy of the system for the dynamical system of amenable group action. 2. The measure entropy of invariant Borel probability measure is less than the Bowen topological entropy on any whole measure subset of invariant measure; 3. The measure entropy of ergodic measure is equal to the Bowen topological entropy of the measure on its common set. In 1973, the classical results are extended to the framework of amenable group action. We also study the exponential growth rate of the measure of some sets related to ergodic average, and obtain some large deviation theorems in the case of amenable group action.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O19

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