周期性Coven自动机的拓扑熵

发布时间：2019-02-23 23:44

【摘要】：元胞自动机是John von Neumann于1951年正式提出的。在数学领域,1969年Hedlund首次将元胞自动机与符号动力系统建立了联系。近年来,越来越多的科研工作者投入到元胞自动机的理论研究中。作为一种特殊的离散符号动力系统,研究空间中点的运动轨迹和计算一个系统的拓扑熵是两个非常重要的课题。论文共分五章。第一章是序言。介绍了元胞自动机的历史起源、发展以及应用,同时也简要的介绍了本文的主要结果。第二章介绍了拓扑动力系统和遍历理论的一些基本概念。第三章介绍了元胞自动机的一些重要性质,同时建立了一个类似于遍历平均的自动机平均,并证明了自动机平均收敛于空间平均。第四章研究一类特殊的非线性元胞自动机-具有周期性的Coven自动机,主要是证明了当周期词的最小周期大于该词长度的一半时,对应的Coven自动机的拓扑熵都等于ln 2,从而是Li-York混沌的,并且它的最大熵测度不是唯一的。第五章主要讨论了其它非线性元胞自动机的拓扑熵。
[Abstract]:Cellular automata (CA) was formally proposed by John von Neumann in 1951. In the field of mathematics, Hedlund first established a relationship between cellular automata and symbolic dynamical systems in 1969. In recent years, more and more researchers have devoted themselves to the theoretical research of cellular automata. As a special discrete symbolic dynamical system, it is very important to study the motion trajectory of a point in space and to calculate the topological entropy of a system. The thesis is divided into five chapters. The first chapter is the preface. This paper introduces the historical origin, development and application of cellular automata, and also briefly introduces the main results of this paper. In chapter 2, some basic concepts of topological dynamical system and ergodic theory are introduced. In chapter 3, some important properties of cellular automata are introduced, and an automaton average similar to ergodic average is established, and it is proved that automata average converges to space average. In chapter 4, we study a special class of nonlinear cellular automata, Coven automata with periodicity. It is proved that the topological entropy of the corresponding Coven automata is equal to ln 2 when the minimum period of the periodic word is greater than half of the length of the word. So it is Li-York chaotic, and its maximum entropy measure is not unique. In chapter 5, the topological entropy of other nonlinear cellular automata is discussed.
【学位授予单位】：武汉大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TP301.1

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