动力系统初值敏感性、序列熵及相关问题的研究
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摘要
本文主要对初值敏感性、序列熵及相关问题进行了研究。具体安排如下：
在引言中，我们先简要的介绍了动力系统和遍历论的起源与主要研究内容，着重介绍了初值敏感性和序列熵研究的背景及发展现状。
在第二章中，我们介绍了本文涉及到的拓扑动力系统和遍历理论的一些基本概念和结论。
在第三章中，我们对熊金城教授提出的n初值敏感性，特别是对极小系统的n初值敏感性进行了研究。证明了一个极小系统为n初值敏感的当且仅当n局部proximal关系Q_n包含了一个坐标互异的元素。进一步的，我们给出了n初值敏感但不是(n+1)初值敏感(n>1)的极小系统的结构定理，说明这样的极小系统其实就是它极大等度连续因子的有限对一扩充。
在第四章中，我们引入了相对于集合初值敏感，初值敏感集(S-集)和局部proximal集(Q-集)这三个概念，并证明了：一个传递系统是初值敏感的当且仅当它有一个初值敏感集S且Card(S)≥2；传递系统中每一个S-集均为Q-集，并且当系统为极小时，其逆命题也成立；任一传递系统均存在极大几乎等度连续因子。根据S-集的势的不同，我们可以对传递系统进行更细致的分类。我们分别在极小和传递非极小的情形下给出了这种分类的刻画和具体的例子。我们证明了在传递系统中，拓扑熵集都是S-集，因此，一个传递系统如果没有势为不可数的S-集，则它的拓扑熵必为0。进一步的，我们研究了一些特殊系统的敏感性：一个传递，非极小且极小点稠密的系统必有一个势为无限的S-集；存在一个Devaney系统，它没有势为不可数的S-集。最后，我们构造一个非极小的E系统，它的S-集的势最大不超过4。
在第五章中，我们对保测系统引入了测度n初值敏感性的概念，并且证明了：如果(X，B，μ，T)为遍历的且T可逆，则(X，B，μ，T)为测度n初值敏感但非测度n+1初值敏感当且仅当h_μ~*(T)=log n，其中h_μ~*(T)为T的极大pattern熵。
在最后一章，我们研究了可数紧致度量空间上的拓扑序列熵的性质，并证明了：当空间的导集度数≤1时，则定义在其上的任意系统都是拓扑null的。而当空间X的导集度数≥2时，存在系统(X，T)使得X~d为拓扑序列熵集。
In this thesis, we mainly study the properties of sensitivity, sequence entropy andrelated problems in dynamical systems.
The thesis is divided into 6 chapters and is organized as follows:
In the Introduction, the origin, developments and main contents of the topological dynamical system and ergodic theory are presented.
In Chapter 2, the basic notions and properties on topological dynamical system and ergodic theory are recalled.
In Chapter 3, the properties of n-sensitivity, which was introduced by Xiong, are investigated, especially in the minimal case. It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct. Moreover, the structure of a minimal system which is n-sensitive but not (n + 1)-sensitive (n≥2) is determined.
In Chapter 4, notions of sensitive sets (S-sets) and regionally proximal sets (Q-sets) are introduced. It is shown that a transitive system is sensitive if and only if there is an S-set with Card(S)≥2, and for a transitive system each S-set is a Q-set. Moreover, the converse holds when (X, T) is minimal. It turns out that each transitive (X, T) has a maximal almost equicontinuous factor. According to the cardinalities of the S-sets, transitive systems are divided into several classes. Characterizations and examples are given for this classification both in minimal and transitive non-minimal settings. It is proved that for a transitive system any entropy set is an S-set, and consequently, a transitive system which has no uncountable S-sets has zero topological entropy. Moreover, it is shown that a transitive, non-minimal system with dense set ofminimal points has an infinite S-set, and there exists a Devaney chaotic system which has no uncountable S-set. Finally, a non-minimal sensitive E-system is constructed such that each its S-set has cardinality at most 4.
In Chapter 5, the notion of measurable n-sensitivity for measure preserving systems is introduced, and the relation between measurable n-sensitivity and the maximal pattern entropy is studied. It is proved that, when (X, B,μ, T) is ergodic and T is invertible, (X, B,μ, T) is measurable n-sensitive but not measurable n+1-sensitive if and only if h_μ~* (T) = log n, where h_μ~* (T) is the maximal pattern entropy of T.
In the last Chapter, the properties of topological sequence entropy for TDSs on countable compact metric spaces are discussed. It is proved that when d(X)≤1, h_(top)~s(T) = 0; and when d(X)≥2, there exists a homeomorphism T on X such that X~d is the sequence entropy set of (X, T), where d(X) and X~d are the derived degree of X and the set of all accumulation points of X respectively.
在引言中，我们先简要的介绍了动力系统和遍历论的起源与主要研究内容，着重介绍了初值敏感性和序列熵研究的背景及发展现状。
在第二章中，我们介绍了本文涉及到的拓扑动力系统和遍历理论的一些基本概念和结论。
在第三章中，我们对熊金城教授提出的n初值敏感性，特别是对极小系统的n初值敏感性进行了研究。证明了一个极小系统为n初值敏感的当且仅当n局部proximal关系Q_n包含了一个坐标互异的元素。进一步的，我们给出了n初值敏感但不是(n+1)初值敏感(n>1)的极小系统的结构定理，说明这样的极小系统其实就是它极大等度连续因子的有限对一扩充。
在第四章中，我们引入了相对于集合初值敏感，初值敏感集(S-集)和局部proximal集(Q-集)这三个概念，并证明了：一个传递系统是初值敏感的当且仅当它有一个初值敏感集S且Card(S)≥2；传递系统中每一个S-集均为Q-集，并且当系统为极小时，其逆命题也成立；任一传递系统均存在极大几乎等度连续因子。根据S-集的势的不同，我们可以对传递系统进行更细致的分类。我们分别在极小和传递非极小的情形下给出了这种分类的刻画和具体的例子。我们证明了在传递系统中，拓扑熵集都是S-集，因此，一个传递系统如果没有势为不可数的S-集，则它的拓扑熵必为0。进一步的，我们研究了一些特殊系统的敏感性：一个传递，非极小且极小点稠密的系统必有一个势为无限的S-集；存在一个Devaney系统，它没有势为不可数的S-集。最后，我们构造一个非极小的E系统，它的S-集的势最大不超过4。
在第五章中，我们对保测系统引入了测度n初值敏感性的概念，并且证明了：如果(X，B，μ，T)为遍历的且T可逆，则(X，B，μ，T)为测度n初值敏感但非测度n+1初值敏感当且仅当h_μ~*(T)=log n，其中h_μ~*(T)为T的极大pattern熵。
在最后一章，我们研究了可数紧致度量空间上的拓扑序列熵的性质，并证明了：当空间的导集度数≤1时，则定义在其上的任意系统都是拓扑null的。而当空间X的导集度数≥2时，存在系统(X，T)使得X~d为拓扑序列熵集。
In this thesis, we mainly study the properties of sensitivity, sequence entropy andrelated problems in dynamical systems.
The thesis is divided into 6 chapters and is organized as follows:
In the Introduction, the origin, developments and main contents of the topological dynamical system and ergodic theory are presented.
In Chapter 2, the basic notions and properties on topological dynamical system and ergodic theory are recalled.
In Chapter 3, the properties of n-sensitivity, which was introduced by Xiong, are investigated, especially in the minimal case. It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct. Moreover, the structure of a minimal system which is n-sensitive but not (n + 1)-sensitive (n≥2) is determined.
In Chapter 4, notions of sensitive sets (S-sets) and regionally proximal sets (Q-sets) are introduced. It is shown that a transitive system is sensitive if and only if there is an S-set with Card(S)≥2, and for a transitive system each S-set is a Q-set. Moreover, the converse holds when (X, T) is minimal. It turns out that each transitive (X, T) has a maximal almost equicontinuous factor. According to the cardinalities of the S-sets, transitive systems are divided into several classes. Characterizations and examples are given for this classification both in minimal and transitive non-minimal settings. It is proved that for a transitive system any entropy set is an S-set, and consequently, a transitive system which has no uncountable S-sets has zero topological entropy. Moreover, it is shown that a transitive, non-minimal system with dense set ofminimal points has an infinite S-set, and there exists a Devaney chaotic system which has no uncountable S-set. Finally, a non-minimal sensitive E-system is constructed such that each its S-set has cardinality at most 4.
In Chapter 5, the notion of measurable n-sensitivity for measure preserving systems is introduced, and the relation between measurable n-sensitivity and the maximal pattern entropy is studied. It is proved that, when (X, B,μ, T) is ergodic and T is invertible, (X, B,μ, T) is measurable n-sensitive but not measurable n+1-sensitive if and only if h_μ~* (T) = log n, where h_μ~* (T) is the maximal pattern entropy of T.
In the last Chapter, the properties of topological sequence entropy for TDSs on countable compact metric spaces are discussed. It is proved that when d(X)≤1, h_(top)~s(T) = 0; and when d(X)≥2, there exists a homeomorphism T on X such that X~d is the sequence entropy set of (X, T), where d(X) and X~d are the derived degree of X and the set of all accumulation points of X respectively.
引文
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[2] E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, Convergence in Er-godic Theory and Probability (Columbus, OH, 1993) (Ohio University Math. Res. Inst. Pub., 5). de Gruyter, Berlin, 1996, pp. 25-40.
[3] E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. d'Anal. Math., 84 (2001),243-286.
[4] E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16(2003), 1421-1433.
[5] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc, 114(1965), 309-319.
[6] L.M. Abramov and V.A. Rokhlin, The entropy of a skew product of measure preserving transformations, Amer. Math. Soc. Transl. (ser 2), 48(1965), 225-265.
[7] W. Ambrose, P.R. Halmos and S. Kakutani, The decomposition of measures II, Duke Math. J.,9(I942),43-47.
[8] J. Auslander, Minimal flows and their extensions, (North-Holland Mathematics Studies, 153). North-Holland, Amsterdam, 1988.
[9] J. Auslander, On the proximal relation in topological dynamics. Proc. Amer. Math. Soc., 11 (1960), 890-895.
[10] J. Banks, Regular periodic decompositions for topologically transitive maps, Erg. Th. and Dynam. Sys., 17(1997), 505-529.
[11] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334.
[12] F. Blanchard, B. Host and A. Maass, Topological comlexity, Erg. Th. and Dynam. Sys., 20 (2000), 641-662.
[13] G. Birkhoff, Dynamical systems, colloquium publication IX. American Mathematical Society, Providence, Rhode Island, 1927; also 1966.
[14] G. Birkhoff, Proof of the ergodic theorem, Proc Nat. Acac. Sci. U.S.A., 17(1931), 656-660.
[15] F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. France, 121(1993), 565-578.
[16] F. Blanchard and W. Huang, Entropy set, weakly mixing sets and entropy capacity, Dis. and Cont. Dynam. Sys., 20(2008), 275-311.
[17] J. Bobok, Chaos in countable dynamical system, Topology Appl., 126 (2002), no. 1-2, 207-21
[18] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153(1971), 401-414.
[19] B. Cadre and P. Jacob, On pairwise sensitivity, J. Math. Anal. Appl., 309(2005), 375-382.
[20] R. V. Chacon and D. S. Ornstein, A general ergodic theorem, 111. Math. J, 4(1960),153-160.
[21] R. L. Devaney, An introduction to chaotic dynamical systems, Second edition. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
[22] T. Downarowicz and X. Ye, When every point is either transitive or periodic, Colloq. Math., 93(2002), 137-150.
[23] D. Dou, X. Ye and G. Zhang, Entropy sequences and maximal entropy sets, Nonlinearity, 19 (2006), 53-74.
[24] R. Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York 1969.
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