摘要
本文由动力学性质的相对化和局部化两部分构成。首先，我们使用相对化的思想系统地将很多动力学性质的最新进展推广到相对化的情形。其次，一方面，我们细化前人的动力系统熵理论的局部化思想；另一方面，我们考虑更广泛的群作用动力系统的局部熵理论。在这些过程中，很多时候我们借鉴于前人的工作，但也有很多时候在借鉴的同时我们遇到了本质的困难，这迫使我们寻求完全不同的方法来处理。不仅如此，在推广的过程中，我们还得到了很多即使放到绝对情形下来看也是崭新的结果。
本文的具体安排如下：
在第一章中，我们简要地介绍了动力系统(尤其是拓扑动力系统和遍历理论)的起源、发展以及主要研究内容。通过对动力系统发展历史与现状的简单回顾，我们逐步介绍本文的背景知识和主要内容。
在第二章中，我们简单地介绍本文涉及到的拓扑动力系统和遍历理论的预备知识。
第三、四、五章组成本文的第一个主题：动力学性质的相对化。即：给定动力系统间的因子映射，我们分析发生在纤维上的动力学行为。设π∶(X，T)→(Y，S)为动力系统间的因子映射，也就是说，π为从X到Y上与作用相容的连续满射，对每个y∈Y，称π~(-1)(y)为一个纤维。特别的，当(Y，S)平凡时，纤维上动力学行为的研究即成为通常的动力系统动力学行为的研究。
在第三章中，我们研究了复杂性函数和正向等度连续系统在相对化情形下的对应：相对复杂性函数和正向等度连续扩充。我们证明了：对于极小动力系统间的开因子映射，它为正向等度连续扩充当且仅当每个开覆盖具有有界的相对复杂性函数。进一步，基于相对复杂性函数的思想，我们定义了相对复杂性n-串，相对n-扩散(n≥2)和相对扩散。对于极小可逆动力系统间的开因子映射，我们得到了相对于它的最大可逆等度连续因子，并且证明了此时相对扩散蕴含了弱混合。
在第四章中，我们研究了纤维上的混沌行为—相对敏感和相对Mycielski混沌。首先，我们引入了相对敏感性的概念，证明了：极小动力系统间的因子映射或者为相对敏感的，或者为正向等度连续的；动力系统间非平凡的弱混合因子映射为相对敏感的。著名的Glasner-Weiss结果告诉我们：非极小的M-系统一定为敏感的。通过将这个结果推广到相对化情形，某种意义下我们揭示了这个深刻结果的本质。同时，我们证明了：很多时候，相对2-扩散蕴含了相对Mycielski混沌。其次，当考虑可逆动力系统间的因子映射时，我们证明了：正的条件熵不仅蕴含了纤维上真的渐近对的存在性，还蕴含了相对Mycielski混沌。事实上，我们还指出；这种情况下，纤维上存在拓扑意义下‘很大’的混沌集，即，纤维上混沌集拓扑熵的上确界恰好等于因子映射的条件拓扑熵。即使在绝对情形下来看，这个结果也是崭新而又深刻的，以至于审稿人在评审意见中强调，应当将绝对情形下的这个结果单独在文章中凸显出来。
在第五章中，我们建立了有限开覆盖的局部相对变分原理。作为应用，我们定义了相对拓扑熵串和相对测度熵串，建立了两者之间的变分关系，并由此得到了相对的拓扑Pinsker因子，这回答了[111]中提出的某些问题。同时，基于相对拓扑熵对的思想，我们引入了相对一致正熵扩充和相对完全正熵扩充，讨论了它们的基本性质和有限乘积。上述的这些结果将十五年来建立起来的动力系统局部熵理论完全推广到相对化的情形。尤其需要指出的是，在建立有限开覆盖的局部相对变分原理这一过程中，一方面，在[12]中起到关键作用的组合引理[12，引理1]似乎很难推广到相对化情形，因此我们需要用新的完全不同于前人的方法来解决；另一方面，对于有限Borel覆盖，我们在测度意义下引入两种条件熵，并最终建立了两者之间的相等关系，特别地，回到绝对情形下，我们在文献中第一次指出了两者的相等关系，且它在后面第七章的可数离散amenable群作用动力系统局部熵理论的建立过程中起到了重要作用。注意到，这里两种条件熵的引入及它们的相等关系均是对有限Borel覆盖进行的，而不必要求它为开覆盖。
第六、七章组成本文的第二个主题：动力学性质的局部化。
在第六章中，我们细化前人有关动力系统熵理论的局部化思想。一方面，我们在拓扑和测度意义下同时定义C-熵点，并建立了两者之间的变分关系；另一方面，我们定义并研究了一致熵点。一致熵点研究的一个也许有点惊奇的副产品是：每个动力系统里存在一个可数闭子集，使得它的拓扑熵等于整个系统的拓扑熵。注意将这个结果与如下的经典结果做比较：每个紧致可数空间上的同胚均具有零拓扑熵。事实上，通过进一步探讨一致熵点研究的思想，我们在[82]中得到了更多深刻的结果。
在第七章中，我们致力于分析比Z要更加广泛的一类群—可数离散的amenable群(包括所有可数离散的可解群)—作用的动力系统(即，紧致度量空间上的同胚构成一个可数离散的amenable群)的熵理论。在这个背景下我们重新构建十五年来建立起来的Z作用动力系统的局部熵理论(参见E．Glasner和叶向东教授最近撰写的综述性文章[57])。首先，我们对它建立有限开覆盖的局部变分原理，并由此简洁地得到了可数离散amenable群作用动力系统的经典变分原理。接着，做为应用，我们在拓扑和测度意义下同时定义熵串，并建立两者之间的变分关系。最后，基于拓扑熵对的思想，我们引入并讨论了一致正熵和完全正熵的概念，指出：一致正熵蕴含了很强的传递属性，完全正熵蕴含了动力系统为满支撑的，且一致正熵和完全正熵在有限乘积下均保持不变。在局部变分原理的建立过程中，对于有限Borel覆盖，我们在测度意义下也同时引入了两种熵。然而，不同于Z作用及其相对化的情形，为了证明它们关于不变测度的上半连续性，我们需要寻找隐藏在可数离散amenable群身上未知的深刻性质；为了证明两者的等价性，由于这时万有的Rohlin引理及著名的G-W定理的对应形式尚不清楚，我们不得不借助于A．I．Danilenko针对这种情形熵理论而发展起来的轨道理论，同时，我们还使用到Z作用下这两种熵为相等的这个最近刚刚得到的结果。因此，寻求万有的Rohlin引理及著名的G-W定理在这种情形下的对应形式或许成为一个有意思且充满挑战的公开问题。
The thesis consists of two topics of relativization and localization of dynamical properties. In the first part, we generalize systematically to the relative case many developments obtained recently in topological dynamics, including the local entropy theory built in the past fifteen years. In the second part, on one hand, we introduce new methods of localization to more deeply understand the complexity of a dynamical system; on the other hand, we build the local entropy theory of a general group actions, especially a countable discrete amenable group actions, on a compact metric space. Especially, we emphasize the parallelism between ergodic theory and topological dynamics and the applications of ergodic theory in the study of topological dynamics.
The thesis is organized as follows:
In Chapter 1, the origin, developments and main objective and contents of ergodic theory and topological dynamics are recalled, and then the backgrounds and main results of the thesis are summarized as three parts, respectively.
In Chapter 2, some preliminary notions and results in ergodic theory and topological dynamics, which will be used in the thesis, are reviewed.
Chapters 3, 4 and 5 focus on the first topic of the thesis, namely, the relativization of dynamical properties. Precisely speaking, letπ: (X, T)→(Y, S) be a factor map between dynamical systems, i.e. a continuous surjective mapping from X onto Y compatible with the actions on X and Y, eachπ~(-1)(y), y∈Y is called a fiber. Given a factor map between dynamical systems, we aim to study the dynamical behaviors on the fibers. In particular, if (Y, S) is a trivial dynamical system then the study of the fibers is just the absolute case of the study of (X, T).
In Chapter 3, the correspondence of the complexity function and equicontinuity in the relative case, the relative complexity function and the positively equicontinuous extension, are studied. It is shown that, for a given open factor map between minimal dynamical systems, the factor map is positively equicontinuous if and only if each open cover has a bounded relative complexity function. Then based on the idea of the relative complexity function, the notions of relative complexity n-tuples, relative n-scattering (n≥2) and relative scattering are introduced. When considering an open factor map between minimal invertible dynamical systems, it is proved the existence of the maximal invertible equicontinuous factor of the given factor map and that relative scattering implies weak mixing.
In Chapter 4, chaotic behaviors on the fibers, relative sensitivity and relative My-cielski chaos, are introduced. First, the notion of relative sensitivity is introduced. It is proved that: for a factor map between minimal dynamical systems, it is either relative sensitive or positively equicontinuous; any non-trivial weakly mixing factor map between dynamical systems is relative sensitive. The known Glasner-Weiss result tells us that any M-system, which is not minimal, is sensitive. It is generalized to the relative case. Meanwhile, it is proved that, in many cases, relative 2-scattering implies relative Myciel-ski chaos. Then, when considering a factor map between invertible dynamical systems, it is shown that the positivity of relative entropy implies the existence of proper asymptotic pairs on fibers and relative Mycielski chaos. Moreover, it turns out that the scrambled subsets on fibers are topologically very big, i.e. the supremum of the topological entropy of the scrambled subsets on fibers equals the relative entropy of the given factor map. Even in the absolute case, this is a new and deep result.
In Chapter 5, the local entropy theory of a dynamical system is generalized completely to the relative case. Precisely speaking, the relative local variational principle of a finite open cover is established. Then as its application, the notion of relative entropy tuples is introduced both in topological and measure-theoretic settings. The variational relation between these two kinds of relative entropy tuples is interpreted. And it is proved the existence of relative topological Pinsker factor, which answers affirmatively some question mentioned in [111]. Moreover, based on the idea of relative topological entropy pairs, the notion of relative uniformly positive entropy and relative completely positive entropy extensions is introduced. Some basic properties and the finite production is studied. In the process, on one hand, it seems that the key lemma [12, Lemma 1] is very difficult to be generalized to the relative case, and so some new method is needed; on the other hand, in the measure-theoretic setting two kinds of relative entropy for a finite Borel cover are introduced and turn out to be the same. Note that the equivalence of these two kinds entropy will play an important role in the building of the local entropy theory for a countable discrete amenable group actions on a compact metric space, and the equivalence of them for a finite open cover has just been obtained recently.
Chapters 6 and 7 aim to another topic of the thesis, namely, the localization of dynamical properties.
In Chapter 6, in order to more deeply understand the complexity of a dynamical system, some new methods of localization are applied. On one hand, along the line of entropy pairs, tuples, sequences and sets, the notion of C-entropy point both in topological and measure-theoretic settings is introduced. The structure of the set of C-entropy points is studied, and the variational relation between these two kinds of C-entropy points is established. On the other hand, using the idea of Bowen's separated and spanning subsets, the notion of uniformly entropy point is introduced and studied. A maybe unexpected byproduct of the study of uniformly entropy point is that, in each dynamical system, there exists a countable closed subset such that its topological entropy equals the topological entropy of the origin system. Note that it is known that each homeo-morphism on a compact countable space has zero topological entropy. Moreover, more interesting and deep results are obtained in [82].
In Chapter 7, the local entropy theory of a countable discrete amenable group actions, which is more general than (?) actions, on a compact metric space is built, including all countable discrete solvable group actions. This generalizes completely to this general group actions the local entropy theory set up in the past fifteen years (see for example a survey being prepared by Profs. E. Glasner and X. Ye [57]). First, the local variational principle of a finite open cover is established in this setting, which can be used to obtain the classic variational principle by some standard arguments. Then, as its application, the notion of entropy tuples both in topological and measure-theoretic settings is introduced, the structure of the set of entropy tuples is studied and the variational relation between these two kinds of entropy tuples is interpreted. Finally, based on the idea of topological entropy pairs, topological analogue of Kolmogorov systems in ergodic theory, namely uniformly positive entropy and completely positive entropy, are introduced and studied. It turns out that: uniformly positive entropy implies some strong transitivity properties, completely positive entropy implies the existence of invariant measures with full support and they are both preserved under the finite production. In the establishing of the local variational principle, in the measure-theoretic setting two kinds of entropy for a finite Borel cover are introduce, studied and turn out to be the same. Whereas, different from the case of (?) actions, to prove the upper semi-continuity of these two kinds of entropy with respect to invariant measures, it is discovered some deep property underlying a countable discrete amenable group; to prove the equivalence of them, besides of the results obtained in Chapter 5, it becomes inevitable the orbital pointview of entropy theory in this setting developed by A. I. Danilenko in [23].
引文
[1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114(1965), 309-319.
[2] E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic? Convergence in Ergodic Theory and Probability, (Ohio State Univ. Math. Res. Inst. Publ. 5), dc Gruytcr, Berlin, 1996, 25-40.
[3] I. Assani, Wiener Wintner Ergodic Theorems, World Scientific Publishing Co., Inc., River Edge, N J, 2003.
[4] J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies 153, North-Holland Publishing Co., Amsterdam, 1988.
[5] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99(1992), no. 4, 332-334.
[6] V. Bergelson, Ergodic Ramsey theory — an update, Ergodic theory of Z~d Actions, London Math. Soc. Lecture Notes Ser. 228, Cambridge Univ. Press, 1996, 1-61.
[7] V. Bergelson, Combinatorial and Diophantine applications of ergodic theory (Appendix A by A. Leibman and Appendix B by A. Quas and M. Wierdl), Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006, 745-869.
[8] G. Birkhoff, Proof of the ergodic theorem, Proc. Nat. Acad. Sci. U.S.A., 17(1931), 656-660.
[9] G. Birkhoff, Dynamical Systems, Reprinting of the original edition [New York, 1927], AMS Colloquium Publications, Vol. Ⅸ, American Mathematical Society, 1966.
[10] F. Blanchard, Fully positive topological entropy and topological mixing, Symbolic Dynamics and its Applications, AMS Contemporary Mathematics, 135(1992), 95-105.
[11] F. Blanchard, A disjointness theorem involving topological entropy, Bull. de la Soc. Math. de France, 121(1993), no. 4, 465-478.
[12] F. Blanchard, E. Glasner and B. Host, A variation on the variational principle and applications to entropy pairs, Ergod. Th. & Dynam. Sys., 17(1997), no. 1, 29-43.
[13] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547(2002), 51-68.
[14] F. Blanchard, B. Host and A. Maass, Topological complexity, Ergod. Th. & Dynam. Sys., 20(2000), no. 3, 641-662.
[15] F. Blanchard, B. Host, A. Maass, S. Martinez and D. Rudolph, Entropy pairs for a measure, Ergod. Th. & Dynam. Sys., 15(1995), no. 4, 621-632.
[16] F. Blanchard, B. Host and S. Ruette, Asymptotic pairs in positive-entropy systems, Ergod. Th. & Dynam. Sys., 22(2002), no. 3, 671-686.
[17] F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity, to appear in Discrete Continuous Dynam. Sys..
[18] F. Blanchard and Y. Lacroix, Zero entropy factors of topological flows, Proc. Amer. Math. Soc., 119(1993), no. 3, 985-992.
[19] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153(1971), 401-414.
[20] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184(1973), 125-136.
[21] M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math., 156(2004), no. 1, 119-161.
[22] A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergod. Th. & Dynam. Sys., 1(1981), no. 4, 431-450.
[23] A. I. Danilenko, Entropy theory from the orbital point of view, Monatsh. Math., 134(2001), no. 2, 121-141.
[24] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecturc Notes in Math. 527, Springer-Verlag, Berlin-New York, 1976.
[25] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Second edition, AddisonWesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
[26] A. H. Dooley and V. Ya. Golodets, The spectrum of completely positive entropy actions of countable amenable groups, J. Funct. Anal., 196(2002), no. 1, 1-18.
[27] D. Dou, Research on Entropy of Dynamical System and It's Application, Doctoral Thesis of University of Science and Technology of China (in Chinese), 2006.
[28] D. Dou, X. Ye and G. H. Zhang, Entropy sequences and maximal entropy sets, Nonlinearity, 19(2006), no. 1, 53-74.
[29] T. Downarowicz and J. Serafin, Fiber entropy and conditional variational principles in compact non-metrizable spaces, Fund. Math., 172(2002), no. 3, 217-247.
[30] H. A. Dye, On groups of measure preserving transformation. Ⅰ, Amer. J. Math., 81(1959), no. 1, 119-159.
[31] H. A. Dye, On groups of measure preserving transformation. Ⅱ, Amer. J. Math., 85(1963), no. 4, 551-576.
[32] R. Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc., 94(1960), 272-281.
[33] R. Ellis, Lectures in Topological Dynamics, W, A. Benjamin, New York, 1969.
[34] R. Ellis, The Veech structure theorem, Trans. Amer. Math. Soc., 186(1973), 203-218.
[35] R. Ellis, The Furstenberg structure theorem, Pacific J. Math., 76(1978), no. 2, 345-349.
[36] R. Ellis, S. Glasner and L. P. Shapiro, Proximal-isometric flows, Advances in Math., 17(1975), no. 3, 213-260.
[37] W. R. Emerson, The pointwise ergodic theorem for amenable groups, Amer. J. Math., 96(1974), no. 3, 472-278.
[38] J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and yon Ncumann algebras. Ⅰ, Trans. Amer. Math. Soc., 234(1977), no. 2, 289-324.
[39] J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and yon Ncumann algebras. Ⅱ, Trans. Amer. Math. Soc., 234(1977), no. 2, 325-359.
[40] S. Ferenczi, Complexity of sequences and dynamical systems, Combinatorics and Number Theory, Discrete Math., 206(1999), no. 1-3, 145-154.
[41] H. Furstenberg, The structure of distal flows, Amer. J. Math., 85(1963), 477-515.
[42] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1(1967), 1-49.
[43] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.
[44] E. Glasner, A simple characterization of the set of p-entropy pairs and applications, Israel J. Math., 102(1997), 13-27.
[45] E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs 101, American Mathematical Society, 2003.
[46] E. Glasner, Topological weak mixing and quasi-Bohr systems, Probability in Mathematics, Israel J. Math., 148(2005), 277-304.
[47] E. Glasner, On tame dynamical systems, Colloq. Math., 105(2006), no. 2, 283-295.
[48] E. Glasner, The structure of tame minimal dynamical systems, Preprint 2006.
[49] E. Glasner and M. Megrelishvili, Hereditarily non-sensitive dynamical systems and linear representations, Colloq. Math., 104(2006), no. 2, 223-283.
[50] E. Glasner, M. Megrelishvili and V. V. Uspenskij, On metrizable enveloping semigroups, to appear in Israel J. Math..
[51] E. Glasner, J. P. Thouvenot and B. Weiss, Entropy theory without a past, Ergod. Th. & Dynam. Sys., 20(2000), no. 5, 1355-1370.
[52] E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6(1993), no. 6, 1067-1075.
[53] E. Glasner and B. Weiss, Strictly ergodic, uniform positive entropy models, Bull. de la Soc. Math. de France, 122(1994), no. 3, 399-412.
[54] E. Glasner and B. Weiss, Topological entropy of extensions, Ergodic Theory and its Connections with Harmonic Analysis, London Math. Soc. Lecture Note Ser. 205, Cambridge Univ. Press, 1995, 299-307.
[55] E. Glasner and B. Weiss, Locally equicontinuous dynamical systems, Dedicated to the Memory of Anzelm Iwanik, Colloq. Math., 84/85(2000), part 2, 345-361.
[56] E. Glasner and B. Weiss, On the interplay between measurable and topological dynamics, Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006, 597-648.
[57] E. Glasner and X. Ye, Local entropy theory, In preparation.
[58] T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soe., 3(1971), 176-180.
[59] T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc. (3), 29(1974), no. 2, 331-350.
[60] L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23(1969), 679-688.
[61] W. Gottschalk, Orbit-closure decompositions and almost periodic properties, Bull. Amer. Math. Soc., 50(1944), 915-919.
[62] W. Gottschalk and G. Hedlund, Topological Dynamics, AMS Colloquium Publications, Vol. 36, American Mathematical Society, 1955.
[63] B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, to appear in Ann. Math. (2).
[64] P. R. Halmos, The decomposition of measures, Duke Math. J., 8(1941), 386-392.
[65] P. R. Halmos, Measurable transformations, Bull. Amer. Math. Soc., 55(1949), 1015-1034.
[66] P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. Ⅱ, Ann. Math. (2), 43(1942), 332-350.
[67] E. Hopf, Ergodentheorie, Chelsea, New Yorke, 1948.
[68] B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. Math. (2), 161(2005), no. 1, 397-488.
[69] W. Huang, Complexity of Dynamical Systems via Tuples, Doctoral Thesis of University of Science and Technology of China (in Chinese), 2003.
[70] W. Huang, Tame systems and scrambled pairs under an abelian group action, Ergod. Th. & Dynam. Sys., 26(2006), no. 5, 1549-1567.
[71] W. Huang, S. Li, S. Shao and X. Ye, Null systems and sequence entropy pairs, Ergod. Th. & Dynam. Sys., 23(2003), no. 5, 1505-1523.
[72] W. Huang, A. Maass, P. P. Romagnoli and X. Ye, Entropy pairs and a local Abramov formula for a measure theoretical entropy of open covers, Ergod. Th. & Dynam. Sys., 24(2004), no. 4, 1127-1153.
[73] W. Huang, A. Maass and X. Ye, Sequence entropy pairs and complexity pairs for a measure, Ann. Inst. Fourier (Grenoble), 54(2004), no. 4, 1005-1028.
[74] W. Huang, S. Shao and X. Ye, Mixing via sequence entropy, Algebraic and Topological Dynamics, AMS Contemporary Mathematics, 385(2005), 101-122.
[75] W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology Appl., 117(2002), no. 3, 259-272.
[76] W. Huang and X. Ye, Topological complexity, return times and weak disjointness, Ergod. Th. & Dynam. Sys., 24(2004), no. 3, 825-846.
[77] W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151(2006), 237-279.
[78] W. Huang and X. Ye, Combinatorial lemmas and applications, Preprint 2007.
[79] W. Huang, X. Ye and G. H. Zhang, A local variational principle for conditional entropy, Ergod. Th. & Dynamo Sys., 26(2006), no. 1, 219-245.
[80] W. Huang, X. Ye and G. H. Zhang, Relative entropy tuples, relative u.p.e, and c.p.e. extensions, to appear in Israel J. Math..
[81] W. Huang, X. Ye and G. H. Zhang, Local entropy theory of a countable discrete amenable group action, Preprint 2007.
[82] W. Huang, X. Ye and G. H. Zhang, Lowering topological entropy over subsets, Preprint 2007.
[83] P. Hulse, Sequence entropy relative to an invariant a-algebra, J. London Math. Soc. (2), 33(1986), no. 1, 59-72.
[84] S. Kakutani, Ergodic theory, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., Vol. 2, 1950, 128-142.
[85] A. B. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. Math. I.H.E.S., 51(1980), 137-173.
[86] A. Y. Khintchine, Zur Birkhoffs Losung des ergodenproblems, Math. Ann., 107(1933), no. 1, 485-488.
[87] J. Kieffer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space, Ann. Prob., 3(1975), no. 6, 1031-1037.
[88] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, (Russian) Dokl. Akad. Nauk SSSR (N. S.), 119(1958), 861-864.
[89] B. Kra, The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view, Bull. Amer. Math. Soc. (N. S.), 43(2006), no. 1, 3-23.
[90] B. Kra, From combinatorics to ergodic theory and back again, Proceedings of the International Congress of Mathematicians, Madrid, Vol. Ⅲ, 2006, 57-76.
[91] R. Kuang, Ergodic Theory and Furstenberg Families, Doctoral Thesis of University of Science and Technology of China (in Chinese), 2007.
[92] F. Ledrappier, A variational principle for the topological conditional entropy, Ergodic Theory, Lecture Notes in Math. 729, Springer Verlag, 1979, 78-88.
[93] F. Ledrappier and P. Waiters, A relativised variational principle for continuous transformations, J. London Math. Soc. (2), 16(1977), no. 3, 568-576.
[94] M. Lemanczyk and A. Siemaszko, A note on the existence of a largest topological factor with zero entropy, Proc. Amer. Math. Soc., 129(2001), no. 2, 475-482.
[95] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82(1975), no. 10, 985-992.
[96] E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146(2001), no. 2, 259-295.
[97] M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bull. Acad. Pol. Sci., 21(1973), 903-910.
[98] M. Misiurewicz, A short proof of the variational principle for a Z_+~n action on a compact space, International Conference on Dynamical Systems in Mathematical Physics, Astérisque, no. 40, Soc. Math. Prance, Paris, 1976, 145-157.
[99] J. Mycielski, Independent sets in topological algebras, Fund. Math., 55(1964), 139-147.
[100] J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. U.S.A., 18(1932), 70-82.
[101] J. von Neumann, Zur operatoren methode in der klassischen mechanik, Ann. Math., 33(1932), 587-642.
[102] A. Nevo, Pointwise ergodic theorems for actions of groups, Handbook of dynamical systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006, 871-982.
[103] I. S. Newton, Philosophiae Naturalis Principia Mathematica, (Latin) William Dawson & sons, Ltd., London, updated.
[104] J. M. Ollagnier, Ergodic Theory and Statistical Mechanics, Lecture Notes in Math. 1115, Springer-Verlag, Berlin, 1985.
[105] J. M. Ollagnier and D. Pinchon, The variational principle, Studia Math., 72(1982), no. 2, 151-159.
[106] D. S. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math., 4(1970), 337-352.
[107] D. S. Ornstein, A K automorphism with no square root and Pinsker's conjecture, Advances in Math., 10(1973), 89-102.
[108] D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. d'Anal. Math., 48(1987), 1-141.
[109] D. S. Ornstein and B. Weiss, Subsequence ergodic theorems for amenable groups, Israel J. Math., 79(1992), no. 1, 113-127.
[110] J. C. Oxtoby, Measure and Category, A Surevey of the Analogies between Topological and Measure Spaces, Second edition, Graduate Texts in Mathematics 2, Springer-Verlag, New York-Berlin, 1980.
[111] K. K. Park and A. Siemaszko, Relative topological Pinsker factors and entropy pairs, Monatsh. Math., 134(2001), no. 1, 67-79.
[112] W. Parry, Topics in Ergodic Theory, Cambridge Tracks in Mathematics, Cambridge Univ. Press, Cambridge-New York, 1981.
[113] K. Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics 2, Cambridge Univ. Press, Cambridge, 1983.
[114] H. Poincaré, Sur le problème des trois corps et les équations de la dynamiquc, Acta Math., 13(1890), 1-270.
[115] H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vols. Ⅰ, Ⅱ, Ⅲ, GauthierVillars, Paris, 1892, 1893, 1899.
[116] F. Przytycki and M. Urbanski, Fractals in the Plane — the Ergodic Theory Methods, Preprint 2006.
[117] V. A. Rohlin, On the fundamental ideas of measure theory, Math. Sb. (N. S.), 25(67) (1949), 107-150. Engl. Transl., Amer. Math. Soc. Translations (1), 10(1962), 1-54.
[118] V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Amer. Math. Soc. Translations (2), 39(1964), 1-36.
[119] V. A. Rohlin, Lectures on ergodic theory, Russian Math. Surverys, 22(1967), 1-52.
[120] V. A. Rohlin and Yakov G. Sinai, Construction and properties of invariant measurable partitions, (Russian) Dokl. Akad. Nauk SSSR, 141(1967), 1038-1041.
[121] P. Romagnoli, A local variational principle for the topological entropy, Ergod. Th. & Dynam. Sys., 23(2003), no. 5, 1601-1610.
[122] D. J. Rudolph, Fundamentals of Measurable Dynamics, Ergodic Theory on Lebesgue Spaces, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990
[123] D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. Math. (2), 151(2000), no. 3, 1119-1150.
[124] C. Shannon, A mathematical theory of communication, Bell System Tech. J., 27(1948), 379-423, 623-656.
[125] S. Shao, Dynamical Systems and Families, Doctoral Thesis of University of Science and Technology of China (in Chinese), 2005.
[126] A. M. Stepin and A. T. Tagi-Zade, Variational characterization of topological pressure of the amenable groups of transformations, (Russian) Dokl. Akad. Nauk SSSR, 254(1980), no. 3, 545-549.
[127] E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith., 27(1975), 199-245.
[128] W. A. Veech, Point-distal flows, Amer. J. Math., 92(1970), 205-242.
[129] W. A. Veech, Topological dynamics, Bull. Amer. Math. Soc., 83(1977), no. 5, 775-830.
[130] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, Springer-Verlag, New York-Berlin, 1982.
[131] T. Ward and Q. Zhang, The Abramov-Rokhlin entropy addition formula for amenable group actions, Monatsh. Math., 114(1992), no. 3-4, 317-329.
[132] B. L. van der Waerden, Beweis einer baudetschen vermutung, Nieuw Arch. Wisk., 15(1927), 212-216.
[133] B. Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc. (N. S.), 13(1985), no. 2, 143-146.
[134] B. Weiss, Actions of amenable groups, Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser. 310, Cambridge Univ. Press, Cambridge, 2003, 226-262.
[135] X. Ye, W. Huang and S. Shao, An Introduction to Topological Dynamics (a book in Chinese), Preprint 2006.
[136] X. Ye and G. H. Zhang, Entropy points and applications, to appear in Trans. Amer. Math. Soc..
[137] X. Ye and R. Zhang, On sensitive sets in topological dynamics, Preprint 2006.
[138] K. Yosida and S. Kakutani, Birkhoff's ergodic theorem and the maximal ergodic theorem, Pro. Imp. Acad. Tokoyo, 15(1939), 165-168.
[139] G. H. Zhang, Relative entropy, asymptotic pairs and chaos, J. London Math. Soc. (2), 73(2006), no. 1, 157-172.
[140] G. H. Zhang, Relativization of complexity and sensitivity, to appear in Ergod. Th. & Dynam. Sys..
[141] R. J. Zimmer, Extensions of ergodic group actions, Ill. J. Math., 20(1976), no. 3, 373-409.
[142] R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Ill. J. Math., 20(1976), no. 4, 555-588.
[143] R. J. Zimmer, Hyperfinite factors and amenable ergodic actions, Invent. Math., 41(1977), no. 1, 23-31.
