Abstract
In the past twenty years, great achievements have been made by many researchers in the studies of chaotic behavior and local entropy theory of dynamical systems. Most of the results have been generalized to the relative case in the sense of a given factor map. In this survey we offer an overview of these developments.
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References
Adler R L, Konheim A G, McAndrew M H. Topological entropy. Trans Amer Math Soc, 1965, 114: 309–319
Ahn Y, Lee J, Park K K. Relative sequence entropy pairs for a measure and relative topological Kronecker factor. J Korean Math Soc, 2005, 42: 857–869
Akin E, Glasner E, Huang W, et al. Sufficient conditions under which a transtive system is chaotic. Ergod Th & Dynam Sys, 2010, 30: 1277–1310
Auslander J. Minimal Flows and Their Extensions. In: North-Holland Mathematics Studies, vol. 153. Amsterdam: Elsevier, 1988
Banks J, Brooks J, Cairns G, et al. On Devaney’s definition of chaos. Amer Math Monthly, 1992, 99: 332–334
Blanchard F. Fully positive topological entropy and topological mixing. In: Symbolic Dynamics and its Applications, vol. 135. Providence, RI: Amer Math Soc, 1992, 95–105
Blanchard F. A disjointness theorem involving topological entropy. Bull Soc Math France, 1993, 121: 465–478
Blanchard F, Glasner E, Host B. A variation on the variational principle and applications to entropy pairs. Ergod Th & Dynam Sys, 1997, 17: 29–43
Blanchard F, Glasner E, Kolyada S, et al. On Li-Yorke pairs, J Reine Angew Math, 2002, 547: 51–68
Blanchard F, Host B, Maass A, et al. Entropy pairs for a measure. Ergod Th & Dynam Sys, 1995, 15: 621–632
Blanchard F, Host B, Ruette S. Asymptotic pairs in positive-entropy systems. Ergod Th & Dynam Sys, 2002, 22: 671–686
Blanchard F, Huang W. Entropy sets, weakly mixing sets and entropy capacity. Discrete Contin Dyn Syst, 2008, 20: 275–311
Blanchard F, Lacroix Y. Zero-entropy factors of topological flows. Proc Amer Math Soc, 1993, 119: 985–992
Bowen R. Entropy for group endomorphisms and homogeneous spaces. Trans Amer Math Soc, 1971, 153: 401–414
Boyle M, Fiebig D, Fiebig U. Residual entropy, conditional entropy and subshift covers. Forum Math, 2002, 14: 713–757
Bryant B F. On expansive homeomorphisms. Pacific J Math, 1960, 10: 1163–1167
Chung N P, Li H. Homoclinic group, IE group, and expansive algebraic actions. Preprint, 2011
Devaney R L. An Introduction to Chaotic Dynamical Systems (Second Edition). New York: Addison-Wesley, 1989
Dinaburg E I. A correction between topological and metric entropy. Dokl Akad Nauk SSSR, 1970, 190: 13–16
Dooley A, Zhang G H. Co-induction in dynamical systems. Ergod Th & Dynam Sys, to appear
Dooley A, Zhang G H. Local entropy theory of a random dynamical system. Preprint, 2011
Dou D, Ye X, Zhang G H. Entropy sequences and maximal entropy sets. Nonlinearity, 2006, 19: 53–74
Downarowicz T, J Serafin. Fiber entropy and conditional variational principles in compact non-metrizable spaces. Fund Math, 2002, 172: 217–247
Furstenberg H. The structure of distal flows. Amer J Math, 1963, 85: 477–515
Furstenberg H. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math Systems Th, 1967, 1: 1–49
Furstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton: Princeton Univ Press, 1981
Glasner E. A simple characterization of the set of μ-entropy pairs and applications. Israel J Math, 1997, 102: 13–27
Glasner E. Ergodic Theory via Joinings. In: Mathematical Surveys and Monographs, vol. 101. Providence, RI: Amer Math Soc, 2003
Glasner E, Thouvenot J P, Weiss B. Entropy theory without a past. Ergod Th & Dynam Sys, 2000, 20: 1355–1370
Glanser E, Weiss B. Strictly ergodic uniform positive entropy models. Bull Soc Math France, 1994, 122: 399–412
Glasner E, Weiss B. Topological entropy of extensions. Ergodic Theory and its Connections with Harmonic Analysis. Cambridge: Cambridge University Press, 1995
Glasner E, Weiss B. On the interplay between measurable and topological dynamics. In: Handbook of Dynamical Systems, vol. 1B. Amsterdam: Elsevier, 2006
Glasner E, Ye X. Local entropy theory. Ergod Th & Dynam Sys, 2009, 29: 321–356
Goodman T N T. Relating topological entropy and measure entropy. Bull London Math Soc, 1971, 3: 176–180
Goodwyn L W. Topological entropy bounds measure-theoretic entropy. Proc Amer Math Soc, 1969, 23: 679–688
Huang W. Stable sets and ∈-stable sets in positive-entropy systems. Comm Math Phys, 2008, 279: 535–557
Huang W. personal communications, 2010
Huang W, Li S, Shao S, et al. Null systems and sequence entropy pairs. Ergod Th & Dynam Sys, 2003, 23: 1505–1523
Huang W, Maass A, Romagnoli P P, et al. Entropy pairs and a local Abramov formula for a measure theoretical entropy of open covers. Ergod Th & Dynam Sys, 2004, 24: 1127–1153
Huang W, Maass A, Ye X. Sequence entropy pairs and complexity pairs for a measure. Ann Inst Fourier (Grenoble), 2004, 54: 1005–1028
Huang W, Ye X. Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos. Topology Appl, 2002, 117: 259–272
Huang W, Ye X. Topological complexity, return times and weak disjointness. Ergod Th & Dynam Sys, 2004, 24: 825–846
Huang W, Ye X. A local variational relation and applications. Israel J Math, 2006, 151: 237–279
Huang W, Ye X, Zhang G H. A local variational principle for conditional entropy. Ergod Th & Dynam Sys, 2006, 26: 219–245
Huang W, Ye X, Zhang G H. Relative entropy tuples, relative u.p.e. and c.p.e. extensions. Israel J Math, 2007, 158: 249–283
Huang W, Ye X, Zhang G H. Local entropy theory for a countable discrete amenable group action. J Funct Anal, 2011, 261: 1028–1082
Huang W, Ye X, Zhang G H. Lowering topological entropy over subsets. II. Preprint, 2011
Huang W, Ye X, Zhang G H. Lowering topological entropy over subsets. Ergod Th & Dynam Sys, 2010, 30: 181–209
Huang W, Yi Y. A local variational principle of pressure and its applications to equilibrium states. Israel J Math, 2007, 161: 249–283
Kerr D, Li H. Dynamical entropy in Banach spaces. Invent Math, 2005, 162: 649–686
Kerr D, Li H. Independence in topological and C*-dynamics. Math Ann, 2007, 338: 869–926
Kerr D, Li H. Combinatorial independence in measurable dynamics. J Funct Anal, 2009, 256: 1341–1386
Kolmogorov A N. A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces (in Russian). Dokl Akad Sci SSSR, 1958, 119: 861–864
Kolyada S F. Li-Yorke sensitivity and other concepts of chaos. Ukrainian Math J, 2004, 56: 1242–1257
Kolyada S F, Snoha L. Some aspects of topological transitivity—a survey. Grazer Math Ber, 1997, 334: 3–35
Lemanczyk M, Siemaszko A. A note on the existence of a largest topological factor with zero entropy. Proc Amer Math Soc, 2001, 129: 475–482
Ledrappier F, Walters P. A relativised variational principle for continuous transformations. J London Math Soc, 1977, 16: 568–576
Li T Y, Yorke J A. Period three implies chaos. Amer Math Monthly, 1975, 82: 985–992
Lind D, Schimidt K. Homoclinic points of algebraic ℤd-actions. J Amer Math Soc, 1999, 12: 953–980
Ma X F, Chen E C, Zhang A H. A relative local variational principle for topological pressure. Sci China Math, 2010, 53: 1491–1506
Misiurewicz M. A short proof of the variational principle for a Z n+ action on a compact space. Astérisque, 1976, 40: 227–262
Mycielski J. Independent sets in topological algebras. Fund Math, 1964, 55: 139–147
Ornstein D S, Weiss B. Entropy and isomorphism theorems for actions of amenable groups. J d’Anal Math, 1987, 48: 1–141
Oprocha P, Zhang G H. On local aspects of topological weak mixing in dimension one and beyond. Studia Mathematica, 2011, 202: 261–288
Oprocha P, Zhang G H. On local aspects of topological weak mixing, sequence entropy and chaos. Preprint, 2010
Oprocha P, Zhang G H. On sets with recurrence properties, their topological structure and entropy. Topology Appl, to appear
Park K K, Siemaszko A. Relative topological Pinsker factors and entropy pairs. Monatsh Math, 2001, 134: 67–79
Rohlin V A. On the fundamental ideas of measure theory. Amer Math Soc Translation, 1952, 71: 1–54
Romagnoli P P. A local variational principle for the topological entropy. Ergod Th & Dynam Sys, 2003, 23: 1601–1610
Shapira U. Measure theoretical entropy for covers. Israel J Math, 2007, 158: 225–247
Song B, Ye X. A minimal completely positive entropy non-uniformly positive entropy example. J Difference Equ Appl, 2009, 15: 87–95
Tan F. The set of sequence entropies for graph maps. Topology Appl, 2011, 158: 533–541
Tan F, Ye X, Zhang R F. The set of sequence entropies for a given space. Nonlinearity, 2010, 23: 159–178
Walters P. An Introduction to Ergodic Theory. In: Graduate Texts in Mathematics, vol. 79. New York-Berlin: Springer-Verlag, 1982
Weiss B. Strictly ergodic models for dynamical systems. Bull Amer Math Soc, 1985, 13: 143–146
Xiong J C, Yang Z G. Chaos caused by a topologically mixing map. Dynamical Systems and Related Topics (Nagoya, 1990). River Edge, NJ: World Sci Publ, 1991
Ye X, Zhang G H. Entropy points and applications. Trans Amer Math Soc, 2007, 359: 6167–6186
Zhang G H. Relative entropy, asymptotic pairs and chaos. J London Math Soc, 2006, 73: 157–172
Zhang G H. Relativization of complexity and sensitivity. Ergod Th & Dynam Sys, 2007, 27: 1349–1371
Zhang G H. Relativization and Localization of Dynamical Systems. Ph.D. Thesis, University of Science and Technology of China, 2007 (available at http://homepage.fudan.edu.cn/zhanggh/files/2011/06/english.pdf)
Zhang G H. Variational principles of pressure. Discrete Contin Dyn Syst, 2009, 24: 1409–1435
Zimmer R J. Extensions of ergodic group actions. Illinois J Math, 1976, 20: 373–409
Zimmer R J. Ergodic actions with generalized discrete spectrum. Illinois J Math, 1976, 20: 555–588
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Zhang, G. Relativization of dynamical properties. Sci. China Math. 55, 913–936 (2012). https://doi.org/10.1007/s11425-011-4332-4
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DOI: https://doi.org/10.1007/s11425-011-4332-4
Keywords
- relative entropy
- asymptotic pair
- scrambled set
- local variational principles
- relative entropy tuple
- relative topological Pinsker factor
- relative UPE
- relative CPE