Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-14T02:25:07.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Local entropy theory

Published online by Cambridge University Press:  01 April 2009

ELI GLASNER  and
XIANGDONG YE
ELI GLASNER
Affiliation:
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel (email: [email protected])
XIANGDONG YE
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this survey we offer an overview of the so-called local entropy theory, which has been in development since the early 1990s. While doing so, we emphasize the connections between the topological dynamics and the ergodic theory points of view.

Type
SURVEY
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 321 - 356
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
[2]Akin, E., Auslander, J. and Berg, K.. When is a transitive map chaotic? Convergence in Ergodic Theory and Probability (Columbus, OH, 1993) (Ohio State University Mathematical Research Institute Publications, 5). de Gruyter, Berlin, 1996, pp. 25–40.Google Scholar
[3]Akin, E. and Glasner, E.. Residual properties and almost equicontinuity. J. Anal. Math. 84 (2001), 243–286.CrossRefGoogle Scholar
[4]Auslander, J.. Minimal Flows and their Extensions (North-Holland Mathematics Studies, 153). North-Holland, Amsterdam, 1988.Google Scholar
[5]Blanchard, F.. Fully positive topological entropy and topological mixing. Symbolic Dynamics and its Applications, 135 (Contemporary Mathematics). American Mathematical Society, Providence, RI, 1992, pp. 95–105.CrossRefGoogle Scholar
[6]Blanchard, F.. A disjointness theorem involving topological entropy. Bull. Math. Soc. France 121 (1993), 565–578.Google Scholar
[7]Blanchard, F. and Huang, W.. Entropy set, extremely chaotic and entropy capacity. Discrete Contin. Dyn. Syst. 20(2) (2008), 275–311.CrossRefGoogle Scholar
[8]Blanchard, F. and Lacroix, Y.. Zero-entropy factors of topological flows. Proc. Amer. Math. Soc. 119 (1993), 985–992.CrossRefGoogle Scholar
[9]Blanchard, F., Host, B., Maass, A., Martinez, S. and Rudolph, D.. Entropy pairs for a measure. Ergod. Th. & Dynam. Sys. 15 (1995), 621–632.CrossRefGoogle Scholar
[10]Blanchard, F., Host, B. and Ruette, S.. Asymptotic pairs in positive-entropy systems. Ergod. Th. & Dynam. Sys. 22(3) (2002), 671–686.CrossRefGoogle Scholar
[11]Blanchard, F., Glasner, E. and Host, B.. A variation on the variational principle and applications to entropy pairs. Ergod. Th. & Dynam. Sys. 17 (1997), 29–43.CrossRefGoogle Scholar
[12]Blanchard, F., Glasner, E., Kolyada, S. and Maass, A.. On Li–Yorke pairs. J. Reine Angew. Math. 547 (2002), 51–68.Google Scholar
[13]Blanchard, F., Host, B. and Maass, A.. Topological complexity. Ergod. Th. & Dynam. Sys. 20 (2000), 641–662.CrossRefGoogle Scholar
[14]Block, L. and Coppel, W.. Dymanics in One Dimension (Lecure Notes in Mathematics, 1513). Springer, Berlin, 1992.CrossRefGoogle Scholar
[15]Bourgain, J., Fremlin, D. H. and Talagrand, M.. Pointwise compact sets of Baire-measurable functions. Amer. J. Math. 100 (1978), 845–886.CrossRefGoogle Scholar
[16]Brin, M. and Katok, A.. On local entropy. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 30–38.CrossRefGoogle Scholar
[17]Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
[18]Cánovas, J. S.. Topological sequence entropy of interval maps. Nonlinearity 17 (2004), 49–56.CrossRefGoogle Scholar
[19]Denker, M., Grillenberger, C. and Sigmund, C.. Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer, New York, 1976.CrossRefGoogle Scholar
[20]Denker, M. and Urbanski, M.. Ergodic theory of equilibrium states for rational maps. Nonlinearity 4 (1991), 103–134.CrossRefGoogle Scholar
[21]Dinaburg, E. I.. The relation between topological and metric entropy. Dokl. Akad. Nauk SSSR 190 (1970), 13–16. Soviet Math. Dokl. 11 (1970).Google Scholar
[22]Dou, D., Huang, W. and Ye, X.. Null flows and null functions on R. J. Dynam. Differential Equations 18 (2006), 197–221.CrossRefGoogle Scholar
[23]Dou, D., Ye, X. and Zhang, G.. Entropy sequences and maximal entropy sets. Nonlinearity 19 (2006), 53–74.CrossRefGoogle Scholar
[24]van Dulst, D.. Characterization of Banach Spaces not Containing l 1. Centrum voor Wiskunde en Informatica, Amsterdam, 1989.Google Scholar
[25]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1–49.CrossRefGoogle Scholar
[26]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory (M. B. Porter Lectures). Princeton University Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
[27]Furstenberg, H. and Weiss, B.. On almost 1-1 extensions. Israel J. Math. 65 (1989), 311–322.CrossRefGoogle Scholar
[28]Franzová, N. and Smital, J.. Positive sequence topological entropy charaterizes chaotic maps. Proc. Amer. Math. Soc. 112 (1991), 1083–1086.CrossRefGoogle Scholar
[29]Geller, W. and Misiurewicez, M.. Rotation and entropy. Trans. Amer. Math. Soc. 251 (1999), 2927–2984.CrossRefGoogle Scholar
[30]Glasner, E.. A simple characterization of the set of μ-entropy pairs and applications. Israel J. Math. 192 (1997), 13–27.CrossRefGoogle Scholar
[31]Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[32]Glasner, E.. Classifying dynamical systems by their recurrence properties. Topol. Methods Nonlinear Anal. 24 (2004), 21–40.CrossRefGoogle Scholar
[33]Glasner, E.. On tame dynamical systems. Colloq. Math. 105 (2006), 283–295.CrossRefGoogle Scholar
[34]Glasner, E.. The structure of tame minimal dynamical systems. Ergod. Th. & Dynam. Sys. 27 (2007), 1819–1837.CrossRefGoogle Scholar
[35]Glasner, E.. Enveloping semigroups in topological dynamics. Topology Appl. 154 (2007), 2344–2363.CrossRefGoogle Scholar
[36]Glasner, E. and Maon, D.. Rigidity in topological dynamics. Ergod. Th. & Dynam. Sys. 9 (1989), 309–320.CrossRefGoogle Scholar
[37]Glasner, E. and Megrelishvili, M.. Hereditarily non-sensitive dynamical systems and linear representations. Colloq. Math. 104 (2006), 223–283.CrossRefGoogle Scholar
[38]Glasner, E., Megrelishvili, M. and Uspenskij, V. V.. On metrizable enveloping semigroups. Israel J. Math. 164 (2008), 317–332.CrossRefGoogle Scholar
[39]Glasner, E. and Weiss, B.. Strictly ergodic uniform positive entropy models. Bull. Soc. Math. France 122 (1994), 399–412.CrossRefGoogle Scholar
[40]Glasner, E. and Weiss, B.. Quasi-factors of zero-entropy systems. J. Amer. Math. Soc. 8 (1995), 665–686.Google Scholar
[41]Glasner, E. and Weiss, B.. Sensitive dependence on initial conditions. Nonlinearity 6 (1993), 1067–1075.CrossRefGoogle Scholar
[42]Glasner, E. and Weiss, B.. Topological entropy of extensions. Ergodic Theory and its Connections with Harmonic Analysis. Cambridge University Press, Cambridge, 1995, pp. 299–307.CrossRefGoogle Scholar
[43]Glasner, E. and Weiss, B.. On the Interplay Between Measurable and Topological Dynamics (Handbook of Dynamical Systems, 1B). Eds. B. Hasselblatt and A. Katok. Elsevier, Amsterdam, 2006, pp. 597–648.Google Scholar
[44]Goodman, T. N. T.. Relating topological entropy with measure theoretical entropy. Bull. London Math. Soc. 3 (1971), 176–180.CrossRefGoogle Scholar
[45]Goodman, T. N. T.. Topological sequence entropy. Proc. London Math. Soc. 29(3) (1974), 331–350.CrossRefGoogle Scholar
[46]Goodwyn, L. W.. Topological entropy bounds measure theoretical entropy. Proc. Amer. Math. Soc. 23 (1969), 679–688.CrossRefGoogle Scholar
[47]Huang, W.. Tame systems and scrambled pairs under an Abelian group action. Ergod. Th. & Dynam. Sys. 26 (2006), 1549–1567.CrossRefGoogle Scholar
[48]Huang, W., Li, S., Shao, S. and Ye, X.. Null systems and sequence entropy pairs. Ergod. Th. & Dynam. Sys. 23 (2003), 1505–1523.CrossRefGoogle Scholar
[49]Huang, W., Lu, P. and Ye, X.. Measure-theoretical sensitivity and equicontinuity. Preprint, 2007.Google Scholar
[50]Huang, W., Maass, A. and Ye, X.. Sequence entropy pairs and complexity pairs for a measure. Ann. Inst. Fourier (Grenoble) 54 (2004), 1005–1028.CrossRefGoogle Scholar
[51]Huang, W., Maass, A., Romagnoli, P. and Ye, X.. Entropy pairs and a local Abramov formula for a measure theoretical entropy of open covers. Ergod. Th. & Dynam. Sys. 24 (2004), 1127–1153.CrossRefGoogle Scholar
[52]Huang, W., Park, K. and Ye, X.. Dynamical systems disjoint from all minimal systems with zero entropy. Bull. Math. Soc. France, to appear.Google Scholar
[53]Huang, W., Shao, S. and Ye, X.. Mixing and proximal cells along sequences. Nonlinearity 17 (2004), 1245–1260.CrossRefGoogle Scholar
[54]Huang, W., Shao, S. and Ye, X.. Mixing via sequence entropy. Contemp. Math. 385 (2005), 101–122.CrossRefGoogle Scholar
[55]Huang, W. and Ye, X.. An explicit scattering, non-weakly mixing example and weak disjointness. Nonlinearity 15 (2002), 1–14.CrossRefGoogle Scholar
[56]Huang, W. and Ye, X.. Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl. 117 (2002), 259–272.CrossRefGoogle Scholar
[57]Huang, W. and Ye, X.. Topological complexity, return times and weak disjointness. Ergod. Th. & Dynam. Sys. 24 (2004), 825–846.CrossRefGoogle Scholar
[58]Huang, W. and Ye, X.. Dynamical systems disjoint from all minimal systems. Trans. Amer. Math. Soc. 357 (2005), 669–694.CrossRefGoogle Scholar
[59]Huang, W. and Ye, X.. A local variational relation and applications. Israel J. Math. 151 (2006), 237–280.CrossRefGoogle Scholar
[60]Huang, W. and Ye, X.. Combinatorial lemmas and applications. Preprint, 2006.Google Scholar
[61]Huang, W. and Yi, Y.. A local variational principle of presure and its applications to equilibrium states. Israel J. Math. 161 (2007), 29–74.CrossRefGoogle Scholar
[62]Huang, W., Ye, X. and Zhang, G. H.. Relative entropy tuples, relative u.p.e. and c.p.e. extensions. Israel. J. Math. 158 (2007), 249–283.CrossRefGoogle Scholar
[63]Huang, W., Ye, X. and Zhang, G. H.. A local variational principle for conditional entropy. Ergod. Th. & Dynam. Sys. 26 (2006), 219–245.CrossRefGoogle Scholar
[64]Huang, W., Ye, X. and Zhang, G. H.. Local entropy theory for countable discrete amenable group actions. Preprint, 2007.Google Scholar
[65]Hulse, P.. Sequence entropy and subsequence generators. J. London Math. Soc. 26 (1982), 441–450.CrossRefGoogle Scholar
[66]Kamae, T.. Maximal pattern complexity as a topological invariant. Preprint, 2002.Google Scholar
[67]Kamae, T. and Zamboni, L.. Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys. 22 (2002), 1191–1199.CrossRefGoogle Scholar
[68]Kamae, T. and Zamboni, L.. Maximal pattern complexity for discrete systems. Ergod. Th. & Dynam. Sys. 22 (2002), 1201–1214.CrossRefGoogle Scholar
[69]Kaminski, B., Siemaszko, A. and Szymanski, J.. The determinism and the Kolmogorov property in topological dynamics. Bull. Polish Acad. Sci. Math. 51 (2003), 401–417.Google Scholar
[70]Kaminski, B., Siemaszko, A. and Szymanski, J.. Extreme relations for topological flows. Bull. Polish Acad. Sci. Math. 53 (2005), 17–24.CrossRefGoogle Scholar
[71]Karpovsky, M. G. and Milman, V. D.. Coordinate density of sets of vectors. Discrete Math. 24 (1978), 177–184.CrossRefGoogle Scholar
[72]Katok, A. B.. Lyapunov exponents, entropy and periodic points for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
[73]Kerr, D. and Li, H.. Dynamical entropy in Banach spaces. Invent. Math. 162 (2005), 649–686.CrossRefGoogle Scholar
[74]Kerr, D. and Li, H.. Independence in topological and C *-dynamics. Math. Ann. 338 (2007), 869–926.CrossRefGoogle Scholar
[75]Kerr, D. and Li, H.. Combinatorial independence in measurable dynamics, Preprint math.DS (math.FA). arXiv:0705.3424.Google Scholar
[76]Köhler, A.. Enveloping semigroups for flows. Proc. R. Irish Acad. 95A (1995), 179–191.Google Scholar
[77]Kolmogorov, A. N.. A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces. Dokl. Akad. Sci. SSSR. 119 (1958), 861–864 (in Russian).Google Scholar
[78]Kuang, R. and Ye, X.. The return times set and mixing for mpt. Discrete. Contin. Dyn. Syst. 18 (2007), 817–827.CrossRefGoogle Scholar
[79]Kushnirenko, A. G.. On metric invariants of entropy type. Russian Math. Surveys 22(5) (1967), 53–61.CrossRefGoogle Scholar
[80]Lemanczyk, M. and Siemaszko, A.. A note on the existence of a largest topological factor with zero entropy. Proc. Amer. Math. Soc. 129 (2001), 475–482.CrossRefGoogle Scholar
[81]Maass, A. and Shao, S.. Structure of bounded topological-sequence-entropy minimal systems. J. London Math. Soc. (2) 76 (2007), 702–718.CrossRefGoogle Scholar
[82]Parry, W.. Entropy and Generators in Ergodic Theory. Benjamin, New York, 1969.Google Scholar
[83]Park, K. K. and Siemaszko, A.. Relative topological Pinsker factors and entropy pairs. Monatsh. Math. 134 (2001), 67–79.CrossRefGoogle Scholar
[84]Ornstein, D. S.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math 4 (1970), 337–352.CrossRefGoogle Scholar
[85]Petersen, K.. Disjointness and weak mixing of minimal sets. Proc. Amer. Math. Soc. 24 (1970), 278–280.CrossRefGoogle Scholar
[86]Rohklin, V. A.. Lectures on ergodic theory. Russian Math. Surveys 22 (1967), 1–52.Google Scholar
[87]Rohklin, V. A. and Sinai, Ya. G.. Construction and properties of measurable partitions. Dokl. Akad. Nauk SSSR 141 (1961), 1038–1041 (in Russian).Google Scholar
[88]Romagnoli, P.. A local variational principle for the topological entropy. Ergod. Th. & Dynam. Sys. 23 (2003), 1601–1610.CrossRefGoogle Scholar
[89]Rosenthal, H. P.. A characterization of Banach spaces containing ℓ 1. Proc. Natl. Acad. Sci. USA 71 (1974), 2411–2413.CrossRefGoogle Scholar
[90]Saleski, A.. Sequence entropy and mixing. J. Math. Anal. Appl. 60 (1977), 58–66.CrossRefGoogle Scholar
[91]Sauer, N.. On the density of families of sets. J. Combin. Theory Ser. A 23 (1972), 145–147.CrossRefGoogle Scholar
[92]Shapira, U.. Measure theoretical entropy for covers. Israel J. Math 158 (2007), 225–247.CrossRefGoogle Scholar
[93]Shelah, S.. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific J. Math. 41 (1972), 247–261.CrossRefGoogle Scholar
[94]Shao, S., Ye, X. and Zhang, R.. Sensitivity and regionally proximal relation in minimal systems. Sci. China Ser. A 51 (2008), 987–994.CrossRefGoogle Scholar
[95]Song, B. and Ye, X.. Minimal c.p.e. not u.p.e. systems. J. Difference Equ. Appl. (a special issue on Combinatorial and Topological Dynamics), to appear.Google Scholar
[96]Todorc̆ević, S.. Topics in Topology (Lecture Notes in Mathematics, 1652). Springer, Berlin, 1997.CrossRefGoogle Scholar
[97]Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar
[98]Weiss, B.. Proc. Conf. Honoring Dorothy Maharam Stone (Countable Generators in Dynamics—Universal Minimal Models) (Contemporary Mathematics, 94). American Mathematical Society, Providence, RI, 1989, pp. 321–326.Google Scholar
[99]Weiss, B.. Strictly ergodic models for dynamical systems. Bull. Amer. Math. Soc. 13 (1985), 143–146.CrossRefGoogle Scholar
[100]Weiss, B.. Single Orbit Dynamics (Regional Conference Series in Mathematics, 95). American Mathematical Society, Providence, RI, 2000.Google Scholar
[101]Weiss, B.. Multiple recurrence and doubly minimal systems. Contemp. Math. 215 (1998), 189–196.CrossRefGoogle Scholar
[102]Xiong, J.. Chaos in topological transitive systems. Sci. China 48 (2005), 929–939.CrossRefGoogle Scholar
[103]Ye, X. and Zhang, G. H.. Entropy points and applications. Trans. Amer. Math. Soc. 359 (2007), 6167–6186.CrossRefGoogle Scholar
[104]Ye, X. and Zhang, R. F.. On sensitive sets in topological dynamics. Nonlinearity, to appear.Google Scholar
[105]Zhang, G. H.. Relative entropy, asymptotic pairs and chaos. J. London Math. Soc. 73 (2006), 157–172.CrossRefGoogle Scholar
[106]Zhang, G. H.. Relativization of complexity and sensitivity. Ergod. Th. & Dynam. Sys. 27 (2007), 1349–1371.CrossRefGoogle Scholar