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Symbolic dynamics for non-uniformly hyperbolic systems

Part of: Dynamical systems with hyperbolic behavior Dynamical systems and ergodic theory Smooth dynamical systems: general theory Topological dynamics

Published online by Cambridge University Press:  17 December 2020

YURI LIMA
YURI LIMA*
Affiliation:
Yuri Lima, Departamento de Matemática, Universidade Federal do Ceará (UFC), Campus do Pici, Bloco 914, CEP 60440-900. Fortaleza – CE, Brazil (e-mail: yurilima@gmail.com)
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Abstract

This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

Keywords

Markov partitionPesin theorysymbolic dynamics

MSC classification

Primary: 37-02: Research exposition (monographs, survey articles) 37B10: Symbolic dynamics 37C05: Smooth mappings and diffeomorphisms 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37C35: Orbit growth 37D35: Thermodynamic formalism, variational principles, equilibrium states

Information

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2591 - 2658
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Adler, R. L.. Symbolic dynamics and Markov partitions. Bull. Amer. Math. Soc. (N.S.) 35(1) (1998), 1–56.CrossRefGoogle Scholar
Anosov, D. V.. Tangential fields of transversal foliations in U-systems. Mat. Zametki 2 (1967), 539–548.Google Scholar
Anosov, D. V.. Geodesic Flows on Closed Riemann Manifolds with Negative Curvature (Proc. Seteklov Inst. Math., 90) (1967). Translated from the Russian by S. Feder. American Mathematical Society, Providence, RI, 1969. Google Scholar
Anosov, D. V.. On certain class of invariant sets of smooth dynamical systems (in Russian). Proc. 5th Int. Conf. on Non-Linear Oscillations. Vol. 2, 1970, pp. 39–45.Google Scholar
Adler, R. L. and Weiss, B.. Entropy, a complete metric invariant for automorphisms of the torus. Proc. Nat/Acad. Sci. USA 57 (1967), 1573–1576.CrossRefGoogle ScholarPubMed
Adler, R. L. and Weiss, B.. Similarity of Automorphisms of the Torus (Memoirs of the American Mathematical Society, 98) American Mathematical Society, Providence, RI, 1970.Google Scholar
Ballmann, W.. Lectures on Spaces of Nonpositive Curvature (DMV Seminar, 25). Birkhäuser, Basel, 1995. With an appendix by Misha Brin.CrossRefGoogle Scholar
Boyle, M. and Buzzi, J.. The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors. J. Eur. Math. Soc. (JEMS) 19(9) (2017), 2739–2782.CrossRefGoogle Scholar
Burns, K., Climenhaga, V., Fisher, T., and Thompson, D. J.. Unique equilibrium states for geodesic flows in nonpositive curvature. Geom. Funct. Anal. 28(5) (2018), 1209–1259.CrossRefGoogle Scholar
Buzzi, J., Crovisier, S., and Lima, Y.. $\chi$ -universal Markov partitions for three dimensional flows. In preparation (2019).Google Scholar
Buzzi, J., Crovisier, S., and Sarig, O.. Measures of maximal entropy for surface diffeomorphisms. Preprint, arXiv:1811.02240, 2018.Google Scholar
Boyle, M. and Downarowicz, T.. The entropy theory of symbolic extensions. Invent. Math. 156(1) (2004), 119–161.CrossRefGoogle Scholar
Baladi, V. and Demers, M. F.. On the measure of maximal entropy for finite horizon Sinai Billiard maps. J. Amer. Math. Soc. 33(2) (2020), 381–449.CrossRefGoogle Scholar
Berg, K. R.. Entropy of torus automorphisms. Topological Dynamics (Symposium, Colorado State Univ., Ft. Collins, CO 1967), Benjamin, New York, 1968, pp. 67–79.Google Scholar
Berg, K. R.. Convolution of invariant measures, maximal entropy. Math. Syst. Theory 3 (1969), 146–150.CrossRefGoogle Scholar
Burns, K. and Gerber, M.. Real analytic Bernoulli geodesic flows on ${S}^2$ . Ergod. Th. & Dynam. Sys. 9(1) (1989), 27–45.CrossRefGoogle Scholar
Burns, K. and Hasselblatt, B.. The Sharkovsky theorem: a natural direct proof. Amer. Math. Monthly 118(3) (2011), 229–244.CrossRefGoogle Scholar
Burns, K., Masur, H., and Wilkinson, A.. The Weil–Petersson geodesic flow is ergodic. Ann. of Math. (2) 175(2) (2012), 835–908.CrossRefGoogle Scholar
Ben Ovadia, S.. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn. 13 (2018), 43–113.CrossRefGoogle Scholar
Ovadia, S. B.. Canonically codable points and irreducible codings. Preprint, 2019, arXiv:1904.01391.Google Scholar
Bowen, R.. Markov partitions for Axiom $\mathrm{A}$ diffeomorphisms. Amer. J. Math. 92 (1970), 725–747.CrossRefGoogle Scholar
Bowen, R.. Topological entropy and axiom $\mathrm{A}$ . Global Analysis (Berkeley, CA, 1968) (Proceedings of Symposia in Pure Mathematics, XIV). American Mathematical Society, Providence, RI, 1970, pp. 23–41.CrossRefGoogle Scholar
Bowen, R.. Periodic points and measures for Axiom $A$ diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
Bowen, R.. The equidistribution of closed geodesics. Amer. J. Math. 94 (1972), 413–423.CrossRefGoogle Scholar
Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30.CrossRefGoogle Scholar
Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–460.CrossRefGoogle Scholar
Bowen, R.. Markov partitions are not smooth. Proc. Amer. Math. Soc. 71(1) (1978), 130–132.CrossRefGoogle Scholar
Bowen, R.. On Axiom A Diffeomorphisms (Regional Conference Series in Mathematics, 35). American Mathematical Society, Providence, RI, 1978.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470), revised edn. Springer, Berlin. 2008. With a preface by David Ruelle, edited by Jean-René Chazottes.CrossRefGoogle Scholar
Bowen, R.. Rufus Bowen’s Notebook, 2017. Available at https://bowen.pims.math.ca/.Google Scholar
Bowen, R.. Bernoulli equilibrium states for Axiom A diffeomorphisms. Math. Syst. Theory 8(4) (1974/75), 289–294.CrossRefGoogle Scholar
Bowen, R.. Some systems with unique equilibrium states. Math. Syst. Theory 8(3) (1974/75), 193–202.CrossRefGoogle Scholar
Barreira, L. and Pesin, Y.. Nonuniform Hyperbolicity (Encyclopedia of Mathematics and its Applications, 115). Cambridge University Press, Cambridge, 2007. Dynamics of systems with nonzero Lyapunov exponents.CrossRefGoogle Scholar
Barreira, L. and Pesin, Y.. Introduction to Smooth Ergodic Theory (Graduate Studies in Mathematics, 148). American Mathematical Society, Providence, RI, 2013.CrossRefGoogle Scholar
Backes, L., Poletti, M., Varandas, P., and Lima, Y.. Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 40(11) (2020), 2947–2969.CrossRefGoogle Scholar
Bowen, R. and Ruelle, D.. The ergodic theory of axiom A flows. Invent. Math. 29(3) (1975), 181–202.CrossRefGoogle Scholar
Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, 2002.CrossRefGoogle Scholar
Buzzi, J. and Sarig, O.. Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergod. Th. & Dynam. 23(5) (2003), 1383–1400.CrossRefGoogle Scholar
Bunimovich, L. A., Sinaĭ, Y. G., and Chernov, N. I.. Markov partitions for two-dimensional hyperbolic billiards. Uspekhi Mat. Nauk 45(3) (1990), 97–134.Google Scholar
Bunimovič, L. A.. Billiards that are close to scattering billiards. Mat. Sb. 94(136) (1974), 49–73, 159.Google Scholar
Bunimovič, L. A.. The ergodic properties of certain billiards. Funktsional. Anal. i Prilozhen 8(3) (1974), 73–74.Google Scholar
Bunimovich, L. A.. On the ergodic properties of nowhere dispersing billiards. Comm. Math. Phys. 65(3) (1979), 295–312.CrossRefGoogle Scholar
Burguet, D.. ${C}^2$ surface diffeomorphisms have symbolic extensions. Invent. Math. 186(1) (2011), 191–236.CrossRefGoogle Scholar
Buzzi, J.. The degree of Bowen factors and injective codings of diffeomorphisms. J. Mod. Dyn. 16 (2020), 1–36.CrossRefGoogle Scholar
Chernov, N. I.. Topological entropy and periodic points of two-dimensional hyperbolic billiards. Funktsional. Anal. i Prilozhen 25(1) (1991), 50–57.CrossRefGoogle Scholar
Coornaert, M. and Knieper, G.. Growth of conjugacy classes in Gromov hyperbolic groups. Geom. Funct. Anal. 12(3) (2002), 464–478.CrossRefGoogle Scholar
Chen, D., Kao, L.-Y., and Park, K.. Unique equilibrium states for geodesic flows over surfaces without focal points. Nonlinearity 33(3) (2020), 1118–1155.CrossRefGoogle Scholar
Climenhaga, V., Knieper, G., and War, K.. Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points. Preprint, 2019, arXiv:1903.09831.Google Scholar
Chernov, N. and Markarian, R.. Chaotic Billiards (Mathematical Surveys and Monographs, 127). American Mathematical Society, Providence, RI, 2006.CrossRefGoogle Scholar
Donnay, V. J.. Geodesic flow on the two-sphere. I. Positive measure entropy. Ergod. Th. & Dynam. Sys. 8(4) (1988), 531–553.CrossRefGoogle Scholar
Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18). Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
Duarte, P.. Plenty of elliptic islands for the standard family of area preserving maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 11(4) (1994), 359–409.CrossRefGoogle Scholar
Eberlein, P.. Geodesic flows in manifolds of nonpositive curvature. Smooth Ergodic Theory and its Applications (Seattle, WA, 1999) (Proceedings of Symposia in Pure Mathematics, 69) American Mathematical Society, Providence, RI, 2001, pp. 525–571.CrossRefGoogle Scholar
Einsiedler, M., Lindenstrauss, E., and Ward, T.. Entropy in dynamics, 2011. Available at http://maths.dur.ac.uk/tpcc68/entropy/welcome.html.Google Scholar
Ferenczi, S.. Complexity of sequences and dynamical systems. Discrete Math. 206(1–3) (1999), 145–154.CrossRefGoogle Scholar
Fathi, A., Herman, M.-R., and Yoccoz, J.-C.. A proof of Pesin’s stable manifold theorem. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 177–215.Google Scholar
Filip, S.. Notes on the multiplicative ergodic theorem. Ergod. Th. & Dynam. Sys. 39(5) (2019), 1153–1189.CrossRefGoogle Scholar
Gaspard, P. and Baras, F.. Chaotic scattering and diffusion in the Lorentz gas. Phys. Rev. E (3) 51(6, part A) (1995), 5332–5352.CrossRefGoogle ScholarPubMed
Gelfert, K. and Ruggiero, R. O.. Geodesic flows modelled by expansive flows. Proc. Edinb. Math. Soc. (2) 62(1) (2019), 61–95.CrossRefGoogle Scholar
Gurevič, B. M.. Topological entropy of a countable Markov chain. Dokl. Akad. Nauk 187 (1969), 715–718.Google Scholar
Gurevič, B. M.. Shift entropy and Markov measures in the space of paths of a countable graph. Dokl. Akad. Nauk 192 (1970), 963–965.Google Scholar
Hadamard, J.. Sur la forme des lignes géodésiques à l’infini et sur les géodésiques des surfaces réglées du second ordre. Bull. Soc. Math. France 26 (1898), 195–216.CrossRefGoogle Scholar
Hedlund, G. A.. The dynamics of geodesic flows. Bull. Amer. Math. Soc. 45(4) (1939), 241–260.CrossRefGoogle Scholar
Hofbauer, F.. $\beta$ -shifts have unique maximal measure. Monatsh. Math. 85(3) (1978), 189–198.CrossRefGoogle Scholar
Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. 34(3) (1979), 213–237.CrossRefGoogle Scholar
Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. II. Israel J. Math. 38(1–2) (1981), 107–115.CrossRefGoogle Scholar
Hopf, E.. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304.Google Scholar
Hopf, E.. Statistik der Lösungen geodätischer Probleme vom unstabilen Typus. II. Math. Ann. 117 (1940), 590–608.CrossRefGoogle Scholar
Huber, H.. Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen. Math. Ann. 138 (1959), 1–26.CrossRefGoogle Scholar
Kaloshin, V. Y.. Generic diffeomorphisms with superexponential growth of number of periodic orbits. Comm. Math. Phys. 211(1) (2000), 253–271.CrossRefGoogle Scholar
Katok, A.. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2) 110(3) (1979), 529–547.CrossRefGoogle Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
Katok, A.. Entropy and closed geodesics. Ergod. Th. & Dynam. Sys. 2(3–4) (1983), 339–365.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza.CrossRefGoogle Scholar
Kitchens, B. P.. Symbolic Dynamics. Springer, Berlin, 1998.CrossRefGoogle Scholar
Katok, A. and Mendoza, L.. Dynamical Systems with Non-Uniformly Hyperbolic Behavior. Cambridge University Press, Cambridge, 1995. Supplement to Introduction to the modern theory of dynamical systems.Google Scholar
Knieper, G.. Das Wachstum der Äquivalenzklassen geschlossener Geodätischer in kompakten Mannigfaltigkeiten. Arch. Math. (Basel) 40(6) (1983), 559–568.CrossRefGoogle Scholar
Knieper, G.. On the asymptotic geometry of nonpositively curved manifolds. Geom. Funct. Anal. 7(4) (1997), pp. 755–782.CrossRefGoogle Scholar
Knieper, G.. The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds. Ann. of Math. (2) 148(1) (1998), 291–314.CrossRefGoogle Scholar
Knieper, G.. Hyperbolic dynamics and Riemannian geometry. Handbook of Dynamical Systems. Vol. 1A. North-Holland, Amsterdam, 2002, pp. 453–545.Google Scholar
Knieper, G.. The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank. Israel J. Math. 149 (2005), 171–183.CrossRefGoogle Scholar
Katok, A., Strelcyn, J.-M., Ledrappier, F., and Przytycki, F.. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222). Springer, Berlin, 1986.Google Scholar
Katok, S. and Ugarcovici, I.. Symbolic dynamics for the modular surface and beyond. Bull. Amer. Math. Soc. (N.S.) 44(1) (2007), 87–132.CrossRefGoogle Scholar
Ledrappier, F., Lima, Y., and Sarig, O.. Ergodic properties of equilibrium measures for smooth three dimensional flows. Comment. Math. Helv. 91(1) (2016), 65–106.CrossRefGoogle Scholar
Lima, Y. and Matheus, C.. Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities. Ann. Sci. Éc. Norm. Supér. (4) 51(1) (2018), 1–38.CrossRefGoogle Scholar
Lima, Y. and Sarig, O. M.. Symbolic dynamics for three-dimensional flows with positive topological entropy. J. Eur. Math. Soc. (JEMS) 21(1) (2019), 199–256.CrossRefGoogle Scholar
Margulis, G. A.. Certain applications of ergodic theory to the investigation of manifolds of negative curvature. Funksional. Anal. i Priložen. 3(4) (1969), 89–90.Google Scholar
Morse, M.. Does instability imply transitivity? Proc. Natl Acad. Sci. USA 20(1) (1934), 46–50.CrossRefGoogle ScholarPubMed
Milnor, J. and Thurston, W.. On iterated maps of the interval. Dynamical Systems (College Park, MD, 1986–87) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 465–563.Google Scholar
Newhouse, S. E.. Continuity properties of entropy. Ann. of Math. (2) 129(2) (1989), 215–235.CrossRefGoogle Scholar
Oseledec, V. I.. A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. Trudy Moskov. Mat. Obšč. 19 (1968), 179–210.Google Scholar
Parry, W.. Intrinsic Markov chains. Trans. Amer. Math. Soc. 112 (1964), 55–66.CrossRefGoogle Scholar
Pesin, J. B.. Families of invariant manifolds that correspond to nonzero characteristic exponents. Izv. Akad. Nauk SSSR Ser. Mat. 40(6) (1976), 1332–1379, 1440.Google Scholar
Pesin, J. B.. Characteristic Ljapunov exponents, and smooth ergodic theory. Uspehi Mat. Nauk 32(4(196)) (1977), 55–112, 287.Google Scholar
Pesin, J. B.. Geodesic flows in closed Riemannian manifolds without focal points. Izv. Akad. Nauk SSSR Ser. Mat. 41(6) (1977), 1252–1288, 1447.Google Scholar
Parry, W. and Pollicott, M.. An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. of Math. (2) 118(3) (1983), 573–591.CrossRefGoogle Scholar
Pollicott, M. and Sharp, R.. Error terms for closed orbits of hyperbolic flows. Ergod. Th. & Dynam. Sys. 21(2) (2001), 545–562.CrossRefGoogle Scholar
Plante, J. F. and Thurston, W. P.. Anosov flows and the fundamental group. Topology 11 (1972), 147–150.CrossRefGoogle Scholar
Ratner, M. E.. Markov decomposition for an U-flow on a three-dimensional manifold. Mat. Zametki 6 (1969), 693–704.Google Scholar
Ratner, M.. Markov partitions for Anosov flows on $n$ -dimensional manifolds. Israel J. Math. 15 (1973), 92–114.CrossRefGoogle Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A., Tahzibi, A., and Ures, R.. Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys. 306(1) (2011), 35–49.CrossRefGoogle Scholar
Sarig, O. M.. Bernoulli equilibrium states for surface diffeomorphisms. J. Mod. Dyn. 5(3) (2011), 593–608.CrossRefGoogle Scholar
Sarig, O. M.. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc. 26(2) (2013), 341–426.CrossRefGoogle Scholar
Shub, M.. Global Stability of Dynamical Systems. Springer, New York, 1987.CrossRefGoogle Scholar
Sinaĭ, J.. G.. Asymptotic behavior of closed geodesics on compact manifolds with negative curvature. Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 1275–1296.Google Scholar
Sinaĭ, J. G.. Construction of Markov partitionings. Funkcional. Anal. i Priložen . 2(3) (1968), 70–80 (Loose errata).Google Scholar
Sinaĭ, J. G.. Markov partitions and U-diffeomorphisms. Funkcional. Anal. i Priložen. 2(1) (1968), 64–89.Google Scholar
Sinaĭ, J. G.. Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat. Nauk 25(2 (152)) 1970, 141–192.Google Scholar
Smale, S.. Finding a horseshoe on the beaches of Rio. Math. Intelligencer 20(1) (1998), 39–44.CrossRefGoogle Scholar
Takahashi, Y.. Isomorphisms of $\beta$ -automorphisms to Markov automorphisms. Osaka J. Math. 10 (1973), 175–184.Google Scholar
Weaver, B. A.. Precise asymptotic growth rate of periodic orbits for a class of non-uniformly hyperbolic geodesic flows. Preprint, 2011.Google Scholar
Yoccoz, J.-C.. A proof of Jakobson’s theorem, 2015, pp. 34. Available at https://www.college-de-france.fr/media/jean-christophe-yoccoz/UPL74162544747766%2098194%20Jakobson%20jcy.pdf.Google Scholar
Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147(3) (1998), 585–650.CrossRefGoogle Scholar