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Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds

This work is a part of a M.Sc. thesis at the Weizmann Institute of Science. The author was partly supported by the ISF grant 199/14.
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  • We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of Sarig [29] for surfaces. These partitions allow us to obtain symbolic coding on invariant sets of full measure for all hyperbolic measures whose Lyapunov exponents are bounded away from zero by a fixed constant. Applications include counting results for hyperbolic periodic orbits, and structure of hyperbolic measures of maximal entropy.

    Mathematics Subject Classification: Primary: 37D25; Secondary: 37B10.

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  • Figure 1.  Illustration of the discussed tangent vectors

  • [1] R. L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus, Proc. Nat. Acad. Sci. U.S.A., 57 (1967), 1573-1576.  doi: 10.1073/pnas.57.6.1573.
    [2] R. L. Adler and B. Weiss, Similarity of Automorphisms of the Torus, Memoirs of the American Math. Society, No. 98, American Mathematical Society, Providence, R.I., 1970.
    [3] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, volume 115 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.
    [4] M. Boyle and J. Buzzi, The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors, Journal of the European Mathematical Society, 19 (2017), 2739-2782.  doi: 10.4171/JEMS/727.
    [5] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747.  doi: 10.2307/2373370.
    [6] R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.
    [7] R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460.  doi: 10.2307/2373793.
    [8] R. Bowen, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, R.I., 1978.
    [9] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, With a preface by David Ruelle, Edited by J.-R. Chazottes, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, revised edition, 2008.
    [10] M. Brin, Hölder continuity of invariant distributions, in lectures on Lyapunov exponents and smooth ergodic theory, in Smooth Ergodic Theory and Its Applications (Seattle, WA, 1999), Appendix A by M. Brin and Appendix B by D. Dolgopyat, H. Hu and Pesin, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 3-106. doi: 10.1090/pspum/069/1858534.
    [11] L. A. Bunimovich and Ya. G. Sinaĭ, Markov partitions for dispersed billiards, Comm. Math. Phys., 78 (1980/81), 247-280.  doi: 10.1007/BF01942372.
    [12] J. Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1723-1763.  doi: 10.1017/S0143385708000953.
    [13] A. Fathi and M. Shub, Some dynamics of pseudo-anosov diffeomorphisms, Asterisque, 66 (1979), 181-207. 
    [14] B. M. Gurevič, Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR, 187 (1969), 715-718. 
    [15] B. M. Gurevič, Shift entropy and Markov measures in the space of paths of a countable graph, Dokl. Akad. Nauk SSSR, 192 (1970), 963-965. 
    [16] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. 
    [17] A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, 1245-1253.
    [18] A. Katok, Fifty years of entropy in dynamics: 1958-2007, J. Modern Dynamics, 1 (2007), 545-596.  doi: 10.3934/jmd.2007.1.545.
    [19] A. Katok and L. Mendoza, Dynamical Systems with Non-Uniformly Hyperbolic Behavior, supplement to "Introduction to the Modern Theory of Dynamical Systems", Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
    [20] Y. Lima and C. Matheus, Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities, Ann. Sci. Éc. Norm. Supér., 51 (2018), 1-38.  doi: 10.24033/asens.2350.
    [21] Y. Lima and O. Sarig, Symbolic dynamics for three dimensional flows with positive topological entropy, J. Eur. Math. Soc., 21 (2019), 199-256.  doi: 10.4171/JEMS/834.
    [22] E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934), 837-842.  doi: 10.1090/S0002-9904-1934-05978-0.
    [23] O. Perron, Über Stabilität und asymptotisches Verhalten der Lösungen eines Systems endlicher Differenzengleichungen, J. Reine Angew. Math., 161 (1929), 41-64.  doi: 10.1515/crll.1929.161.41.
    [24] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.
    [25] J. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), 1332-1379, 1440. 
    [26] J. B. Pesin, Characteristic lyapunov exponents and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112,287. 
    [27] M. E. Ratner, Markov decomposition for an U-flow on a three-dimensional manifold, Mat. Zametki, 6 (1969), 693-704. 
    [28] M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math., 15 (1973), 92-114.  doi: 10.1007/BF02771776.
    [29] O. M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), 341-426.  doi: 10.1090/S0894-0347-2012-00758-9.
    [30] J. G. Sinaĭ, Construction of Markov partitionings, Funkcional. Anal. i Priložen., 2 (1968), 70-80 (Loose errata).
    [31] J. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89. 
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