# American Institute of Mathematical Sciences

2018, 13: 43-113. doi: 10.3934/jmd.2018013

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

 Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 234 Herzl Street, POB 26, Rehovot, 7610001 Israel

Received  April 23, 2017 Revised  October 27, 2018 Published  December 2018

Fund Project: 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

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.

Citation: Snir Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. Journal of Modern Dynamics, 2018, 13: 43-113. doi: 10.3934/jmd.2018013
##### References:
 [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747. doi: 10.2307/2373370. Google Scholar [6] R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. doi: 10.2307/1995452. Google Scholar [7] R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793. Google Scholar [8] R. Bowen, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, R.I., 1978. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] J. Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1723-1763. doi: 10.1017/S0143385708000953. Google Scholar [13] A. Fathi and M. Shub, Some dynamics of pseudo-anosov diffeomorphisms, Asterisque, 66 (1979), 181-207. Google Scholar [14] B. M. Gurevič, Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR, 187 (1969), 715-718. Google Scholar [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. Google Scholar [16] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662. Google Scholar [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. Google Scholar [26] J. B. Pesin, Characteristic lyapunov exponents and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112,287. Google Scholar [27] M. E. Ratner, Markov decomposition for an U-flow on a three-dimensional manifold, Mat. Zametki, 6 (1969), 693-704. Google Scholar [28] M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math., 15 (1973), 92-114. doi: 10.1007/BF02771776. Google Scholar [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. Google Scholar [30] J. G. Sinaĭ, Construction of Markov partitionings, Funkcional. Anal. i Priložen., 2 (1968), 70-80 (Loose errata). Google Scholar [31] J. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89. Google Scholar

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##### References:
 [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747. doi: 10.2307/2373370. Google Scholar [6] R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. doi: 10.2307/1995452. Google Scholar [7] R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793. Google Scholar [8] R. Bowen, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, R.I., 1978. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] J. Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1723-1763. doi: 10.1017/S0143385708000953. Google Scholar [13] A. Fathi and M. Shub, Some dynamics of pseudo-anosov diffeomorphisms, Asterisque, 66 (1979), 181-207. Google Scholar [14] B. M. Gurevič, Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR, 187 (1969), 715-718. Google Scholar [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. Google Scholar [16] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662. Google Scholar [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. Google Scholar [26] J. B. Pesin, Characteristic lyapunov exponents and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112,287. Google Scholar [27] M. E. Ratner, Markov decomposition for an U-flow on a three-dimensional manifold, Mat. Zametki, 6 (1969), 693-704. Google Scholar [28] M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math., 15 (1973), 92-114. doi: 10.1007/BF02771776. Google Scholar [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. Google Scholar [30] J. G. Sinaĭ, Construction of Markov partitionings, Funkcional. Anal. i Priložen., 2 (1968), 70-80 (Loose errata). Google Scholar [31] J. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89. Google Scholar
Illustration of the discussed tangent vectors
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