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

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

Snir Ben Ovadia

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.

Keywords: Hyperbolic, diffeomorphisms, Markov partition, Lyapunov exponents, non-uniformly hyperbolic, symbolic dynamics, manifolds of any dimension.
Mathematics Subject Classification: Primary: 37D25; Secondary: 37B10.
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.

[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. 

show all references

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.

[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. 

Figure 1.  Illustration of the discussed tangent vectors
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