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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 |
We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of Sarig [
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.
|

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