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January  2014, 8(1): 15-24. doi: 10.3934/jmd.2014.8.15

On Omri Sarig's work on the dynamics on surfaces (Brin Prize article)

François Ledrappier 1,

1. 

LPMA, Boîte Courrier 188, 4, Place Jussieu, 75252 PARIS cedex 05, France

Published  July 2014

N/A
Keywords: Brin prize, Sarig, Markov partitions, surface diffeomorphisms, horocycle flow..
Mathematics Subject Classification: 37D40, 37D25, 37D4.
Citation: François Ledrappier. On Omri Sarig's work on the dynamics on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 15-24. doi: 10.3934/jmd.2014.8.15
References:
[1]

R. Adler and B. Weiss, Similarities of the Automorphisms of the Torus, Memoirs of the Amer. Math. Soc., 98, Amer. Mat. Soc. Providence, RI, 1970.  Google Scholar

[2]

M. Babillot, On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds, in Random Walks and Geometry (ed. V. A. Kaimanovich), Walter de Gruyter GmbH & Co. KG, Berlin, 2004, 319-335.  Google Scholar

[3]

K. Berg, Convolutions of invariant measures, maximal entropy, Math. Systems Theory, 3 (1969), 146-150. doi: 10.1007/BF01746521.  Google Scholar

[4]

P. Berger, Properties of the maximal entropy measure and geometry of Hénon attractors, arXiv:1202.2822, (2012). Google Scholar

[5]

M. Babillot and F. Ledrappier, Geodesic paths and horocycle flows on abelian covers, in Lie Groups and Ergodic Theory (Mumbai, 1996), Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998, 1-32.  Google Scholar

[6]

R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans.Amer. Math. Soc., 154 (1971), 377-397.  Google Scholar

[7]

M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math., 14 (2002), 713-747. doi: 10.1515/form.2002.031.  Google Scholar

[8]

M. Burger, Horocycle flows on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803. doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar

[9]

D. Burguet, Symbolic extensions in intermediate smoothness on surfaces, Ann. Sci. Éc. Norm. Sup. (4), 45 (2012), 337-362.  Google Scholar

[10]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161. doi: 10.1007/BF02773637.  Google Scholar

[11]

S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J., 51 (1984), 185-194. doi: 10.1215/S0012-7094-84-05110-X.  Google Scholar

[12]

T. Downarowicz and S. Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math., 160 (2005), 453-499. doi: 10.1007/s00222-004-0413-0.  Google Scholar

[13]

N. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances Math., 5 (1970), 365-394. doi: 10.1016/0001-8708(70)90010-1.  Google Scholar

[14]

H. Furstenberg, The unique ergodicity of the horocycle flow, Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Springer Lect. Notes Math., 318, Springer, Berlin, 1972, 95-115.  Google Scholar

[15]

F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math., 34 (1979), 213-237; Israel J. Math., 38 (1981), 107-115. doi: 10.1007/BF02761854.  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]

V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811.  Google Scholar

[18]

F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 163-188.  Google Scholar

[19]

F. Ledrappier and O. Sarig, Invariant measures for horocycle flows on periodic hyperbolic surfaces, Israel J. Math., 160 (2007), 281-317. doi: 10.1007/s11856-007-0064-0.  Google Scholar

[20]

S. Newhouse, Continuity properties of entropy, Ann. Math. (2), 129 (1989), 215-235. doi: 10.2307/1971492.  Google Scholar

[21]

W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[22]

S. J. Patterson, Spectral theory and Fuchsian groups, Math. Proc. Cambridge Phil. Soc., 81 (1977), 59-75. doi: 10.1017/S030500410000027X.  Google Scholar

[23]

S. J. Patterson, Some examples of Fuchsian groups, Proc. London Math. Soc. (3), 39 (1979), 276-298. doi: 10.1112/plms/s3-39.2.276.  Google Scholar

[24]

Y. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114. doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar

[25]

Y. Pesin, On the work of Omri Sarig on infinite Markov chains and thermodynamical formalism, J. Modern Dynamics, (2014). Google Scholar

[26]

M. Ratner, On Raghunathan's measure conjecture, Ann. Math. (2), 134 (1991), 545-607. doi: 10.2307/2944357.  Google Scholar

[27]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N. S.), 95 (2003).  Google Scholar

[28]

O. Sarig, Invariant Radon measures for horocycle flows on Abelian covers, Invent. Math., 157 (2004), 519-551. doi: 10.1007/s00222-004-0357-4.  Google Scholar

[29]

O. Sarig, The horocycle flow and the Laplacian on hyperbolic surfaces of infinite genus, Geom. Funct. Anal., 19 (2010), 1757-1812. doi: 10.1007/s00039-010-0048-9.  Google Scholar

[30]

O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms, J. Modern Dynamics, 5 (2011), 593-608. doi: 10.3934/jmd.2011.5.593.  Google Scholar

[31]

O. 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

[32]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.  Google Scholar

[33]

B. Schapira, Equidistribution of the horocycles of a geomertically finite surface, Int. Mat. Res. Not., (2005), 2447-2471. doi: 10.1155/IMRN.2005.2447.  Google Scholar

[34]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

[35]

Y. G. Sinaĭ, Construction of Markov partitions, Functional Anal. Appl., 2 (1968), 245-253. doi: 10.1007/BF01076126.  Google Scholar

[36]

B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269. doi: 10.1090/S0002-9904-1970-12632-5.  Google Scholar

[37]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. (2), 147 (1988), 585-650. doi: 10.2307/120960.  Google Scholar

show all references

References:
[1]

R. Adler and B. Weiss, Similarities of the Automorphisms of the Torus, Memoirs of the Amer. Math. Soc., 98, Amer. Mat. Soc. Providence, RI, 1970.  Google Scholar

[2]

M. Babillot, On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds, in Random Walks and Geometry (ed. V. A. Kaimanovich), Walter de Gruyter GmbH & Co. KG, Berlin, 2004, 319-335.  Google Scholar

[3]

K. Berg, Convolutions of invariant measures, maximal entropy, Math. Systems Theory, 3 (1969), 146-150. doi: 10.1007/BF01746521.  Google Scholar

[4]

P. Berger, Properties of the maximal entropy measure and geometry of Hénon attractors, arXiv:1202.2822, (2012). Google Scholar

[5]

M. Babillot and F. Ledrappier, Geodesic paths and horocycle flows on abelian covers, in Lie Groups and Ergodic Theory (Mumbai, 1996), Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998, 1-32.  Google Scholar

[6]

R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans.Amer. Math. Soc., 154 (1971), 377-397.  Google Scholar

[7]

M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math., 14 (2002), 713-747. doi: 10.1515/form.2002.031.  Google Scholar

[8]

M. Burger, Horocycle flows on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803. doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar

[9]

D. Burguet, Symbolic extensions in intermediate smoothness on surfaces, Ann. Sci. Éc. Norm. Sup. (4), 45 (2012), 337-362.  Google Scholar

[10]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161. doi: 10.1007/BF02773637.  Google Scholar

[11]

S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J., 51 (1984), 185-194. doi: 10.1215/S0012-7094-84-05110-X.  Google Scholar

[12]

T. Downarowicz and S. Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math., 160 (2005), 453-499. doi: 10.1007/s00222-004-0413-0.  Google Scholar

[13]

N. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances Math., 5 (1970), 365-394. doi: 10.1016/0001-8708(70)90010-1.  Google Scholar

[14]

H. Furstenberg, The unique ergodicity of the horocycle flow, Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Springer Lect. Notes Math., 318, Springer, Berlin, 1972, 95-115.  Google Scholar

[15]

F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math., 34 (1979), 213-237; Israel J. Math., 38 (1981), 107-115. doi: 10.1007/BF02761854.  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]

V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811.  Google Scholar

[18]

F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 163-188.  Google Scholar

[19]

F. Ledrappier and O. Sarig, Invariant measures for horocycle flows on periodic hyperbolic surfaces, Israel J. Math., 160 (2007), 281-317. doi: 10.1007/s11856-007-0064-0.  Google Scholar

[20]

S. Newhouse, Continuity properties of entropy, Ann. Math. (2), 129 (1989), 215-235. doi: 10.2307/1971492.  Google Scholar

[21]

W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[22]

S. J. Patterson, Spectral theory and Fuchsian groups, Math. Proc. Cambridge Phil. Soc., 81 (1977), 59-75. doi: 10.1017/S030500410000027X.  Google Scholar

[23]

S. J. Patterson, Some examples of Fuchsian groups, Proc. London Math. Soc. (3), 39 (1979), 276-298. doi: 10.1112/plms/s3-39.2.276.  Google Scholar

[24]

Y. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114. doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar

[25]

Y. Pesin, On the work of Omri Sarig on infinite Markov chains and thermodynamical formalism, J. Modern Dynamics, (2014). Google Scholar

[26]

M. Ratner, On Raghunathan's measure conjecture, Ann. Math. (2), 134 (1991), 545-607. doi: 10.2307/2944357.  Google Scholar

[27]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N. S.), 95 (2003).  Google Scholar

[28]

O. Sarig, Invariant Radon measures for horocycle flows on Abelian covers, Invent. Math., 157 (2004), 519-551. doi: 10.1007/s00222-004-0357-4.  Google Scholar

[29]

O. Sarig, The horocycle flow and the Laplacian on hyperbolic surfaces of infinite genus, Geom. Funct. Anal., 19 (2010), 1757-1812. doi: 10.1007/s00039-010-0048-9.  Google Scholar

[30]

O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms, J. Modern Dynamics, 5 (2011), 593-608. doi: 10.3934/jmd.2011.5.593.  Google Scholar

[31]

O. 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

[32]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.  Google Scholar

[33]

B. Schapira, Equidistribution of the horocycles of a geomertically finite surface, Int. Mat. Res. Not., (2005), 2447-2471. doi: 10.1155/IMRN.2005.2447.  Google Scholar

[34]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

[35]

Y. G. Sinaĭ, Construction of Markov partitions, Functional Anal. Appl., 2 (1968), 245-253. doi: 10.1007/BF01076126.  Google Scholar

[36]

B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269. doi: 10.1090/S0002-9904-1970-12632-5.  Google Scholar

[37]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. (2), 147 (1988), 585-650. doi: 10.2307/120960.  Google Scholar

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