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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.

[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.

[3]

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

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[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.

[30]

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

[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.

[32]

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

[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.

[34]

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

[35]

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

[36]

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

[37]

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

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.

[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.

[3]

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

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[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.

[30]

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

[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.

[32]

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

[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.

[34]

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

[35]

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

[36]

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

[37]

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

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