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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

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., (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), (2004), 319.   Google Scholar

[3]

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

[4]

P. Berger, Properties of the maximal entropy measure and geometry of Hénon attractors,, , (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), 1.   Google Scholar

[6]

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

[7]

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

[8]

M. Burger, Horocycle flows on geometrically finite surfaces,, Duke Math. J., 61 (1990), 779.  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.   Google Scholar

[10]

J. Buzzi, Intrinsic ergodicity of smooth interval maps,, Israel J. Math., 100 (1997), 125.  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.  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.  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.  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., (1972), 95.   Google Scholar

[15]

F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy,, Israel J. Math., 34 (1979), 213.  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.   Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

W. Parry, Intrinsic Markov chains,, Trans. Amer. Math. Soc., 112 (1964), 55.  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.  doi: 10.1017/S030500410000027X.  Google Scholar

[23]

S. J. Patterson, Some examples of Fuchsian groups,, Proc. London Math. Soc. (3), 39 (1979), 276.  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.  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.  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.  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.  doi: 10.1007/s00039-010-0048-9.  Google Scholar

[30]

O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms,, J. Modern Dynamics, 5 (2011), 593.  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.  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.  doi: 10.1017/S0143385799146820.  Google Scholar

[33]

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

[34]

M. Shub, Global Stability of Dynamical Systems,, Springer-Verlag, (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.  doi: 10.1007/BF01076126.  Google Scholar

[36]

B. Weiss, Intrinsically ergodic systems,, Bull. Amer. Math. Soc., 76 (1970), 1266.  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.  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., (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), (2004), 319.   Google Scholar

[3]

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

[4]

P. Berger, Properties of the maximal entropy measure and geometry of Hénon attractors,, , (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), 1.   Google Scholar

[6]

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

[7]

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

[8]

M. Burger, Horocycle flows on geometrically finite surfaces,, Duke Math. J., 61 (1990), 779.  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.   Google Scholar

[10]

J. Buzzi, Intrinsic ergodicity of smooth interval maps,, Israel J. Math., 100 (1997), 125.  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.  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.  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.  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., (1972), 95.   Google Scholar

[15]

F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy,, Israel J. Math., 34 (1979), 213.  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.   Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

W. Parry, Intrinsic Markov chains,, Trans. Amer. Math. Soc., 112 (1964), 55.  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.  doi: 10.1017/S030500410000027X.  Google Scholar

[23]

S. J. Patterson, Some examples of Fuchsian groups,, Proc. London Math. Soc. (3), 39 (1979), 276.  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.  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.  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.  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.  doi: 10.1007/s00039-010-0048-9.  Google Scholar

[30]

O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms,, J. Modern Dynamics, 5 (2011), 593.  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.  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.  doi: 10.1017/S0143385799146820.  Google Scholar

[33]

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

[34]

M. Shub, Global Stability of Dynamical Systems,, Springer-Verlag, (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.  doi: 10.1007/BF01076126.  Google Scholar

[36]

B. Weiss, Intrinsically ergodic systems,, Bull. Amer. Math. Soc., 76 (1970), 1266.  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.  doi: 10.2307/120960.  Google Scholar

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