Abstract
We show that $C^\infty $-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal entropy in general, and exactly one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist. To do this we generalize Smale’s spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irreducible countable state Markov shifts.
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author = {Buzzi, J{é}r{ô}me},
title = {{$C^r$} surface diffeomorphisms with no maximal entropy measure},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {34},
year = {2014},
number = {6},
pages = {1770--1793},
issn = {0143-3857},
mrclass = {37B40},
mrnumber = {3272770},
doi = {10.1017/etds.2013.25},
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author = {Buzzi, J{é}r{ô}me},
title = {The almost {B}orel structure of diffeomorphisms with some hyperbolicity},
booktitle = {Hyperbolic Dynamics, Fluctuations and Large Deviations},
series = {Proc. Sympos. Pure Math.},
volume = {89},
pages = {9--44},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2015},
mrclass = {37A35 (37B10 37D25)},
mrnumber = {3309094},
mrreviewer = {Anthony Quas},
doi = {10.1090/pspum/089/01491},
url = {https://doi.org/10.1090/pspum/089/01491},
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J. Buzzi, "The degree of Bowen factors and injective codings of diffeomorphisms," J. Mod. Dyn., vol. 16, pp. 1-36, 2020.
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author = {Buzzi, J{é}r{ô}me},
title = {The degree of {B}owen factors and injective codings of diffeomorphisms},
journal = {J. Mod. Dyn.},
fjournal = {Journal of Modern Dynamics},
volume = {16},
year = {2020},
pages = {1--36},
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author = { Buzzi, J{é}r{ô}me and Crovisier, S. and Sarig, Omri},
title = {Surface diffeomorphisms with infinitely many or no measures maximizing the entropy},
note = {in preparation},
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author = {Buzzi, J{é}r{ô}me and Fisher, Todd},
title = {Entropic stability beyond partial hyperbolicity},
journal = {J. Mod. Dyn.},
fjournal = {Journal of Modern Dynamics},
volume = {7},
year = {2013},
number = {4},
pages = {527--552},
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author = {Buzzi, J{é}r{ô}me and Fisher, Todd and Sambarino, M. and V\'{a}squez, C.},
title = {Maximal entropy measures for certain partially hyperbolic, derived from {A}nosov systems},
journal = {Ergodic Theory Dynam. Systems},
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volume = {32},
year = {2012},
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author = {Buzzi, J{é}r{ô}me and Ruette, Sylvie},
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journal = {Discrete Contin. Dyn. Syst.},
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J. Buzzi and O. Sarig, "Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps," Ergodic Theory Dynam. Systems, vol. 23, iss. 5, pp. 1383-1400, 2003.
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author = {Buzzi, J{é}r{ô}me and Sarig, Omri},
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author = {Climenhaga, Vaughn and Fisher, Todd and Thompson, Daniel J.},
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author = {Climenhaga, Vaughn and Thompson, Daniel J.},
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doi = {10.1017/S0143385700001528},
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} -
[Lima-Matheus-2018] Y. Lima and C. Matheus, "Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities," Ann. Sci. Éc. Norm. Supér. (4), vol. 51, iss. 1, pp. 1-38, 2018.
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author = {Lima, Yuri and Matheus, Carlos},
title = {Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities},
journal = {Ann. Sci. \'{E}c. Norm. Supér. (4)},
fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
volume = {51},
year = {2018},
number = {1},
pages = {1--38},
issn = {0012-9593},
mrclass = {37B10 (37B25 37D30 37D50)},
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[Lima-Sarig]
Y. Lima and O. M. Sarig, "Symbolic dynamics for three-dimensional flows with positive topological entropy," J. Eur. Math. Soc. (JEMS), vol. 21, iss. 1, pp. 199-256, 2019.
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author = {Lima, Yuri and Sarig, Omri M.},
title = {Symbolic dynamics for three-dimensional flows with positive topological entropy},
journal = {J. Eur. Math. Soc. (JEMS)},
fjournal = {Journal of the European Mathematical Society (JEMS)},
volume = {21},
year = {2019},
number = {1},
pages = {199--256},
issn = {1435-9855},
mrclass = {37B10 (37C27 37D25 37D40 53D25)},
mrnumber = {3880208},
mrreviewer = {Nicolas Bedaride},
doi = {10.4171/JEMS/834},
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[Mane-MME-1983]
R. Mañé, "On the uniqueness of the maximizing measure for rational maps," Bol. Soc. Brasil. Mat., vol. 14, iss. 1, pp. 27-43, 1983.
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author = {Ma\~{n}é,
Ricardo},
title = {On the uniqueness of the maximizing measure for rational maps},
journal = {Bol. Soc. Brasil. Mat.},
fjournal = {Boletim da Sociedade Brasileira de Matem\'{a}tica},
volume = {14},
year = {1983},
number = {1},
pages = {27--43},
issn = {0100-3569},
mrclass = {58F11 (30D05)},
mrnumber = {0736567},
mrreviewer = {D. Newton},
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[manning-dimension]
A. Manning, "A relation between Lyapunov exponents, Hausdorff dimension and entropy," Ergodic Theory Dynam. Systems, vol. 1, iss. 4, pp. 451-459, 1981.
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author = {Manning, Anthony},
title = {A relation between {L}yapunov exponents, {H}ausdorff dimension and entropy},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {1},
year = {1981},
number = {4},
pages = {451--459},
issn = {0143-3857},
mrclass = {58F11 (54C70 82A05)},
mrnumber = {0662737},
doi = {10.1017/s0143385700001371},
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} -
[misiurewicz] M. Misiurewicz, "Diffeomorphism without any measure with maximal entropy," Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., vol. 21, pp. 903-910, 1973.
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author = {Misiurewicz, M.},
title = {Diffeomorphism without any measure with maximal entropy},
journal = {Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.},
fjournal = {Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques},
volume = {21},
year = {1973},
pages = {903--910},
issn = {0001-4117},
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[MisiurewiczTCE]
M. Misiurewicz, "Topological conditional entropy," Studia Math., vol. 55, iss. 2, pp. 175-200, 1976.
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author = {Misiurewicz, Micha\l},
title = {Topological conditional entropy},
journal = {Studia Math.},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
volume = {55},
year = {1976},
number = {2},
pages = {175--200},
issn = {0039-3223},
mrclass = {54H20},
mrnumber = {0415587},
mrreviewer = {Timothy N.t Goodman},
doi = {10.4064/sm-55-2-175-200},
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[Newhouse-Homoclinic]
S. E. Newhouse, "Hyperbolic limit sets," Trans. Amer. Math. Soc., vol. 167, pp. 125-150, 1972.
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author = {Newhouse, Sheldon E.},
title = {Hyperbolic limit sets},
journal = {Trans. Amer. Math. Soc.},
fjournal = {Transactions of the American Mathematical Society},
volume = {167},
year = {1972},
pages = {125--150},
issn = {0002-9947},
mrclass = {58F15 (34C35)},
mrnumber = {0295388},
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[newhouse1978] S. E. Newhouse, "Topological entropy and Hausdorff dimension for area preserving diffeomorphisms of surfaces," in Dynamical Systems, Vol. III—Warsaw, Math. Soc. France, Paris, 1978, vol. 51, pp. 323-334.
@INCOLLECTION{newhouse1978,
author = {Newhouse, S. E.},
title = {Topological entropy and {H}ausdorff dimension for area preserving diffeomorphisms of surfaces},
booktitle = {Dynamical Systems, {V}ol. {III}---{W}arsaw},
pages = {323--334},
series = {Astérisque},
volume = {51},
publisher = {Math. Soc. France, Paris},
year = {1978},
mrclass = {58F15 (28A65)},
mrnumber = {0482851},
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} -
[Newhouse-Entropy]
S. E. Newhouse, "Continuity properties of entropy," Ann. of Math. (2), vol. 129, iss. 2, pp. 215-235, 1989.
@ARTICLE{Newhouse-Entropy,
author = {Newhouse, Sheldon E.},
title = {Continuity properties of entropy},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {129},
year = {1989},
number = {2},
pages = {215--235},
issn = {0003-486X},
mrclass = {58F11 (28D20 54C70)},
mrnumber = {0986792},
mrreviewer = {A. I. Danilenko},
doi = {10.2307/1971492},
url = {https://doi.org/10.2307/1971492},
zblnumber = {0676.58039},
} -
[Newhouse1990] S. E. Newhouse, "Entropy in smooth dynamical systems," in Proceedings of the International Congress of Mathematicians, Vol. I, II, 1991, pp. 1285-1294.
@INPROCEEDINGS{Newhouse1990,
author = {Newhouse, Sheldon E.},
title = {Entropy in smooth dynamical systems},
booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {I},
{II}},
venue = {{K}yoto, 1990},
pages = {1285--1294},
publisher = {Math. Soc. Japan, Tokyo},
year = {1991},
mrclass = {58F11 (28D20)},
mrnumber = {1159313},
mrreviewer = {Ladislav Andrey},
zblnumber = {0745.54005},
} -
[Newhouse-On-Hofbauer] S. N. Newhouse, "On some results of Hofbauer on maps of the interval," in Dynamical Systems and Related Topics, World Sci. Publ., River Edge, NJ, 1991, vol. 9, pp. 407-421.
@INCOLLECTION{Newhouse-On-Hofbauer,
author = {Newhouse, Sheldon{ \noopsort{E.}}},
title = {On some results of {H}ofbauer on maps of the interval},
booktitle = {Dynamical Systems and Related Topics},
venue = {{N}agoya, 1990},
series = {Adv. Ser. Dynam. Systems},
volume = {9},
pages = {407--421},
publisher = {World Sci. Publ., River Edge, NJ},
year = {1991},
mrclass = {58F11},
mrnumber = {1164905},
zblnumber = {},
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[Newhouse-LSY-1983]
S. E. NewhousenoopsortXXX and L. Young, "Dynamics of certain skew products," in Geometric Dynamics, Springer, Berlin, 1983, vol. 1007, pp. 611-629.
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author = {Newhouse\noopsort{XXX},
Sheldon E. and Young, Lai-Sang},
title = {Dynamics of certain skew products},
booktitle = {Geometric Dynamics},
venue = {{R}io de {J}aneiro, 1981},
series = {Lecture Notes in Math.},
volume = {1007},
pages = {611--629},
publisher = {Springer, Berlin},
year = {1983},
mrclass = {58F11 (28D20 54C70)},
mrnumber = {0730289},
mrreviewer = {R. L. Adler},
doi = {10.1007/BFb0061436},
url = {https://doi.org/10.1007/BFb0061436},
zblnumber = {0535.58036},
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[palis-viana]
J. Palis and M. Viana, "On the continuity of Hausdorff dimension and limit capacity for horseshoes," in Dynamical systems, Valparaiso 1986, Springer, Berlin, 1988, vol. 1331, pp. 150-160.
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author = {Palis, J. and Viana, M.},
title = {On the continuity of {H}ausdorff dimension and limit capacity for horseshoes},
booktitle = {Dynamical systems, {V}alparaiso 1986},
series = {Lecture Notes in Math.},
volume = {1331},
pages = {150--160},
publisher = {Springer, Berlin},
year = {1988},
mrclass = {58F15},
mrnumber = {0961098},
mrreviewer = {Hans G. Bothe},
doi = {10.1007/BFb0083071},
url = {https://doi.org/10.1007/BFb0083071},
zblnumber = {0661.58023},
} -
[Parry-Entropy]
W. Parry, "Intrinsic Markov chains," Trans. Amer. Math. Soc., vol. 112, pp. 55-66, 1964.
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author = {Parry, William},
title = {Intrinsic {M}arkov chains},
journal = {Trans. Amer. Math. Soc.},
fjournal = {Transactions of the American Mathematical Society},
volume = {112},
year = {1964},
pages = {55--66},
issn = {0002-9947},
mrclass = {60.65},
mrnumber = {0161372},
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[Parry-Pollicott-Asterisque] W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Math. Soc. France, Paris, 1990, vol. 187-188.
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author = {Parry, William and Pollicott, Mark},
title = {Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics},
series = {Astérisque},
publisher = {Math. Soc. France, Paris},
volume = {187-188},
year = {1990},
pages = {268},
issn = {0303-1179},
mrclass = {58F20 (58F11 58F15)},
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[Pavlov-2016]
R. Pavlov, "On intrinsic ergodicity and weakenings of the specification property," Adv. Math., vol. 295, pp. 250-270, 2016.
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author = {Pavlov, Ronnie},
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volume = {295},
year = {2016},
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mrnumber = {3488036},
mrreviewer = {Dominik Kwietniak},
doi = {10.1016/j.aim.2016.03.013},
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[Pavlov-NUspecification-2017]
R. Pavlov, "On controlled specification and uniqueness of the equilibrium state in expansive systems," Nonlinearity, vol. 32, iss. 7, pp. 2441-2466, 2019.
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title = {On controlled specification and uniqueness of the equilibrium state in expansive systems},
journal = {Nonlinearity},
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volume = {32},
year = {2019},
number = {7},
pages = {2441--2466},
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mrclass = {37D35 (37B10 37B40)},
mrnumber = {3957218},
doi = {10.1088/1361-6544/ab0e5b},
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[Pesin-Izvestia-1976] J. B. Pesin, "Families of invariant manifolds that correspond to nonzero characteristic exponents," Izv. Akad. Nauk SSSR Ser. Mat., vol. 40, iss. 6, pp. 1332-1379, 1976.
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author = {Pesin, Ja. B.},
title = {Families of invariant manifolds that correspond to nonzero characteristic exponents},
journal = {Izv. Akad. Nauk SSSR Ser. Mat.},
fjournal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
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number = {6},
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[Pinto-Rand]
A. A. Pinto and D. A. Rand, "Smoothness of holonomies for codimension 1 hyperbolic dynamics," Bull. London Math. Soc., vol. 34, iss. 3, pp. 341-352, 2002.
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author = {Pinto, A. A. and Rand, D. A.},
title = {Smoothness of holonomies for codimension 1 hyperbolic dynamics},
journal = {Bull. London Math. Soc.},
fjournal = {The Bulletin of the London Mathematical Society},
volume = {34},
year = {2002},
number = {3},
pages = {341--352},
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mrclass = {37D20 (37C05 37C15)},
mrnumber = {1887706},
mrreviewer = {Fernando Fl\'{a}vio Ferreira},
doi = {10.1112/S0024609301008670},
url = {https://doi.org/10.1112/S0024609301008670},
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[Hertz-Hertz-Tahzibi-Ures-2012]
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, and R. Ures, "Maximizing measures for partially hyperbolic systems with compact center leaves," Ergodic Theory Dynam. Systems, vol. 32, iss. 2, pp. 825-839, 2012.
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author = {Rodriguez Hertz, F. and Rodriguez Hertz, M. A. and Tahzibi, A. and Ures, R.},
title = {Maximizing measures for partially hyperbolic systems with compact center leaves},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
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[Rodriguez-Hertz-Squared-Tahzibi-Ures-CMP]
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, and R. Ures, "Uniqueness of SRB measures for transitive diffeomorphisms on surfaces," Comm. Math. Phys., vol. 306, iss. 1, pp. 35-49, 2011.
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author = {Rodriguez Hertz, F. and Rodriguez Hertz, M. A. and Tahzibi, A. and Ures, R.},
title = {Uniqueness of {SRB} measures for transitive diffeomorphisms on surfaces},
journal = {Comm. Math. Phys.},
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year = {2011},
number = {1},
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mrreviewer = {Dante Carrasco-Olivera},
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[RokhlinLectures] V. A. Rohlin, "Lectures on the entropy theory of transformations with invariant measure," Uspehi Mat. Nauk, vol. 22, iss. 5 (137), pp. 3-56, 1967.
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author = {Rohlin, V. A.},
title = {Lectures on the entropy theory of transformations with invariant measure},
journal = {Uspehi Mat. Nauk},
fjournal = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
volume = {22},
year = {1967},
number = {5 (137)},
pages = {3--56},
issn = {0042-1316},
mrclass = {28.70},
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mrreviewer = {D. Newton},
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} -
[Ruelle-Entropy-Inequality]
D. Ruelle, "An inequality for the entropy of differentiable maps," Bol. Soc. Brasil. Mat., vol. 9, iss. 1, pp. 83-87, 1978.
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author = {Ruelle, David},
title = {An inequality for the entropy of differentiable maps},
journal = {Bol. Soc. Brasil. Mat.},
fjournal = {Boletim da Sociedade Brasileira de Matem\'{a}tica},
volume = {9},
year = {1978},
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pages = {83--87},
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[Ruette-Note-2015]
S. Ruette, "Transitive topological Markov chains of given entropy and period with or without measure of maximal entropy," Pacific J. Math., vol. 303, iss. 1, pp. 317-323, 2019.
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author = {Ruette, Sylvie},
title = {Transitive topological {M}arkov chains of given entropy and period with or without measure of maximal entropy},
journal = {Pacific J. Math.},
fjournal = {Pacific Journal of Mathematics},
volume = {303},
year = {2019},
number = {1},
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} -
@ARTICLE{Sard-1958,
author = {Sard, Arthur},
title = {Images of critical sets},
journal = {Ann. of Math. (2)},
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volume = {68},
year = {1958},
pages = {247--259},
issn = {0003-486X},
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[Sarig-thermodynamics]
O. M. Sarig, "Thermodynamic formalism for countable Markov shifts," Ergodic Theory Dynam. Systems, vol. 19, iss. 6, pp. 1565-1593, 1999.
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author = {Sarig, Omri M.},
title = {Thermodynamic formalism for countable {M}arkov shifts},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {19},
year = {1999},
number = {6},
pages = {1565--1593},
issn = {0143-3857},
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mrreviewer = {Irene Hueter},
doi = {10.1017/S0143385799146820},
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[Sarig-Bernoulli-JMD]
O. M. Sarig, "Bernoulli equilibrium states for surface diffeomorphisms," J. Mod. Dyn., vol. 5, iss. 3, pp. 593-608, 2011.
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author = {Sarig, Omri M.},
title = {Bernoulli equilibrium states for surface diffeomorphisms},
journal = {J. Mod. Dyn.},
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volume = {5},
year = {2011},
number = {3},
pages = {593--608},
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[Sarig-JAMS]
O. M. Sarig, "Symbolic dynamics for surface diffeomorphisms with positive entropy," J. Amer. Math. Soc., vol. 26, iss. 2, pp. 341-426, 2013.
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author = {Sarig, Omri M.},
title = {Symbolic dynamics for surface diffeomorphisms with positive entropy},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the American Mathematical Society},
volume = {26},
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number = {2},
pages = {341--426},
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} -
@BOOK{Shub-Book,
author = {Shub, Michael},
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publisher = {Springer-Verlag, New York},
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pages = {xii+150},
isbn = {0-387-96295-6},
mrclass = {58F10 (34Dxx)},
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[Sinai-Gibbs] Y. G. Sinaui, "Gibbs measures in ergodic theory," Uspehi Mat. Nauk, vol. 27, iss. 4(166), pp. 21-64, 1972.
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author = {Sinaĭ,
Ya. G.},
title = {Gibbs measures in ergodic theory},
journal = {Uspehi Mat. Nauk},
fjournal = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
volume = {27},
year = {1972},
number = {4(166)},
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[Smale-Diff-Dynam-Syst]
S. Smale, "Differentiable dynamical systems," Bull. Amer. Math. Soc., vol. 73, pp. 747-817, 1967.
@ARTICLE{Smale-Diff-Dynam-Syst,
author = {Smale, S.},
title = {Differentiable dynamical systems},
journal = {Bull. Amer. Math. Soc.},
fjournal = {Bulletin of the American Mathematical Society},
volume = {73},
year = {1967},
pages = {747--817},
issn = {0002-9904},
mrclass = {57.47 (34.00)},
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mrreviewer = {M. M. Peixoto},
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pages = {321--332},
publisher = {Amer. Math. Soc., Providence, RI},
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doi = {10.1007/BF02766215},
url = {https://doi.org/10.1007/BF02766215},
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}