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The 2015 Michael Brin Prize in Dynamical Systems
The work of Federico Rodriguez Hertz on ergodicity of dynamical systems (Brin Prize article)
1. | Department of Mathematics, University of Maryland, College Park, MD 20742, United States |
References:
[1] |
J. Aaronson, An Introduction to Infinite Ergodic Theory, Math. Surv. & Monographs, 50, AMS, Providence, RI, 1997.
doi: 10.1090/surv/050. |
[2] |
D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp. |
[3] |
D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems, Russian Math. Surveys, 22 (1967), 103-167.
doi: 10.1070/RM1967v022n05ABEH001228. |
[4] |
A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189.
doi: 10.1007/s00222-010-0243-1. |
[5] |
L. Barreira and Ya. B Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia Math., Appl., 115, Cambridge Univ. Press, Cambridge, 2007.
doi: 10.1017/CBO9781107326026. |
[6] |
M. Benedicks and M. Viana, Solution of the basin problem for Hénon-like attractors, Invent. Math., 143 (2001), 375-434.
doi: 10.1007/s002220000109. |
[7] |
Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, J. AMS, 5 (1992), 33-74.
doi: 10.2307/2152750. |
[8] |
C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encycl. Math. Sci., 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005. |
[9] |
C. Bonatti, X. Gómez-Mont and M. Viana, Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré Anal., 20 (2003), 579-624.
doi: 10.1016/S0294-1449(02)00019-7. |
[10] |
R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747.
doi: 10.2307/2373370. |
[11] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, 2nd revised edition, Lecture Notes Math., 470, Springer-Verlag, Berlin, 2008. |
[12] |
M. I. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Priložen., 9 (1975), 9-19. |
[13] |
M. I. Brin and Ya. B. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. |
[14] |
A. W. Brown and F. Rodriguez Hertz, Measure rigidity for random dynamics on surfaces and related skew products, arXiv:1506.06826. |
[15] |
K. Burns and A. Wilkinson, Stable ergodicity of skew products, Ann. ENS., 32 (1999), 859-889.
doi: 10.1016/S0012-9593(00)87721-6. |
[16] |
K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489.
doi: 10.4007/annals.2010.171.451. |
[17] |
N. Chernov and R. Markarian, Chaotic Billiards, Math. Surveys & Monographs, 127, AMS, Providence, RI, 2006.
doi: 10.1090/surv/127. |
[18] |
E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré, 15 (1998), 539-579.
doi: 10.1016/S0294-1449(98)80001-2. |
[19] |
D. Damjanović, Hamilton's theorem for smooth Lie group actions, in Ergodic Theory and Dynamical Systems (ed. Idris Assani), De Gruyter Proc. in Math., De Gruyter, Berlin, 2014, 117-127. |
[20] |
P. Didier, Stability of accessibility, Ergodic Th. Dyn. Syst., 23 (2003), 171-1731.
doi: 10.1017/S0143385702001785. |
[21] |
D. Dolgopyat, H. Hu and Ya. B. Pesin, An example of a smooth hyperbolic measure with countably many ergodic components, Proc. Symp. Pure Math., 69 (2001), 95-106. |
[22] |
D. Dolgopyat and R. Krikorian, On simultaneous linearization of diffeomorphisms of the sphere, Duke Math. J., 136 (2007), 475-505. |
[23] |
H. Furstenberg, Noncommuting random products, Trans. AMS, 108 (1963), 377-428.
doi: 10.1090/S0002-9947-1963-0163345-0. |
[24] |
M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2), 140 (1994), 295-329.
doi: 10.2307/2118602. |
[25] |
B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations, Erg. Th. Dynam. Sys., 19 (1999), 643-656.
doi: 10.1017/S0143385799133868. |
[26] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes Math., 583, Springer-Verlag, Berlin-New York, 1977. |
[27] |
E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91 (1939), 261-304. |
[28] |
S. Kakutani, Random ergodic theorems and Markoff processes with a stable distribution, in Proc. 2nd Berkeley Symposium on Math. Stat. & Prob., Univ. of California Press, Berkeley-Los Angeles, 1951, 247-261. |
[29] |
I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. AMS (N.S.), 31 (1994), 68-74.
doi: 10.1090/S0273-0979-1994-00507-5. |
[30] |
A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Math. Res. Lett., 3 (1996), 191-210.
doi: 10.4310/MRL.1996.v3.n2.a6. |
[31] |
F. Ledrappier, Quelques propriétés des exposants caractéristiques, Lecture Notes Math., 1097, Springer, Berlin, 1984, 305-396.
doi: 10.1007/BFb0099434. |
[32] |
R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys & Monographs, 91, AMS, Providence, RI, 2002. |
[33] |
W. Parry and M. Pollicott, Skew products and Livsic theory, in Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, AMS Transl. Ser. 2, 217, AMS, Providence, RI, 2006, 139-165. |
[34] |
Ya. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114.
doi: 10.1070/RM1977v032n04ABEH001639. |
[35] |
Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties, Erg. Th. Dynam. Sys., 12 (1992), 123-151.
doi: 10.1017/S0143385700006635. |
[36] |
C. C. Pugh and M. Shub, Ergodic attractors, Trans. AMS, 312 (1989), 1-54.
doi: 10.1090/S0002-9947-1989-0983869-1. |
[37] |
C. C. Pugh and M. Shub, Stable ergodicity and stable accessibility, in Differential Equations and Applications (Hangzhou, 1996), Int. Press, Cambridge, MA, 258-268. |
[38] |
C. C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity, 13 (1997), 125-179.
doi: 10.1006/jcom.1997.0437. |
[39] |
C. Pugh, M. Shub and A. Starkov, Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms, Discrete Contin. Dyn. Syst., 14 (2006), 845-855.
doi: 10.3934/dcds.2006.14.845. |
[40] |
D. Repovš, A. B. Skopenkov and E. V. Ščepin, $C^1$-homogeneous compacta in $\mathbb{R}^{N}$ are $C^1$-submanifolds of $\mathbb{R}^{N}$, Proc. AMS, 124 (1996), 1219-1226.
doi: 10.1090/S0002-9939-96-03157-7. |
[41] |
F. Rodriguez Hertz, Stable ergodicity of certain linear automorphisms of the torus, Ann. of Math. (2), 162 (2005), 65-107.
doi: 10.4007/annals.2005.162.65. |
[42] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures, Uniqueness of SRB measures for transitive diffeomorphisms on surfaces, Comm. Math. Phys., 306 (2011), 35-49.
doi: 10.1007/s00220-011-1275-0. |
[43] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and nonuniform hyperbolicity, Duke Math. J., 160 (2011), 599-629.
doi: 10.1215/00127094-1444314. |
[44] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three, J. Mod. Dyn., 2 (2008), 187-208.
doi: 10.3934/jmd.2008.2.187. |
[45] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381.
doi: 10.1007/s00222-007-0100-z. |
[46] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and abundance of ergodicity in dimension three: A survey, Publ. Mat. Urug., 12 (2011), 177-198. |
[47] |
D. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.
doi: 10.2307/2373810. |
[48] |
Ya. G. Sinai, Construction of Markov partitionings, Funct. An., Appl., 2 (1968), 64-89, 70-80. |
[49] |
Ya. G. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64. |
[50] |
S. Smale, Differentiable dynamical systems, Bull. AMS, 73 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
[51] |
R. Spatzier, On the work of Rodriguez Hertz on rigidity in dynamics, J. Mod. Dyn., 10 (2016), 191-207.
doi: 10.3934/jmd.2016.10.191. |
[52] |
M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math. (2), 167 (2008), 643-680.
doi: 10.4007/annals.2008.167.643. |
[53] |
A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Erg. Th. Dynam. Sys., 18 (1998), 1545-1587.
doi: 10.1017/S0143385798117984. |
[54] |
A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75-165. |
[55] |
L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754.
doi: 10.1023/A:1019762724717. |
show all references
References:
[1] |
J. Aaronson, An Introduction to Infinite Ergodic Theory, Math. Surv. & Monographs, 50, AMS, Providence, RI, 1997.
doi: 10.1090/surv/050. |
[2] |
D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp. |
[3] |
D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems, Russian Math. Surveys, 22 (1967), 103-167.
doi: 10.1070/RM1967v022n05ABEH001228. |
[4] |
A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189.
doi: 10.1007/s00222-010-0243-1. |
[5] |
L. Barreira and Ya. B Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia Math., Appl., 115, Cambridge Univ. Press, Cambridge, 2007.
doi: 10.1017/CBO9781107326026. |
[6] |
M. Benedicks and M. Viana, Solution of the basin problem for Hénon-like attractors, Invent. Math., 143 (2001), 375-434.
doi: 10.1007/s002220000109. |
[7] |
Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, J. AMS, 5 (1992), 33-74.
doi: 10.2307/2152750. |
[8] |
C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encycl. Math. Sci., 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005. |
[9] |
C. Bonatti, X. Gómez-Mont and M. Viana, Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré Anal., 20 (2003), 579-624.
doi: 10.1016/S0294-1449(02)00019-7. |
[10] |
R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747.
doi: 10.2307/2373370. |
[11] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, 2nd revised edition, Lecture Notes Math., 470, Springer-Verlag, Berlin, 2008. |
[12] |
M. I. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Priložen., 9 (1975), 9-19. |
[13] |
M. I. Brin and Ya. B. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. |
[14] |
A. W. Brown and F. Rodriguez Hertz, Measure rigidity for random dynamics on surfaces and related skew products, arXiv:1506.06826. |
[15] |
K. Burns and A. Wilkinson, Stable ergodicity of skew products, Ann. ENS., 32 (1999), 859-889.
doi: 10.1016/S0012-9593(00)87721-6. |
[16] |
K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489.
doi: 10.4007/annals.2010.171.451. |
[17] |
N. Chernov and R. Markarian, Chaotic Billiards, Math. Surveys & Monographs, 127, AMS, Providence, RI, 2006.
doi: 10.1090/surv/127. |
[18] |
E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré, 15 (1998), 539-579.
doi: 10.1016/S0294-1449(98)80001-2. |
[19] |
D. Damjanović, Hamilton's theorem for smooth Lie group actions, in Ergodic Theory and Dynamical Systems (ed. Idris Assani), De Gruyter Proc. in Math., De Gruyter, Berlin, 2014, 117-127. |
[20] |
P. Didier, Stability of accessibility, Ergodic Th. Dyn. Syst., 23 (2003), 171-1731.
doi: 10.1017/S0143385702001785. |
[21] |
D. Dolgopyat, H. Hu and Ya. B. Pesin, An example of a smooth hyperbolic measure with countably many ergodic components, Proc. Symp. Pure Math., 69 (2001), 95-106. |
[22] |
D. Dolgopyat and R. Krikorian, On simultaneous linearization of diffeomorphisms of the sphere, Duke Math. J., 136 (2007), 475-505. |
[23] |
H. Furstenberg, Noncommuting random products, Trans. AMS, 108 (1963), 377-428.
doi: 10.1090/S0002-9947-1963-0163345-0. |
[24] |
M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2), 140 (1994), 295-329.
doi: 10.2307/2118602. |
[25] |
B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations, Erg. Th. Dynam. Sys., 19 (1999), 643-656.
doi: 10.1017/S0143385799133868. |
[26] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes Math., 583, Springer-Verlag, Berlin-New York, 1977. |
[27] |
E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91 (1939), 261-304. |
[28] |
S. Kakutani, Random ergodic theorems and Markoff processes with a stable distribution, in Proc. 2nd Berkeley Symposium on Math. Stat. & Prob., Univ. of California Press, Berkeley-Los Angeles, 1951, 247-261. |
[29] |
I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. AMS (N.S.), 31 (1994), 68-74.
doi: 10.1090/S0273-0979-1994-00507-5. |
[30] |
A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Math. Res. Lett., 3 (1996), 191-210.
doi: 10.4310/MRL.1996.v3.n2.a6. |
[31] |
F. Ledrappier, Quelques propriétés des exposants caractéristiques, Lecture Notes Math., 1097, Springer, Berlin, 1984, 305-396.
doi: 10.1007/BFb0099434. |
[32] |
R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys & Monographs, 91, AMS, Providence, RI, 2002. |
[33] |
W. Parry and M. Pollicott, Skew products and Livsic theory, in Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, AMS Transl. Ser. 2, 217, AMS, Providence, RI, 2006, 139-165. |
[34] |
Ya. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114.
doi: 10.1070/RM1977v032n04ABEH001639. |
[35] |
Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties, Erg. Th. Dynam. Sys., 12 (1992), 123-151.
doi: 10.1017/S0143385700006635. |
[36] |
C. C. Pugh and M. Shub, Ergodic attractors, Trans. AMS, 312 (1989), 1-54.
doi: 10.1090/S0002-9947-1989-0983869-1. |
[37] |
C. C. Pugh and M. Shub, Stable ergodicity and stable accessibility, in Differential Equations and Applications (Hangzhou, 1996), Int. Press, Cambridge, MA, 258-268. |
[38] |
C. C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity, 13 (1997), 125-179.
doi: 10.1006/jcom.1997.0437. |
[39] |
C. Pugh, M. Shub and A. Starkov, Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms, Discrete Contin. Dyn. Syst., 14 (2006), 845-855.
doi: 10.3934/dcds.2006.14.845. |
[40] |
D. Repovš, A. B. Skopenkov and E. V. Ščepin, $C^1$-homogeneous compacta in $\mathbb{R}^{N}$ are $C^1$-submanifolds of $\mathbb{R}^{N}$, Proc. AMS, 124 (1996), 1219-1226.
doi: 10.1090/S0002-9939-96-03157-7. |
[41] |
F. Rodriguez Hertz, Stable ergodicity of certain linear automorphisms of the torus, Ann. of Math. (2), 162 (2005), 65-107.
doi: 10.4007/annals.2005.162.65. |
[42] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures, Uniqueness of SRB measures for transitive diffeomorphisms on surfaces, Comm. Math. Phys., 306 (2011), 35-49.
doi: 10.1007/s00220-011-1275-0. |
[43] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and nonuniform hyperbolicity, Duke Math. J., 160 (2011), 599-629.
doi: 10.1215/00127094-1444314. |
[44] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three, J. Mod. Dyn., 2 (2008), 187-208.
doi: 10.3934/jmd.2008.2.187. |
[45] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381.
doi: 10.1007/s00222-007-0100-z. |
[46] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and abundance of ergodicity in dimension three: A survey, Publ. Mat. Urug., 12 (2011), 177-198. |
[47] |
D. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.
doi: 10.2307/2373810. |
[48] |
Ya. G. Sinai, Construction of Markov partitionings, Funct. An., Appl., 2 (1968), 64-89, 70-80. |
[49] |
Ya. G. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64. |
[50] |
S. Smale, Differentiable dynamical systems, Bull. AMS, 73 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
[51] |
R. Spatzier, On the work of Rodriguez Hertz on rigidity in dynamics, J. Mod. Dyn., 10 (2016), 191-207.
doi: 10.3934/jmd.2016.10.191. |
[52] |
M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math. (2), 167 (2008), 643-680.
doi: 10.4007/annals.2008.167.643. |
[53] |
A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Erg. Th. Dynam. Sys., 18 (1998), 1545-1587.
doi: 10.1017/S0143385798117984. |
[54] |
A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75-165. |
[55] |
L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754.
doi: 10.1023/A:1019762724717. |
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Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048 |
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