The work of Federico Rodriguez Hertz on ergodicity of dynamical systems

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The 2015 Michael Brin Prize in Dynamical Systems

2016, 10: 175-189. doi: 10.3934/jmd.2016.10.175

The work of Federico Rodriguez Hertz on ergodicity of dynamical systems (Brin Prize article)

Dmitry Dolgopyat ^1,

1.	Department of Mathematics, University of Maryland, College Park, MD 20742, United States

Received March 2016 Published June 2016

Abstract
Related Papers
Under a Creative Commons license

We review recent advances on ergodicity of partially and nonuniformly hyperbolic systemsdescribing, in particular, important contributions of Federico Rodriguez Hertz and his collaborators.

Keywords: ergodicity, Hopf argument., Partial hyperbolicity, nonuniform hyperbolicity, accessibility.

Mathematics Subject Classification: Primary: 37D25, 37D30; Secondary: 37C4.

Citation: Dmitry Dolgopyat. The work of Federico Rodriguez Hertz on ergodicity of dynamical systems. Journal of Modern Dynamics, 2016, 10: 175-189. doi: 10.3934/jmd.2016.10.175

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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