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

The work of Federico Rodriguez Hertz on ergodicity of dynamical systems

Dmitry Dolgopyat 1,

1. 

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

Received  March 2016 Published  June 2016

We review recent advances on ergodicity of partially and nonuniformly hyperbolic systems describing, 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, (1997).  doi: 10.1090/surv/050.  Google Scholar

[2]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967).   Google Scholar

[3]

D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems,, Russian Math. Surveys, 22 (1967), 103.  doi: 10.1070/RM1967v022n05ABEH001228.  Google Scholar

[4]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications,, Invent. Math., 181 (2010), 115.  doi: 10.1007/s00222-010-0243-1.  Google Scholar

[5]

L. Barreira and Ya. B Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,, Encyclopedia Math., (2007).  doi: 10.1017/CBO9781107326026.  Google Scholar

[6]

M. Benedicks and M. Viana, Solution of the basin problem for Hénon-like attractors,, Invent. Math., 143 (2001), 375.  doi: 10.1007/s002220000109.  Google Scholar

[7]

Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables,, J. AMS, 5 (1992), 33.  doi: 10.2307/2152750.  Google Scholar

[8]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,, Encycl. Math. Sci., (2005).   Google Scholar

[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.  doi: 10.1016/S0294-1449(02)00019-7.  Google Scholar

[10]

R. Bowen, Markov partitions for Axiom A diffeomorphisms,, Amer. J. Math., 92 (1970), 725.  doi: 10.2307/2373370.  Google Scholar

[11]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, 2nd revised edition, (2008).   Google Scholar

[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.   Google Scholar

[13]

M. I. Brin and Ya. B. Pesin, Partially hyperbolic dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170.   Google Scholar

[14]

A. W. Brown and F. Rodriguez Hertz, Measure rigidity for random dynamics on surfaces and related skew products,, , ().   Google Scholar

[15]

K. Burns and A. Wilkinson, Stable ergodicity of skew products,, Ann. ENS., 32 (1999), 859.  doi: 10.1016/S0012-9593(00)87721-6.  Google Scholar

[16]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems,, Ann. of Math. (2), 171 (2010), 451.  doi: 10.4007/annals.2010.171.451.  Google Scholar

[17]

N. Chernov and R. Markarian, Chaotic Billiards,, Math. Surveys & Monographs, (2006).  doi: 10.1090/surv/127.  Google Scholar

[18]

E. Colli, Infinitely many coexisting strange attractors,, Ann. Inst. H. Poincaré, 15 (1998), 539.  doi: 10.1016/S0294-1449(98)80001-2.  Google Scholar

[19]

D. Damjanović, Hamilton's theorem for smooth Lie group actions,, in Ergodic Theory and Dynamical Systems (ed. Idris Assani), (2014), 117.   Google Scholar

[20]

P. Didier, Stability of accessibility,, Ergodic Th. Dyn. Syst., 23 (2003), 171.  doi: 10.1017/S0143385702001785.  Google Scholar

[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.   Google Scholar

[22]

D. Dolgopyat and R. Krikorian, On simultaneous linearization of diffeomorphisms of the sphere,, Duke Math. J., 136 (2007), 475.   Google Scholar

[23]

H. Furstenberg, Noncommuting random products,, Trans. AMS, 108 (1963), 377.  doi: 10.1090/S0002-9947-1963-0163345-0.  Google Scholar

[24]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms,, Ann. of Math. (2), 140 (1994), 295.  doi: 10.2307/2118602.  Google Scholar

[25]

B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations,, Erg. Th. Dynam. Sys., 19 (1999), 643.  doi: 10.1017/S0143385799133868.  Google Scholar

[26]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes Math., (1977).   Google Scholar

[27]

E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung,, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91 (1939), 261.   Google Scholar

[28]

S. Kakutani, Random ergodic theorems and Markoff processes with a stable distribution,, in Proc. 2nd Berkeley Symposium on Math. Stat. & Prob., (1951), 247.   Google Scholar

[29]

I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin,, Bull. AMS (N.S.), 31 (1994), 68.  doi: 10.1090/S0273-0979-1994-00507-5.  Google Scholar

[30]

A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems,, Math. Res. Lett., 3 (1996), 191.  doi: 10.4310/MRL.1996.v3.n2.a6.  Google Scholar

[31]

F. Ledrappier, Quelques propriétés des exposants caractéristiques,, Lecture Notes Math., (1097), 305.  doi: 10.1007/BFb0099434.  Google Scholar

[32]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications,, Math. Surveys & Monographs, (2002).   Google Scholar

[33]

W. Parry and M. Pollicott, Skew products and Livsic theory,, in Representation Theory, 217 (2006), 139.   Google Scholar

[34]

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

[35]

Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties,, Erg. Th. Dynam. Sys., 12 (1992), 123.  doi: 10.1017/S0143385700006635.  Google Scholar

[36]

C. C. Pugh and M. Shub, Ergodic attractors,, Trans. AMS, 312 (1989), 1.  doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar

[37]

C. C. Pugh and M. Shub, Stable ergodicity and stable accessibility,, in Differential Equations and Applications (Hangzhou, (1996), 258.   Google Scholar

[38]

C. C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity,, J. Complexity, 13 (1997), 125.  doi: 10.1006/jcom.1997.0437.  Google Scholar

[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.  doi: 10.3934/dcds.2006.14.845.  Google Scholar

[40]

D. Repovš, A. B. Skopenkov and E. V. Ščepin, $C^1$-homogeneous compacta in $\mathbbR^n$ are $C^1$-submanifolds of $\mathbbR^n$,, Proc. AMS, 124 (1996), 1219.  doi: 10.1090/S0002-9939-96-03157-7.  Google Scholar

[41]

F. Rodriguez Hertz, Stable ergodicity of certain linear automorphisms of the torus,, Ann. of Math. (2), 162 (2005), 65.  doi: 10.4007/annals.2005.162.65.  Google Scholar

[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.  doi: 10.1007/s00220-011-1275-0.  Google Scholar

[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.  doi: 10.1215/00127094-1444314.  Google Scholar

[44]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three,, J. Mod. Dyn., 2 (2008), 187.  doi: 10.3934/jmd.2008.2.187.  Google Scholar

[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.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[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.   Google Scholar

[47]

D. Ruelle, A measure associated with axiom-A attractors,, Amer. J. Math., 98 (1976), 619.  doi: 10.2307/2373810.  Google Scholar

[48]

Ya. G. Sinai, Construction of Markov partitionings,, Funct. An., 2 (1968), 64.   Google Scholar

[49]

Ya. G. Sinai, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21.   Google Scholar

[50]

S. Smale, Differentiable dynamical systems,, Bull. AMS, 73 (1967), 747.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[51]

R. Spatzier, On the work of Rodriguez Hertz on rigidity in dynamics,, J. Mod. Dyn., 10 (2016), 191.  doi: 10.3934/jmd.2016.10.191.  Google Scholar

[52]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents,, Ann. of Math. (2), 167 (2008), 643.  doi: 10.4007/annals.2008.167.643.  Google Scholar

[53]

A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow,, Erg. Th. Dynam. Sys., 18 (1998), 1545.  doi: 10.1017/S0143385798117984.  Google Scholar

[54]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Astérisque, 358 (2013), 75.   Google Scholar

[55]

L.-S. Young, What are SRB measures, and which dynamical systems have them?,, J. Stat. Phys., 108 (2002), 733.  doi: 10.1023/A:1019762724717.  Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory,, Math. Surv. & Monographs, (1997).  doi: 10.1090/surv/050.  Google Scholar

[2]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967).   Google Scholar

[3]

D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems,, Russian Math. Surveys, 22 (1967), 103.  doi: 10.1070/RM1967v022n05ABEH001228.  Google Scholar

[4]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications,, Invent. Math., 181 (2010), 115.  doi: 10.1007/s00222-010-0243-1.  Google Scholar

[5]

L. Barreira and Ya. B Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,, Encyclopedia Math., (2007).  doi: 10.1017/CBO9781107326026.  Google Scholar

[6]

M. Benedicks and M. Viana, Solution of the basin problem for Hénon-like attractors,, Invent. Math., 143 (2001), 375.  doi: 10.1007/s002220000109.  Google Scholar

[7]

Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables,, J. AMS, 5 (1992), 33.  doi: 10.2307/2152750.  Google Scholar

[8]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,, Encycl. Math. Sci., (2005).   Google Scholar

[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.  doi: 10.1016/S0294-1449(02)00019-7.  Google Scholar

[10]

R. Bowen, Markov partitions for Axiom A diffeomorphisms,, Amer. J. Math., 92 (1970), 725.  doi: 10.2307/2373370.  Google Scholar

[11]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, 2nd revised edition, (2008).   Google Scholar

[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.   Google Scholar

[13]

M. I. Brin and Ya. B. Pesin, Partially hyperbolic dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170.   Google Scholar

[14]

A. W. Brown and F. Rodriguez Hertz, Measure rigidity for random dynamics on surfaces and related skew products,, , ().   Google Scholar

[15]

K. Burns and A. Wilkinson, Stable ergodicity of skew products,, Ann. ENS., 32 (1999), 859.  doi: 10.1016/S0012-9593(00)87721-6.  Google Scholar

[16]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems,, Ann. of Math. (2), 171 (2010), 451.  doi: 10.4007/annals.2010.171.451.  Google Scholar

[17]

N. Chernov and R. Markarian, Chaotic Billiards,, Math. Surveys & Monographs, (2006).  doi: 10.1090/surv/127.  Google Scholar

[18]

E. Colli, Infinitely many coexisting strange attractors,, Ann. Inst. H. Poincaré, 15 (1998), 539.  doi: 10.1016/S0294-1449(98)80001-2.  Google Scholar

[19]

D. Damjanović, Hamilton's theorem for smooth Lie group actions,, in Ergodic Theory and Dynamical Systems (ed. Idris Assani), (2014), 117.   Google Scholar

[20]

P. Didier, Stability of accessibility,, Ergodic Th. Dyn. Syst., 23 (2003), 171.  doi: 10.1017/S0143385702001785.  Google Scholar

[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.   Google Scholar

[22]

D. Dolgopyat and R. Krikorian, On simultaneous linearization of diffeomorphisms of the sphere,, Duke Math. J., 136 (2007), 475.   Google Scholar

[23]

H. Furstenberg, Noncommuting random products,, Trans. AMS, 108 (1963), 377.  doi: 10.1090/S0002-9947-1963-0163345-0.  Google Scholar

[24]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms,, Ann. of Math. (2), 140 (1994), 295.  doi: 10.2307/2118602.  Google Scholar

[25]

B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations,, Erg. Th. Dynam. Sys., 19 (1999), 643.  doi: 10.1017/S0143385799133868.  Google Scholar

[26]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes Math., (1977).   Google Scholar

[27]

E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung,, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91 (1939), 261.   Google Scholar

[28]

S. Kakutani, Random ergodic theorems and Markoff processes with a stable distribution,, in Proc. 2nd Berkeley Symposium on Math. Stat. & Prob., (1951), 247.   Google Scholar

[29]

I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin,, Bull. AMS (N.S.), 31 (1994), 68.  doi: 10.1090/S0273-0979-1994-00507-5.  Google Scholar

[30]

A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems,, Math. Res. Lett., 3 (1996), 191.  doi: 10.4310/MRL.1996.v3.n2.a6.  Google Scholar

[31]

F. Ledrappier, Quelques propriétés des exposants caractéristiques,, Lecture Notes Math., (1097), 305.  doi: 10.1007/BFb0099434.  Google Scholar

[32]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications,, Math. Surveys & Monographs, (2002).   Google Scholar

[33]

W. Parry and M. Pollicott, Skew products and Livsic theory,, in Representation Theory, 217 (2006), 139.   Google Scholar

[34]

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

[35]

Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties,, Erg. Th. Dynam. Sys., 12 (1992), 123.  doi: 10.1017/S0143385700006635.  Google Scholar

[36]

C. C. Pugh and M. Shub, Ergodic attractors,, Trans. AMS, 312 (1989), 1.  doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar

[37]

C. C. Pugh and M. Shub, Stable ergodicity and stable accessibility,, in Differential Equations and Applications (Hangzhou, (1996), 258.   Google Scholar

[38]

C. C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity,, J. Complexity, 13 (1997), 125.  doi: 10.1006/jcom.1997.0437.  Google Scholar

[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.  doi: 10.3934/dcds.2006.14.845.  Google Scholar

[40]

D. Repovš, A. B. Skopenkov and E. V. Ščepin, $C^1$-homogeneous compacta in $\mathbbR^n$ are $C^1$-submanifolds of $\mathbbR^n$,, Proc. AMS, 124 (1996), 1219.  doi: 10.1090/S0002-9939-96-03157-7.  Google Scholar

[41]

F. Rodriguez Hertz, Stable ergodicity of certain linear automorphisms of the torus,, Ann. of Math. (2), 162 (2005), 65.  doi: 10.4007/annals.2005.162.65.  Google Scholar

[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.  doi: 10.1007/s00220-011-1275-0.  Google Scholar

[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.  doi: 10.1215/00127094-1444314.  Google Scholar

[44]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three,, J. Mod. Dyn., 2 (2008), 187.  doi: 10.3934/jmd.2008.2.187.  Google Scholar

[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.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[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.   Google Scholar

[47]

D. Ruelle, A measure associated with axiom-A attractors,, Amer. J. Math., 98 (1976), 619.  doi: 10.2307/2373810.  Google Scholar

[48]

Ya. G. Sinai, Construction of Markov partitionings,, Funct. An., 2 (1968), 64.   Google Scholar

[49]

Ya. G. Sinai, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21.   Google Scholar

[50]

S. Smale, Differentiable dynamical systems,, Bull. AMS, 73 (1967), 747.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[51]

R. Spatzier, On the work of Rodriguez Hertz on rigidity in dynamics,, J. Mod. Dyn., 10 (2016), 191.  doi: 10.3934/jmd.2016.10.191.  Google Scholar

[52]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents,, Ann. of Math. (2), 167 (2008), 643.  doi: 10.4007/annals.2008.167.643.  Google Scholar

[53]

A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow,, Erg. Th. Dynam. Sys., 18 (1998), 1545.  doi: 10.1017/S0143385798117984.  Google Scholar

[54]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Astérisque, 358 (2013), 75.   Google Scholar

[55]

L.-S. Young, What are SRB measures, and which dynamical systems have them?,, J. Stat. Phys., 108 (2002), 733.  doi: 10.1023/A:1019762724717.  Google Scholar

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