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

The work of Federico Rodriguez Hertz on ergodicity of dynamical systems

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.
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.

[2]

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

[3]

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

[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.

[5]

L. Barreira and Ya. B Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,, Encyclopedia Math., (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. 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. 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., (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. doi: 10.1016/S0294-1449(02)00019-7.

[10]

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

[11]

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

[13]

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

[14]

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

[15]

K. Burns and A. Wilkinson, Stable ergodicity of skew products,, Ann. ENS., 32 (1999), 859. 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. doi: 10.4007/annals.2010.171.451.

[17]

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

[18]

E. Colli, Infinitely many coexisting strange attractors,, Ann. Inst. H. Poincaré, 15 (1998), 539. 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), (2014), 117.

[20]

P. Didier, Stability of accessibility,, Ergodic Th. Dyn. Syst., 23 (2003), 171. 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.

[22]

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

[23]

H. Furstenberg, Noncommuting random products,, Trans. AMS, 108 (1963), 377. 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. doi: 10.2307/2118602.

[25]

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

[26]

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

[27]

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

[28]

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

[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.

[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.

[31]

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

[32]

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

[33]

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

[34]

Ya. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. 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. doi: 10.1017/S0143385700006635.

[36]

C. C. Pugh and M. Shub, Ergodic attractors,, Trans. AMS, 312 (1989), 1. 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), 258.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[47]

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

[48]

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

[49]

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

[50]

S. Smale, Differentiable dynamical systems,, Bull. AMS, 73 (1967), 747. 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. 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. 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. doi: 10.1017/S0143385798117984.

[54]

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

[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.

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory,, Math. Surv. & Monographs, (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).

[3]

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

[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.

[5]

L. Barreira and Ya. B Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,, Encyclopedia Math., (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. 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. 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., (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. doi: 10.1016/S0294-1449(02)00019-7.

[10]

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

[11]

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

[13]

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

[14]

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

[15]

K. Burns and A. Wilkinson, Stable ergodicity of skew products,, Ann. ENS., 32 (1999), 859. 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. doi: 10.4007/annals.2010.171.451.

[17]

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

[18]

E. Colli, Infinitely many coexisting strange attractors,, Ann. Inst. H. Poincaré, 15 (1998), 539. 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), (2014), 117.

[20]

P. Didier, Stability of accessibility,, Ergodic Th. Dyn. Syst., 23 (2003), 171. 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.

[22]

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

[23]

H. Furstenberg, Noncommuting random products,, Trans. AMS, 108 (1963), 377. 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. doi: 10.2307/2118602.

[25]

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

[26]

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

[27]

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

[28]

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

[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.

[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.

[31]

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

[32]

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

[33]

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

[34]

Ya. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. 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. doi: 10.1017/S0143385700006635.

[36]

C. C. Pugh and M. Shub, Ergodic attractors,, Trans. AMS, 312 (1989), 1. 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), 258.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[47]

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

[48]

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

[49]

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

[50]

S. Smale, Differentiable dynamical systems,, Bull. AMS, 73 (1967), 747. 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. 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. 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. doi: 10.1017/S0143385798117984.

[54]

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

[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.

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