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Seifert manifolds admitting partially hyperbolic diffeomorphisms
Continuity of Lyapunov exponents for cocycles with invariant holonomies
1. | Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, CEP 91509-900, Porto Alegre, RS, Brazil |
2. | Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL 60637, USA |
We prove a conjecture of Viana which states that Lyapunov exponents vary continuously when restricted to $GL(2,\mathbb{R})$-valued cocycles over a subshift of finite type which admit invariant holonomies that depend continuously on the cocycle.
References:
[1] |
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. |
[2] |
A. Avila, M. Viana and A. Eskin, Continuity of Lyapunov Exponents of Random Matrix Products, In preparation. |
[3] |
L. Barreira and Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Enc. of Mathematics and its Applications, 115, Cambridge University Press, 2007.
doi: 10.1017/CBO9781107326026. |
[4] |
J. Bochi, Discontinuity of the Lyapunov exponents for non-hyperbolic cocycles, Preprint, http://www.mat.uc.cl/~jairo.bochi/. |
[5] |
J. Bochi,
Genericity of zero Lyapunov exponents, Ergodic Theory and Dynamical Systems, 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
[6] |
C. Bocker-Neto and M. Viana, Continuity of lyapunov exponents for Random 2D matrices, Preprint, arXiv: 1012.0872, 2010. |
[7] |
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. Non Linéaire, 20 (2003), 579-624.
doi: 10.1016/S0294-1449(02)00019-7. |
[8] |
C. Bonatti and M. Viana,
Lyapunov exponents with multiplicity 1 for deterministic products of matrices, Ergodic Theory Dynam. Systems(5), 24 (2004), 1295-1330.
doi: 10.1017/S0143385703000695. |
[9] |
J. Bourgain,
Positivity and continuity of the Lyapounov exponent for shifts on $\mathbb{T}^d$ with arbitrary frequency vector and real analytic potential, J. Anal. Math., 96 (2005), 313-355.
doi: 10.1007/BF02787834. |
[10] |
J. Bourgain and S. Jitomirskaya,
Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Statist. Phys., 108 (2002), 1203-1218.
doi: 10.1023/A:1019751801035. |
[11] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, 1975. |
[12] |
C. Butler,
Discontinuity of Lyapunov exponents near fiber bunched cocycles, Ergodic Theory Dynam. Systems, 38 (2018), 523-539.
doi: 10.1017/etds.2016.56. |
[13] |
P. Duarte and S. Klein, An abstract continuity theorem of the Lyapunov exponents, Preprint, arXiv: 1410.0699, 2014. |
[14] |
H. Furstenberg and Y. Kifer,
Random matrix products and measures in projective spaces, Israel J. Math., 46 (1983), 12-32.
doi: 10.1007/BF02760620. |
[15] |
B. Kalinin,
Livšic theorem for matrix cocycles, Ann. Math., 173 (2011), 1025-1042.
doi: 10.4007/annals.2011.173.2.11. |
[16] |
B. Kalinin and V. Sadovskaya,
Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 36 (2016), 245-259.
doi: 10.3934/dcds.2016.36.245. |
[17] |
Yu. Kifer,
Perturbations of random matrix products, Z. Wahrsch. Verw. Gebiete, 61 (1982), 83-95.
doi: 10.1007/BF00537227. |
[18] |
J. Kingman,
The ergodic theorem of subadditive stochastic processes, J. Royal Statist. Soc., 30 (1968), 499-510.
|
[19] |
F. Ledrappier,
Positivity of the exponent for stationary sequences of matrices, Lect. Notes in Math., 1186 (1982), 56-73.
doi: 10.1007/BFb0076833. |
[20] |
F. Ledrappier and L.-S. Young,
The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math.(2), 122 (1985), 509-539.
doi: 10.2307/1971328. |
[21] |
F. Ledrappier and L.-S. Young,
The metric entropy of diffeomorphisms. Ⅱ. Relations between entropy, exponents and dimension, Ann. of Math.(2), 122 (1985), 540-574.
doi: 10.2307/1971329. |
[22] |
É. Le Page, Théorèmes limites pour les produits de matrices aléatoires, in Probability Measures on Groups (Oberwolfach, 1981), Lecture Notes in Math., 928, Springer, 1982,258–303. |
[23] |
E. C. Malheiro and M. Viana, Lyapunov exponents of linear cocycles over Markov shifts, Stoch. Dyn., 15 (2015), 1550020, 27 pp.
doi: 10.1142/S0219493715500203. |
[24] |
R. Leplaideur,
Local product structure for equilibrium states, Trans. Amer. Math. Soc., 352 (2000), 1889-1912.
doi: 10.1090/S0002-9947-99-02479-4. |
[25] |
Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, in Lyapunov Exponents (Oberwolfach, 1990), Lecture Notes in Math., 1486, Springer, 1991, 64–80.
doi: 10.1007/BFb0086658. |
[26] |
J. B. Pesin,
Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), 1332-1379, 1440.
|
[27] |
J. B. Pesin,
Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-112,287.
|
[28] |
D. Ruelle,
An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.
doi: 10.1007/BF02584795. |
[29] |
D. Ruelle,
Analyticity properties of the characteristic exponents of random matrix products, Adv. Math., 32 (1979), 68-80.
doi: 10.1016/0001-8708(79)90029-X. |
[30] |
M. Viana,
Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. Math., 167 (2008), 643-680.
doi: 10.4007/annals.2008.167.643. |
[31] |
M. Viana, Lectures on Lyapunov Exponents, Cambridge University Press, 2014.
doi: 10.1017/CBO9781139976602. |
show all references
References:
[1] |
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. |
[2] |
A. Avila, M. Viana and A. Eskin, Continuity of Lyapunov Exponents of Random Matrix Products, In preparation. |
[3] |
L. Barreira and Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Enc. of Mathematics and its Applications, 115, Cambridge University Press, 2007.
doi: 10.1017/CBO9781107326026. |
[4] |
J. Bochi, Discontinuity of the Lyapunov exponents for non-hyperbolic cocycles, Preprint, http://www.mat.uc.cl/~jairo.bochi/. |
[5] |
J. Bochi,
Genericity of zero Lyapunov exponents, Ergodic Theory and Dynamical Systems, 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
[6] |
C. Bocker-Neto and M. Viana, Continuity of lyapunov exponents for Random 2D matrices, Preprint, arXiv: 1012.0872, 2010. |
[7] |
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. Non Linéaire, 20 (2003), 579-624.
doi: 10.1016/S0294-1449(02)00019-7. |
[8] |
C. Bonatti and M. Viana,
Lyapunov exponents with multiplicity 1 for deterministic products of matrices, Ergodic Theory Dynam. Systems(5), 24 (2004), 1295-1330.
doi: 10.1017/S0143385703000695. |
[9] |
J. Bourgain,
Positivity and continuity of the Lyapounov exponent for shifts on $\mathbb{T}^d$ with arbitrary frequency vector and real analytic potential, J. Anal. Math., 96 (2005), 313-355.
doi: 10.1007/BF02787834. |
[10] |
J. Bourgain and S. Jitomirskaya,
Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Statist. Phys., 108 (2002), 1203-1218.
doi: 10.1023/A:1019751801035. |
[11] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, 1975. |
[12] |
C. Butler,
Discontinuity of Lyapunov exponents near fiber bunched cocycles, Ergodic Theory Dynam. Systems, 38 (2018), 523-539.
doi: 10.1017/etds.2016.56. |
[13] |
P. Duarte and S. Klein, An abstract continuity theorem of the Lyapunov exponents, Preprint, arXiv: 1410.0699, 2014. |
[14] |
H. Furstenberg and Y. Kifer,
Random matrix products and measures in projective spaces, Israel J. Math., 46 (1983), 12-32.
doi: 10.1007/BF02760620. |
[15] |
B. Kalinin,
Livšic theorem for matrix cocycles, Ann. Math., 173 (2011), 1025-1042.
doi: 10.4007/annals.2011.173.2.11. |
[16] |
B. Kalinin and V. Sadovskaya,
Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 36 (2016), 245-259.
doi: 10.3934/dcds.2016.36.245. |
[17] |
Yu. Kifer,
Perturbations of random matrix products, Z. Wahrsch. Verw. Gebiete, 61 (1982), 83-95.
doi: 10.1007/BF00537227. |
[18] |
J. Kingman,
The ergodic theorem of subadditive stochastic processes, J. Royal Statist. Soc., 30 (1968), 499-510.
|
[19] |
F. Ledrappier,
Positivity of the exponent for stationary sequences of matrices, Lect. Notes in Math., 1186 (1982), 56-73.
doi: 10.1007/BFb0076833. |
[20] |
F. Ledrappier and L.-S. Young,
The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math.(2), 122 (1985), 509-539.
doi: 10.2307/1971328. |
[21] |
F. Ledrappier and L.-S. Young,
The metric entropy of diffeomorphisms. Ⅱ. Relations between entropy, exponents and dimension, Ann. of Math.(2), 122 (1985), 540-574.
doi: 10.2307/1971329. |
[22] |
É. Le Page, Théorèmes limites pour les produits de matrices aléatoires, in Probability Measures on Groups (Oberwolfach, 1981), Lecture Notes in Math., 928, Springer, 1982,258–303. |
[23] |
E. C. Malheiro and M. Viana, Lyapunov exponents of linear cocycles over Markov shifts, Stoch. Dyn., 15 (2015), 1550020, 27 pp.
doi: 10.1142/S0219493715500203. |
[24] |
R. Leplaideur,
Local product structure for equilibrium states, Trans. Amer. Math. Soc., 352 (2000), 1889-1912.
doi: 10.1090/S0002-9947-99-02479-4. |
[25] |
Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, in Lyapunov Exponents (Oberwolfach, 1990), Lecture Notes in Math., 1486, Springer, 1991, 64–80.
doi: 10.1007/BFb0086658. |
[26] |
J. B. Pesin,
Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), 1332-1379, 1440.
|
[27] |
J. B. Pesin,
Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-112,287.
|
[28] |
D. Ruelle,
An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.
doi: 10.1007/BF02584795. |
[29] |
D. Ruelle,
Analyticity properties of the characteristic exponents of random matrix products, Adv. Math., 32 (1979), 68-80.
doi: 10.1016/0001-8708(79)90029-X. |
[30] |
M. Viana,
Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. Math., 167 (2008), 643-680.
doi: 10.4007/annals.2008.167.643. |
[31] |
M. Viana, Lectures on Lyapunov Exponents, Cambridge University Press, 2014.
doi: 10.1017/CBO9781139976602. |

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