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  2018, 12: 223-260. doi: 10.3934/jmd.2018009

Continuity of Lyapunov exponents for cocycles with invariant holonomies

Lucas Backes 1, , Aaron Brown 2, and  Clark Butler 2,

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

Received  April 04, 2016 Revised  October 11, 2017 Published  July 2018

Fund Project: CB: Supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144082.

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.

Keywords: Lyapunov exponents, continuity, non-uniform hyperbolicity, random dynamics, linear cocycles.
Mathematics Subject Classification: Primary: 37H15, 37D30; Secondary: 37D25, 37E99.
Citation: Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009
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.

Figure 1.  Mass away from the diagonal
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