Advanced Search
Article Contents
Article Contents

Continuity of Lyapunov exponents for cocycles with invariant holonomies

CB: Supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144082.
Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 37H15, 37D30; Secondary: 37D25, 37E99.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Mass away from the diagonal

  • [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. BonattiX. 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, 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., 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., 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.
  • 加载中



Article Metrics

HTML views(992) PDF downloads(150) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint