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A random cocycle with non Hölder Lyapunov exponent

The first author was supported by Fundação para a Ciência e a Tecnologia (FCT, Portugal) under the projects: UID/MAT/04561/2013 and PTDC/MAT-PUR/29126/2017. The second author was supported by the CNPq research grant 306369/2017-6 (Brazil), by a research productivity grant from his institution (PUC-Rio) and by the FCT grant PTDC/MAT-PUR/29126/2017. The third author was supported by a grant given by the Calouste Gulbenkien Foundation, under the project Programa Novos Talentos em Matemática da Fundação Calouste Gulbenkian

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  • We provide an example of a Schrödinger cocycle over a mixing Markov shift for which the integrated density of states has a very weak modulus of continuity, close to the log-Hölder lower bound established by W. Craig and B. Simon in [6]. This model is based upon a classical example due to Y. Kifer [15] of a random Bernoulli cocycle with zero Lyapunov exponents which is not strongly irreducible. It follows that the Lyapunov exponent of a Bernoulli cocycle near this Kifer example cannot be Hölder or weak-Hölder continuous, thus providing a limitation on the modulus of continuity of the Lyapunov exponent of random cocycles.

    Mathematics Subject Classification: Primary: 37D25, 34D08, 37H15; Secondary: 82B44, 81Q10, 47B39.

    Citation:

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  • Figure 1.  Graph of the Markov chain on Σ

    Table 1.  Pascal’s triangle for the numbers a(n, i)

    $i$ $\cdots$ $-4$ $-3$ $-2$ $-1$ $0$ $+1$ $+2$ $+3$ $+4$ $+5$ $\cdots$
    $a(1, i)$ $1$ $1$
    $a(2, i)$ $1$ $1$ $1$ $1$
    $a(3, i)$ $1$ $1$ $2$ $2$ $1$ $1$
    $a(4, i)$ $1$ $1$ $3$ $3$ $3$ $3$ $1$ $1$
    $a(5, i)$ $1$ $1$ $4$ $4$ $6$ $6$ $4$ $4$ $1$ $1$
     | Show Table
    DownLoad: CSV

    Table 2.  Narayana's cows sequence a(n)

    $n$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $\cdots$
    $a(n)$ $1$ $1$ $1$ $2$ $3$ $4$ $6$ $9$ $13$ $\cdots$
     | Show Table
    DownLoad: CSV

    Table 3.  Quantitative results on the continuity of the LE for $\mathrm{GL}_{2}(\mathbb{R})$ random cocycles with strictly positive LE

    Cocycle class Positive results Negative results
    Non diagonalizable [17,Théorèmes 1,2], [3,Theorem 1] [20,Appendix 3]
    Diagonalizable [10,Theorem 1.2], [22,Theorem B]
     | Show Table
    DownLoad: CSV

    Table 4.  Quantitative results on the continuity of the LE for $\mathrm{GL}_{2}(\mathbb{R})$ random cocycles with zero LE

    Cocycle class Positive results Negative results
    Strongly irreducible This paper, Proposition 11
    Non strongly irreducible [22 Theorem C] This paper, Theorem 2
     | Show Table
    DownLoad: CSV
  • [1] A. Avila, On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators, Comm. Math. Phys., 288 (2009), 907-918.  doi: 10.1007/s00220-008-0667-2.
    [2] A. Avila, Y. Last, M. Shamis and Q. Zhou, On the Abominable Properties of the Almost Mathieu Operator, in preparation.
    [3] A. Baraviera and P. Duarte, Approximating Lyapunov exponents and stationary measures, J Dyn Diff Equat, 31 (2019), 25-48.  doi: 10.1007/s10884-018-9724-5.
    [4] C. Bocker-Neto and M. Viana, Continuity of Lyapunov exponents for random two-dimensional matrices, Ergodic Theory Dynam. Systems, 37 (2017), 1413-1442.  doi: 10.1017/etds.2015.116.
    [5] W. Craig, Pure point spectrum for discrete almost periodic Schrödinger operators, Comm. Math. Phys., 88 (1983), 113-131.  doi: 10.1007/BF01206883.
    [6] W. Craig and B. Simon, Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices, Comm. Math. Phys., 90 (1983), 207-218.  doi: 10.1007/BF01205503.
    [7] D. Damanik, Schrödinger operators with dynamically defined potentials, Ergodic Theory Dynam. Systems, 37 (2017), 1681-1764.  doi: 10.1017/etds.2015.120.
    [8] P. Duarte and  S. KleinLyapunov Exponents of Linear Cocycles,, Continuity via large deviations. Atlantis Studies in Dynamical Systems, 3. Atlantis Press, Paris, 2016.  doi: 10.2991/978-94-6239-124-6.
    [9] ____, Continuity of the Lyapunov Exponents of Linear Cocycles, Publicações Matemáticas, $31^\circ$ Colóquio Brasileiro de Matemática, IMPA, 2017, https://impa.br/wp-content/uploads/2017/08/31CBM_02.pdf
    [10] ____, Large deviations for products of random two dimensional matrices, preprint, 2018.
    [11] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469.  doi: 10.1214/aoms/1177705909.
    [12] H. Furstenberg and Y. Kifer, Random matrix products and measures on projective spaces, Israel J. Math., 46 (1983), 12-32.  doi: 10.1007/BF02760620.
    [13] H. Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc., 108 (1963), 377-428.  doi: 10.1090/S0002-9947-1963-0163345-0.
    [14] M. Goldstein and W. Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math., (2) 154 (2001), 155-203. doi: 10.2307/3062114.
    [15] Y. Kifer, Perturbations of random matrix products, Z. Wahrsch. Verw. Gebiete, 61 (1982), 83-95.  doi: 10.1007/BF00537227.
    [16] H. Krüger and Z. Gan, Optimality of log Hölder continuity of the integrated density of states, Math. Nachr., 284 (2011), 1919-1923.  doi: 10.1002/mana.200910139.
    [17] É. Le Page, Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications, Ann. Inst. H. Poincaré Probab. Statist., 25 (1989), 109-142. 
    [18] E. C. Malheiro and M. Viana, Lyapunov exponents of linear cocycles over Markov shifts, Stoch. Dyn., 15 (2015), 1550020, 27pp. doi: 10.1142/S0219493715500203.
    [19] J. Pöschel, Examples of discrete Schrödinger operators with pure point spectrum, Comm. Math. Phys., 88 (1983), 447-463.  doi: 10.1007/BF01211953.
    [20] B. Simon and M. Taylor, Harmonic analysis on SL (2, R) and smoothness of the density of states in the one-dimensional Anderson model, Comm. Math. Phys., 101 (1985), 1-19.  doi: 10.1007/BF01212354.
    [21] N. J. A. Sloane and  S. PlouffeThe Encyclopedia of Integer Sequences, Academic Press, Inc., San Diego, CA, 1995. 
    [22] E. H. Y. Tall and M. Viana, Moduli of Continuity for Lyapunov Exponents of Random GL(2) Cocycles, preprint, 2018, http://w3.impa.br/~viana/out/holder.pdf.
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    [24] M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602.
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