# American Institute of Mathematical Sciences

August  2019, 39(8): 4841-4861. doi: 10.3934/dcds.2019197

## A random cocycle with non Hölder Lyapunov exponent

 1 Departamento de Matemática and CMAFCIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, Piso 2, 1749-016 Lisboa, Portugal 2 Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rua Marquês de São Vicente 225, Rio de Janeiro, RJ, 22430-060, Brazil 3 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal

Received  November 2018 Revised  February 2019 Published  May 2019

Fund Project: 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

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.

Citation: Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197
##### References:
 [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. Klein, Lyapunov 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. Plouffe, The 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. [23] T. Tao, Topics in Random Matrix Theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/132. [24] M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602.

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##### References:
 [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. Klein, Lyapunov 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. Plouffe, The 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. [23] T. Tao, Topics in Random Matrix Theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/132. [24] M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602.
Graph of the Markov chain on Î£
Pascalâ€™s triangle for the numbers a(n, i)
 $i$ $\cdots$ $-4$ $-3$ $-2$ $-1$ $0$ $+1$ $+2$ $+3$ $+4$ $+5$ $\cdots$ $a(1, i)$ $1$ $1$ $a(2, i)$ $1$ $1$ $1$ $1$ $a(3, i)$ $1$ $1$ $2$ $2$ $1$ $1$ $a(4, i)$ $1$ $1$ $3$ $3$ $3$ $3$ $1$ $1$ $a(5, i)$ $1$ $1$ $4$ $4$ $6$ $6$ $4$ $4$ $1$ $1$
 $i$ $\cdots$ $-4$ $-3$ $-2$ $-1$ $0$ $+1$ $+2$ $+3$ $+4$ $+5$ $\cdots$ $a(1, i)$ $1$ $1$ $a(2, i)$ $1$ $1$ $1$ $1$ $a(3, i)$ $1$ $1$ $2$ $2$ $1$ $1$ $a(4, i)$ $1$ $1$ $3$ $3$ $3$ $3$ $1$ $1$ $a(5, i)$ $1$ $1$ $4$ $4$ $6$ $6$ $4$ $4$ $1$ $1$
Narayana's cows sequence a(n)
 $n$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $\cdots$ $a(n)$ $1$ $1$ $1$ $2$ $3$ $4$ $6$ $9$ $13$ $\cdots$
 $n$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $\cdots$ $a(n)$ $1$ $1$ $1$ $2$ $3$ $4$ $6$ $9$ $13$ $\cdots$
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]
 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]
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
 Cocycle class Positive results Negative results Strongly irreducible This paper, Proposition 11 Non strongly irreducible [22 Theorem C] This paper, Theorem 2
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