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Convergence and center manifolds for differential equations driven by colored noise
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 |
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 [
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. |
show all references
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. |

$i$ | $\cdots$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $+1$ | $+2$ | $+3$ | $+4$ | $+5$ | $\cdots$ |
$a(1, i)$ | $1$ | $1$ | ||||||||||
$a(2, i)$ | $1$ | $1$ | $1$ | $1$ | ||||||||
$a(3, i)$ | $1$ | $1$ | $2$ | $2$ | $1$ | $1$ | ||||||
$a(4, i)$ | $1$ | $1$ | $3$ | $3$ | $3$ | $3$ | $1$ | $1$ | ||||
$a(5, i)$ | $1$ | $1$ | $4$ | $4$ | $6$ | $6$ | $4$ | $4$ | $1$ | $1$ |
$i$ | $\cdots$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $+1$ | $+2$ | $+3$ | $+4$ | $+5$ | $\cdots$ |
$a(1, i)$ | $1$ | $1$ | ||||||||||
$a(2, i)$ | $1$ | $1$ | $1$ | $1$ | ||||||||
$a(3, i)$ | $1$ | $1$ | $2$ | $2$ | $1$ | $1$ | ||||||
$a(4, i)$ | $1$ | $1$ | $3$ | $3$ | $3$ | $3$ | $1$ | $1$ | ||||
$a(5, i)$ | $1$ | $1$ | $4$ | $4$ | $6$ | $6$ | $4$ | $4$ | $1$ | $1$ |
$n$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $\cdots$ |
$a(n)$ | $1$ | $1$ | $1$ | $2$ | $3$ | $4$ | $6$ | $9$ | $13$ | $\cdots$ |
$n$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $\cdots$ |
$a(n)$ | $1$ | $1$ | $1$ | $2$ | $3$ | $4$ | $6$ | $9$ | $13$ | $\cdots$ |
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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