A random cocycle with non Hölder Lyapunov exponent

Pedro Duarte; Silvius Klein; Manuel Santos

doi:10.3934/dcds.2019197

Article Contents

2019, Volume 39, Issue 8: 4841-4861. Doi: 10.3934/dcds.2019197

This issue Previous Article Convergence and center manifolds for differential equations driven by colored noise Next Article Regularity and weak comparison principles for double phase quasilinear elliptic equations

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

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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Abstract

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.

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

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Figure 1. Graph of the Markov chain on Î£

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Table 1. Pascalâ€™s triangle for the numbers a(n, i)

$i$	$\cdots$	$-4$	$-3$	$-2$	$-1$	$0$	$+1$	$+2$	$+3$	$+4$	$+5$	$\cdots$
$a(1, i)$						$1$	$1$
$a(2, i)$					$1$	$1$	$1$	$1$
$a(3, i)$				$1$	$1$	$2$	$2$	$1$	$1$
$a(4, i)$			$1$	$1$	$3$	$3$	$3$	$3$	$1$	$1$
$a(5, i)$		$1$	$1$	$4$	$4$	$6$	$6$	$4$	$4$	$1$	$1$

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

$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$\cdots$
$a(n)$	$1$	$1$	$1$	$2$	$3$	$4$	$6$	$9$	$13$	$\cdots$

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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]

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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

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References

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