On SL(2, R) valued cocycles of Hölder class with zero exponent over Kronecker flows

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May 2011, 10(3): 873-884. doi: 10.3934/cpaa.2011.10.873

On $SL(2, R)$ valued cocycles of Hölder class with zero exponent over Kronecker flows

Russell Johnson ^1, and Mahesh G. Nerurkar ^2,

1.	Dipartimento di Sistemi e Informatica, Università di Firenze, 50139 Firenze
2.	Department of Mathematics, Rutgers University, Camden NJ 08102, United States

Received October 2008 Revised March 2009 Published December 2010

Abstract
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We show that a generic $SL(2,R)$ valued cocycle in the class of $C^r$, ($0 < r < 1$) cocycles based on a rotation flow on the $d$-torus, is either uniformly hyperbolic or has zero Lyapunov exponents provided that the components of winding vector $\bar \gamma = (\gamma^1,\cdot \cdot \cdot,\gamma^d)$ of the rotation flow are rationally independent and satisfy the following super Liouvillian condition :

$ |\gamma^i - \frac{p^i_n}{q_n}| \leq Ce^{-q^{1+\delta}_n}, 1\leq i\leq d, n\in N,$

where $C > 0$ and $\delta > 0$ are some constants and $p^i_n, q_n$ are some sequences of integers with $q_n\to \infty$.

Keywords: Lyapunov exponents, Cocycles, irrational rotations, Liouville numbers..

Mathematics Subject Classification: Primary: 37B55, 34A30; Secondary: 58F1.

Citation: Russell Johnson, Mahesh G. Nerurkar. On $SL(2, R)$ valued cocycles of Hölder class with zero exponent over Kronecker flows. Communications on Pure and Applied Analysis, 2011, 10 (3) : 873-884. doi: 10.3934/cpaa.2011.10.873

References:

[1]	J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory and Dynamical Systems, 22 (2002), 1667-1696. doi: doi:10.1017/S0143385702001165.
[2]	J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1485. doi: doi:10.4007/annals.2005.161.1423.
[3]	Roberta Fabbri, "Genericità dell'iperbolicità nei sistemi differenziali lineari di dimensione due," Ph.D. Thesis, Università di Firenze, 1997.
[4]	R. Fabbri and R. Johnson, On the Lyapunov exponent of certain $SL(2,R)$ valued cocycles, Differential Equations and Dynamical Systems, 7 (1999), 349-370.
[5]	R. Fabbri, R. Johnson and R. Pavani, On the nature of the spectrum of the quasi-periodic Schrödinger operator, Nonlinear Analysis: Real World Applications, 3 (2002), 37-59. doi: doi:10.1016/S1468-1218(01)00012-8.
[6]	R. Johnson, Exponential dichotomy, rotation number and linear differential operatorss with bounded coefficients, Jour. Diff. Equn., 61 (1986), 54-78. doi: doi:10.1016/0022-0396(86)90125-7.
[7]	R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438. doi: doi:10.1007/BF01208484.
[8]	R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33. doi: doi:10.1137/0518001.
[9]	J. Moser, An example of a Schrodinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helvetici, 56 (1981), 198-224. doi: doi:10.1007/BF02566210.
[10]	M. Nerurkar, Positive exponents for a dense class of continuous $SL(2,R)$ valued cocycles which arise as solutions to strongly accessible linear differential systems, Contemp. Math., 215 (1998), 265-278.
[11]	M. Nerurkar, Density of positive Lyapunov exponents in the smooth category, preprint (2008).

show all references

References:

[1]	J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory and Dynamical Systems, 22 (2002), 1667-1696. doi: doi:10.1017/S0143385702001165.
[2]	J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1485. doi: doi:10.4007/annals.2005.161.1423.
[3]	Roberta Fabbri, "Genericità dell'iperbolicità nei sistemi differenziali lineari di dimensione due," Ph.D. Thesis, Università di Firenze, 1997.
[4]	R. Fabbri and R. Johnson, On the Lyapunov exponent of certain $SL(2,R)$ valued cocycles, Differential Equations and Dynamical Systems, 7 (1999), 349-370.
[5]	R. Fabbri, R. Johnson and R. Pavani, On the nature of the spectrum of the quasi-periodic Schrödinger operator, Nonlinear Analysis: Real World Applications, 3 (2002), 37-59. doi: doi:10.1016/S1468-1218(01)00012-8.
[6]	R. Johnson, Exponential dichotomy, rotation number and linear differential operatorss with bounded coefficients, Jour. Diff. Equn., 61 (1986), 54-78. doi: doi:10.1016/0022-0396(86)90125-7.
[7]	R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438. doi: doi:10.1007/BF01208484.
[8]	R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33. doi: doi:10.1137/0518001.
[9]	J. Moser, An example of a Schrodinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helvetici, 56 (1981), 198-224. doi: doi:10.1007/BF02566210.
[10]	M. Nerurkar, Positive exponents for a dense class of continuous $SL(2,R)$ valued cocycles which arise as solutions to strongly accessible linear differential systems, Contemp. Math., 215 (1998), 265-278.
[11]	M. Nerurkar, Density of positive Lyapunov exponents in the smooth category, preprint (2008).

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