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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

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 & 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.  doi: doi:10.1017/S0143385702001165.  Google Scholar

[2]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps,, Ann. of Math., 161 (2005), 1423.  doi: doi:10.4007/annals.2005.161.1423.  Google Scholar

[3]

Roberta Fabbri, "Genericità dell'iperbolicità nei sistemi differenziali lineari di dimensione due,", Ph.D. Thesis, (1997).   Google Scholar

[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.   Google Scholar

[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.  doi: doi:10.1016/S1468-1218(01)00012-8.  Google Scholar

[6]

R. Johnson, Exponential dichotomy, rotation number and linear differential operatorss with bounded coefficients,, Jour. Diff. Equn., 61 (1986), 54.  doi: doi:10.1016/0022-0396(86)90125-7.  Google Scholar

[7]

R. Johnson and J. Moser, The rotation number for almost periodic potentials,, Comm. Math. Phys., 84 (1982), 403.  doi: doi:10.1007/BF01208484.  Google Scholar

[8]

R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems,, SIAM J. Math. Anal., 18 (1987), 1.  doi: doi:10.1137/0518001.  Google Scholar

[9]

J. Moser, An example of a Schrodinger equation with almost periodic potential and nowhere dense spectrum,, Comment. Math. Helvetici, 56 (1981), 198.  doi: doi:10.1007/BF02566210.  Google Scholar

[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.   Google Scholar

[11]

M. Nerurkar, Density of positive Lyapunov exponents in the smooth category,, preprint (2008)., (2008).   Google Scholar

show all references

References:
[1]

J. Bochi, Genericity of zero Lyapunov exponents,, Ergodic Theory and Dynamical Systems, 22 (2002), 1667.  doi: doi:10.1017/S0143385702001165.  Google Scholar

[2]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps,, Ann. of Math., 161 (2005), 1423.  doi: doi:10.4007/annals.2005.161.1423.  Google Scholar

[3]

Roberta Fabbri, "Genericità dell'iperbolicità nei sistemi differenziali lineari di dimensione due,", Ph.D. Thesis, (1997).   Google Scholar

[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.   Google Scholar

[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.  doi: doi:10.1016/S1468-1218(01)00012-8.  Google Scholar

[6]

R. Johnson, Exponential dichotomy, rotation number and linear differential operatorss with bounded coefficients,, Jour. Diff. Equn., 61 (1986), 54.  doi: doi:10.1016/0022-0396(86)90125-7.  Google Scholar

[7]

R. Johnson and J. Moser, The rotation number for almost periodic potentials,, Comm. Math. Phys., 84 (1982), 403.  doi: doi:10.1007/BF01208484.  Google Scholar

[8]

R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems,, SIAM J. Math. Anal., 18 (1987), 1.  doi: doi:10.1137/0518001.  Google Scholar

[9]

J. Moser, An example of a Schrodinger equation with almost periodic potential and nowhere dense spectrum,, Comment. Math. Helvetici, 56 (1981), 198.  doi: doi:10.1007/BF02566210.  Google Scholar

[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.   Google Scholar

[11]

M. Nerurkar, Density of positive Lyapunov exponents in the smooth category,, preprint (2008)., (2008).   Google Scholar

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