American Institute of Mathematical Sciences

Advanced
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), 1667-1696. 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), 1423-1485. 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, Università di Firenze, 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), 349-370.  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), 37-59. 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), 54-78. 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), 403-438. 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), 1-33. 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), 198-224. 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), 265-278.  Google Scholar

[11]

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

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.  Google Scholar

[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.  Google Scholar

[3]

Roberta Fabbri, "Genericità dell'iperbolicità nei sistemi differenziali lineari di dimensione due," Ph.D. Thesis, Università di Firenze, 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), 349-370.  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), 37-59. 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), 54-78. 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), 403-438. 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), 1-33. 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), 198-224. 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), 265-278.  Google Scholar

[11]

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

[1]

Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009
[2]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619
[3]

Boris Kalinin, Victoria Sadovskaya. Lyapunov exponents of cocycles over non-uniformly hyperbolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5105-5118. doi: 10.3934/dcds.2018224
[4]

Lucas Backes. On the periodic approximation of Lyapunov exponents for semi-invertible cocycles. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6353-6368. doi: 10.3934/dcds.2017275
[5]

Artur Avila. Density of positive Lyapunov exponents for quasiperiodic SL(2, R)-cocycles in arbitrary dimension. Journal of Modern Dynamics, 2009, 3 (4) : 631-636. doi: 10.3934/jmd.2009.3.631
[6]

Mauricio Poletti. Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5163-5188. doi: 10.3934/dcds.2018228
[7]

Hans Koch. On trigonometric skew-products over irrational circle-rotations. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021084
[8]

Linlin Fu, Jiahao Xu. A new proof of continuity of Lyapunov exponents for a class of $ C^2 $ quasiperiodic Schrödinger cocycles without LDT. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2915-2931. doi: 10.3934/dcds.2019121
[9]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[10]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957
[11]

Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 1989-2009. doi: 10.3934/dcds.2020102
[12]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107
[13]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433
[14]

Gemma Huguet, Rafael de la Llave, Yannick Sire. Fast iteration of cocycles over rotations and computation of hyperbolic bundles. Conference Publications, 2013, 2013 (special) : 323-333. doi: 10.3934/proc.2013.2013.323
[15]

Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071
[16]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004
[17]

Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287
[18]

Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861
[19]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098
[20]

Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences