Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators

Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential

Pointwise existence of the Lyapunov exponent for a quasi-periodic equation

The spectral theory of such quasi-periodic equations is very rich, and the study has generated a vast literature; among the authors are A. Avila, Y. Avron, J. Bellissard, J. Bourgain, V. Buslaev, V. Chulaevsky, D. Damanik, E. Dinaburg, H. Eliasson, A. F., B. Helffer, M. Hermann, S. Jitomirskaya, F. K., R. Krikorian, Y. Last, L.Pastur, J. Puig, M. Shubin, B. Simon, Y. Sinäı, J. Sjöstrand, S. Sor...

Lyapunov Exponents For Some Quasi - Periodic

We consider SL(2; R)-valuedcocycles over rotations of the circle and prove that they are likely to have Lyapunov exponents log if the norms of all of the matrices are. This is proved for suuciently large. The ubiquity of elliptic behavior is also observed. Consider an area preserving diieomorphism f of a compact surface. Assume that f is not uniformly hyperbolic, but that it has obvious hyperbo...

Lyapunov Exponents For Some Quasi-Periodic Cocycles

We consider SL(2,R)-valued cocycles over rotations of the circle and prove that they are likely to have Lyapunov exponents ≈ ± logλ if the norms of all of the matrices are ≈ λ. This is proved for λ sufficiently large. The ubiquity of elliptic behavior is also observed. Consider an area preserving diffeomorphism f of a compact surface. Assume that f is not uniformly hyperbolic, but that it has o...

On the Spectrum and Lyapunov Exponent of Limit Periodic Schrödinger Operators

We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schrödinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero Lebesgue measure. No example with those properties was previously known, even in the larger class of ergodic potentials. We also conclude that the generic limit periodic potential has a spectrum of zero Lebe...