Bounds on the Lyapunov exponent via crude estimates on the density of states

Abstract

We study the Chirikov (standard) map at large coupling $\lambda \gg 1$, and prove that the Lyapounov exponent of the associated Schroedinger operator is of order $\log \lambda$ except for a set of energies of measure $\exp (-c{\lambda }^{\beta })$ for some $1<\beta <2$. We also prove a similar (sharp) lower bound on the Lyapunov exponent (outside a small exceptional set of energies) for a large family of ergodic Schroedinger operators, the prime example being the $d$-dimensional skew shift.

21pp

Cited by in corpus (1)

• Lyapunov exponents for random perturbations of some area-preserving maps including the standard map