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

TL;DR

This paper establishes bounds on the Lyapunov exponent for certain Schrödinger operators, showing it scales with the logarithm of the coupling constant, except for a small measure set of energies, with implications for ergodic systems.

Contribution

It provides new bounds on the Lyapunov exponent for the standard map and ergodic Schrödinger operators, including sharp estimates and measure bounds for exceptional energies.

Findings

Lyapunov exponent scales as log λ for large λ

Exceptional energies form a set of exponentially small measure

Results apply to the standard map and skew shift operators

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.