Uniform positivity of the Lyapunov exponent for $C^1$ monotone potentials generated by the cat map

TL;DR

This paper proves that for a class of $C^1$ monotone potentials generated by Arnold's Cat Map, the Lyapunov exponent remains uniformly positive across all energies when the coupling strength is sufficiently large.

Contribution

It establishes uniform positivity of the Lyapunov exponent for Schrödinger operators with specific $C^1$ monotone potentials derived from the cat map, under large coupling.

Findings

Lyapunov exponent is uniformly positive for all energies

Positivity holds for sufficiently large coupling

Results apply to potentials with bounded directional derivatives in the unstable direction

Abstract

We consider an Arnold's Cat Map generated $C^1$ bounded potential with the directional derivative in the unstable direction bounded away from zero. We show that the Lyapunov exponent for the associated Shr\"odinger Operator is uniformly positive for all energies provided the coupling is sufficiently large.