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Title: Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base
We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew-shift base with a cosine potential and the golden ratio as frequency. For coupling below $1$ , which is the threshold for Herman’s subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large-deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite-volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large-deviation estimates and the avalanche principle, effective.  more » « less
Award ID(s):
1800689 1638352
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Ergodic Theory and Dynamical Systems
Page Range / eLocation ID:
1 to 66
Medium: X
Sponsoring Org:
National Science Foundation
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