Multiscale Analysis for Ergodic Schr\"odinger operators and positivity of Lyapunov exponents

TL;DR

This paper develops a multiscale analysis method for ergodic Schrödinger operators to prove positivity of Lyapunov exponents, with applications to high-dimensional skew-shifts and doubling map potentials.

Contribution

It introduces a new variant of multiscale analysis that establishes positivity of Lyapunov exponents under certain initial conditions and applies it to complex dynamical systems.

Findings

Positivity of Lyapunov exponents for high-dimensional skew-shifts at small coupling.

Positive Lyapunov exponents for potentials from the doubling map, except in a superpolynomially small set.

Development of a multiscale analysis framework tailored for ergodic Schrödinger operators.

Abstract

A variant of multiscale analysis for ergodic Schr\"odinger operators is developed. This enables us to prove positivity of Lyapunov exponents given initial scale estimates and an initial Wegner estimate. This is then applied to high dimensional skew-shifts at small coupling, where initial conditions are checked using the Pastur--Figotin formalism. Furthermore, it is shown that for potentials generated by the doubling map one has positive Lyapunov exponent except in a superpolynomially small set.