Schr\"odinger Operators With Potentials Generated by Hyperbolic Transformations: I. Positivity of the Lyapunov Exponent

TL;DR

This paper investigates the positivity of the Lyapunov exponent for discrete Schr"odinger operators with potentials generated by hyperbolic transformations, establishing conditions under which the exponent is positive across various classes of sampling functions.

Contribution

It provides new results on the positivity of the Lyapunov exponent for Schr"odinger operators with potentials from hyperbolic dynamics, extending to broad classes of functions and transformations.

Findings

Lyapunov exponent is positive away from a discrete set for certain sampling functions.

For residual subsets of H"older continuous functions, the Lyapunov exponent is positive everywhere.

Positivity of the Lyapunov exponent holds outside finite sets for locally constant or fiber bunched functions.

Abstract

We consider discrete one-dimensional Schr\"odinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant H\"older continuous function defined on a subshift of finite type with an ergodic measure admitting a local product structure and a fixed point, then the Lyapunov exponent is positive away from a discrete set of energies. Moreover, for sampling functions in a residual subset of the space of H\"older continuous functions, the Lyapunov exponent is positive everywhere. If we consider locally constant or globally fiber bunched sampling functions, then the Lyapuonv exponent is positive away from a finite set. Moreover, for sampling functions in an open and dense subset of the space in question, the Lyapunov exponent is uniformly positive. Our results can be applied to any subshift of finite type with ergodic measures that are equilibrium states of H\"older continuous potentials. In particular, we apply our results to Schr\"odinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains.