Random M\"obius dynamics on the unit disc and perturbation theory for Lyapunov exponents

TL;DR

This paper studies the effects of random Möbius transformations on the unit disc, determining the invariant measure and exploring implications for Lyapunov exponents in random Schrödinger operators.

Contribution

It provides a detailed analysis of the invariant measure for a perturbed random Möbius dynamics on the unit disc, linking it to Lyapunov exponent perturbation theory.

Findings

Explicit characterization of the invariant measure under perturbations.

Connections established between Möbius dynamics and Lyapunov exponents.

Applications to one-dimensional random Schrödinger operators.

Abstract

Randomly drawn $2\times 2$ matrices induce a random dynamics on the Riemann sphere via the M\"obius transformation. Considering a situation where this dynamics is restricted to the unit disc and given by a random rotation perturbed by further random terms depending on two competing small parameters, the invariant (Furstenberg) measure of the random dynamical system is determined. The results have applications to the perturbation theory of Lyapunov exponents which are of relevance for one-dimensional discrete random Schr\"odinger operators.