A dynamical Thouless formula

TL;DR

This paper develops a dynamical version of the Thouless formula linking Lyapunov exponents and the fibered rotation number for affine $ ext{GL}(2, ext{R})$ cocycles, with applications to continuity properties.

Contribution

It extends the classical Thouless formula to a dynamical setting involving fibered rotation numbers and provides new continuity results for these quantities.

Findings

Established a dynamical Thouless-type formula for affine $ ext{GL}(2, ext{R})$ cocycles.

Derived limitations on the modulus of continuity of random linear cocycles.

Proved H"older-type continuity of the fibered rotation number under various dynamics.

Abstract

In this paper we establish an abstract, dynamical Thouless-type formula for affine families of $\mathrm{GL} (2,\mathbb{R})$ cocycles. This result extends the classical formula relating, via the Hilbert transform, the maximal Lyapunov exponent and the integrated density of states of a Schr\"odinger operator. Here, the role of the integrated density of states will be played by a more geometrical quantity, the fibered rotation number. As an application of this formula we present limitations on the modulus of continuity of random linear cocycles. Moreover, we derive H\"older-type continuity properties of the fibered rotation number for linear cocycles over various base dynamics.