The fibered rotation number

TL;DR

This paper derives an explicit formula for the increment of the fibered rotation number in various dynamical systems, linking stationary measures to spectral properties, and proves H"older continuity of the integrated density of states in the Anderson Model.

Contribution

It provides a new explicit formula for the fibered rotation number increment and establishes H"older regularity of the IDS in the Anderson Model.

Findings

Formula for the increment of fibered rotation number in terms of invariant measures.

Relation between stationary measures and the integrated density of states.

Proof of H"older continuity of the IDS for the Anderson Model.

Abstract

We provide an explicit formula for an increment of the fibered rotation number of a one-parameter family of circle cocycles over any ergodic transformation in terms of invariant measures. As an application, for a family of random dynamical systems on the circle, this gives a formula for an increment of the rotation number in terms of the stationary measures. In the case of projective Schr\"odinger cocycles associated with the Anderson Model, that provides a relation between the properties of the stationary measures on the projective space and the integrated density of states (IDS) of the corresponding family of operators. In particular, it gives a dynamical proof of H\"older regularity of the IDS in Anderson Model. Finally, we prove that the IDS for the Anderson Model with an ergodic background must be H\"older continuous.