Log H\"older continuity of the rotation number

TL;DR

This paper proves that the rotation number of smooth circle cocycles varies in a log-H"older continuous manner with respect to parameters, leading to a dynamical proof of a related spectral regularity result.

Contribution

It establishes the log-H"older regularity of rotation numbers for smooth circle cocycles and applies this to prove a spectral regularity theorem for ergodic Schr"odinger operators.

Findings

Rotation numbers are log-H"older continuous in the parameter.

The integrated density of states of ergodic Schr"odinger operators is log-H"older.

Provides a dynamical proof of the Craig-Simon theorem.

Abstract

We consider one-parameter families of smooth circle cocycles over an ergodic transformation in the base, and show that their rotation numbers must be log-H\"older regular with respect to the parameter. As an immediate application, we get a dynamical proof of 1D version of the Craig-Simon theorem that establishes that the integrated density of states of an ergodic Schr\"odinger operator must be log-H\"older.