H\"older continuity of the integrated density of states for Liouville frequencies

TL;DR

This paper proves H"older continuity of the integrated density of states for certain energies in analytic potentials, based on conditions involving the Lyapunov exponent and Avila's acceleration, advancing understanding in spectral theory.

Contribution

It establishes H"older continuity of the integrated density of states for energies satisfying specific Lyapunov exponent conditions in the context of analytic potentials.

Findings

Proves H"older continuity of the Lyapunov exponent and integrated density of states.

Identifies conditions involving Avila's acceleration for continuity.

Extends results to general analytic potentials.

Abstract

We prove H\"older continuity of the Lyapunov exponent $L(\omega,E)$ and the integrated density of states at energies that satisfy $L(\omega,E)>4\kappa(\omega,E)\cdot \beta(\omega)\geq 0$ for general analytic potentials, with $\kappa(\omega,E)$ being Avila's acceleration.