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Log Hölder continuity of the rotation number

Anton Gorodetski, Victor Kleptsyn

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-Hölder 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 Schrödinger operator must be log-Hölder.

Log Hölder continuity of the rotation number

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-Hölder 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 Schrödinger operator must be log-Hölder.

Paper Structure

This paper contains 6 sections, 6 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Result
  3. Preliminaries
  4. Main result
  5. Proof of the main result
  6. The Craig-Simon Theorem on log-Hölder regularity of the IDS

Key Result

Proposition 2.1

There exists a number $\rho\in \mathbb{R}$ such that for $\mu$-a.e. $\omega\in \frak{M}$ and every $x\in \mathbb{R}$ the limit exists and is equal to $\rho$.

Figures (1)

  • Figure 1: Action of maps $\tilde{g}_{a, \sigma^{n-1}(\omega)}$ and $\tilde{g}_{a', \sigma^{n-1}(\omega)}$.

Theorems & Definitions (14)

  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 4 more