The fibered rotation number

TL;DR

This work derives an explicit formula for the increment of the fibered rotation number for circle cocycles over ergodic bases, expressed via invariant measures and a translation-type functional. It applies this formula to one-parameter random circle dynamics to obtain Hölder continuity results for the rotation number, and connects the rotation-number regularity to the integrated density of states (IDS) in the Anderson model, providing a dynamical proof of IDS Hölder continuity for ergodic backgrounds. The results illuminate the parallel regularity phenomena across dynamical and spectral settings and include an explicit example showing the necessity of a no-invariant-measure hypothesis. Overall, the work unifies dynamical translations and spectral regularity through a measure-theoretic framework.

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 Schrödinger 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 Hölder regularity of the IDS in Anderson Model. Finally, we prove that the IDS for the Anderson Model with an ergodic background must be Hölder continuous.

Figures (5)

• Figure 1: The expression under the integral in \ref{['eq.main']}: the (signed) $\widetilde{\nu}_{E_2,Tx}$-measure of the half-interval drawn in bold.
• Figure 2: Set $K_{E_1,E_2;\omega}$ and a point $(y,z)$ inside it.
• Figure 3: The case of a measure $\nu_1$ supported on a section $\gamma$: the translation value $\mathcal{T}$ is equal to the average measure of the half-open interval between the lift of the section $\gamma$ and its image $\widetilde{F}\gamma$.
• Figure 4: The graph of the composition $\widetilde{g}_{E_2,\omega_n}\circ \dots \circ \widetilde{g}_{E_2,\omega_{j+1}}$ and the effect of changing the parameter at the $j$-th iteration.
• Figure 5: Map $f$ and the system $\{g_{E,1}, g_{E,2}\}$ for $E>0$

Theorems & Definitions (32)

• Proposition 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 2.4
• Corollary 2.5
• Theorem 2.6
• Remark 2.7
• Remark 2.8
• Remark 2.9
• Remark 2.10