Fine dimensional properties of spectral measures

TL;DR

This paper investigates the fine fractal properties of spectral measures in ergodic Schrödinger operators, establishing bounds on their dimensions and constructing explicit examples to demonstrate sharpness.

Contribution

It introduces a complete family of Hausdorff measure functions and extends results on rank one perturbations to analyze spectral measure dimensions.

Findings

Spectral measures of half-line operators with positive Lyapunov exponent have at most logarithmic dimension.

Constructed explicit operator with spectral measure achieving this maximal dimension.

Extended and improved results on generalized Hausdorff dimensions in quantum dynamics.

Abstract

Operators with zero dimensional spectral measures appear naturally in the theory of ergodic Schr\"odinger operators. We develop the concept of a complete family of Hausdorff measure functions in order to analyze and distinguish between these measures with any desired precision. We prove that the dimension of spectral measures of half-line operators with positive upper Lyapunov exponent are at most logarithmic for every possible boundary phase. We show that this is sharp by constructing an explicit operator whose spectral measure obtains this dimension. We also extend and improve some basic results from the theory of rank one perturbations and quantum dynamics to encompass generalized Hausdorff dimensions.