Absolutely Continuous Spectrum for Parabolic Flows/Maps

TL;DR

This paper establishes a general framework for analyzing the spectral properties of parabolic systems, identifying conditions for absolutely continuous spectra, and applies these results to various flows including unipotent, horocycle, and skew product flows.

Contribution

It introduces an abstract framework for spectral analysis of parabolic systems and applies it to multiple classes of flows, extending existing spectral results.

Findings

Conditions for absolutely continuous spectral measures are identified.

Spectral properties of time-changed unipotent flows are characterized.

Results are extended to twisted horocycle flows and skew products.

Abstract

We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive certain spectral results for skew products over translations and Furstenberg transformations.