On the spectrum of shear flows and uniform ergodic theorems

TL;DR

This paper investigates the spectra of shear flows, providing examples of flows with singular continuous spectra, analyzing spectral density, and establishing an ergodic theorem with uniform convergence.

Contribution

It introduces new spectral analysis results for shear flows, including an example with purely singular continuous spectrum and conditions for uniform spectral density.

Findings

Existence of shear flows with purely singular continuous spectrum

Identification of spaces with uniformly bounded spectral density

Proof of an ergodic theorem with uniform convergence

Abstract

The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both $L^2$ spaces and weighted-$L^2$ spaces. As a consequence, an example of a flow admitting a purely singular continuous spectrum is provided. For flows admitting more regular spectra the density of states is analyzed, and spaces on which it is uniformly bounded are identified. As an application, an ergodic theorem with uniform convergence is proved.