Lebesgue spectrum of countable multiplicity for conservative flows on the torus

TL;DR

This paper demonstrates that conservative mixing flows on the 2-torus with strong singularities typically have Lebesgue spectrum of infinite multiplicity, using spectral analysis and geometric criteria.

Contribution

It establishes the Lebesgue nature and infinite multiplicity of the spectrum for a class of flows with singularities, advancing understanding of their spectral properties.

Findings

Spectral measures are Lebesgue with infinite multiplicity for these flows.

Absolute continuity of the maximal spectral type is proven for non-uniformly stretching flows.

A geometric criterion is developed to determine infinite Lebesgue multiplicity.

Abstract

We study the spectral measures of conservative mixing flows on the 2-torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity. For this, we use two main ingredients: 1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows that have an irregular decay of correlations, 2) a geometric criterion that yields infinite Lebesgue multiplicity of the spectrum and that is well adapted to rapidly mixing flows.