On the strong convolution singularity property

TL;DR

This paper introduces a new method to establish the strong convolution singularity property for flows, demonstrating simple spectrum in associated Gaussian systems, with applications to smooth flows on higher-genus surfaces.

Contribution

The paper develops a novel approach for proving strong convolution singularity and applies it to specific smooth flows on surfaces of genus at least 2.

Findings

Established new criteria for strong convolution singularity

Provided examples of smooth flows with simple spectrum in the 3rd chaos

Extended understanding of spectral properties of flows on surfaces

Abstract

We develop a new method for proving that a flow has the so-called strong convolution singularity property, i.e. the Gaussian system induced by its (reduced) maximal spectral type has simple spectrum. We use these methods to give examples of smooth flows on closed orientable surfaces of genus at least 2 with a weaker property: each of their maximal spectral types $\sigma$ is such that the Gaussian system induced by $\sigma$ has simple spectrum on the so-called 3rd chaos (i.e. $V_\sigma^{\odot 3}$ has simple spectrum).