Structure and Spectrum of Nonergodic Nilsystems

TL;DR

This paper extends the spectral decomposition known for ergodic nilsystems to the nonergodic case, analyzing Host-Kra factors and spectral types without ergodicity assumptions.

Contribution

It generalizes the spectral type decomposition of nilsystems from ergodic to nonergodic cases, broadening understanding of their spectral properties.

Findings

Spectral type splits into discrete and Lebesgue components in nonergodic nilsystems

Main result extends ergodic spectral decomposition to nonergodic settings

Analyzes Host-Kra factors without assuming ergodicity

Abstract

We study Host-Kra factors and the spectral type of nilsystems, without assuming ergodicity. In the ergodic case, it is known that the spectral type splits into a discrete component and a Lebesgue component of infinite multiplicity. Our main result extends this decomposition to nonergodic nilsystems.