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Structure and Spectrum of Nonergodic Nilsystems

Felipe Hernández

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. We extend this decomposition to nonergodic nilsystems. As a consequence, we obtain a spectral decomposition for ergodic $\mathbb{Z}^2$-nilsystems.

Structure and Spectrum of Nonergodic Nilsystems

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. We extend this decomposition to nonergodic nilsystems. As a consequence, we obtain a spectral decomposition for ergodic -nilsystems.
Paper Structure (10 sections, 20 theorems, 78 equations)

This paper contains 10 sections, 20 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background from nilsystems
  4. Background from structure theory
  5. Background from spectral theory
  6. Lie algebras and Mal'cev bases
  7. Structure of nonergodic nilsystems
  8. Spectrum of nonergodic nilsystems
  9. The discrete-spectrum and uniform components
  10. Weak mixing and nonuniform component

Key Result

Theorem A

Let $(X=G/\Gamma,\mu,T)$ be a nilsystem. There is a normal rational subgroup $H\subseteq G$ such that the system $(G/H_{k+1}\Gamma, \mu_{G/H_{k+1}\Gamma},T)$ is the $\mathcal{Z}_k$-factor of $X$ for each $k\in \mathbb{N}_0$. In particular, the group $H$ satisfies $\overline{\{ T^nx: n\in \mathbb{N}

Theorems & Definitions (40)

  • Theorem A
  • Theorem B
  • Corollary C
  • Conjecture 1.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • ...and 30 more