Directional ergodicity and weak mixing for actions of $\mathbb R^d$ and $\mathbb Z^d$

TL;DR

This paper introduces and studies directional ergodicity, weak mixing, and mixing for measure-preserving actions of  groups, establishing their spectral invariance and exploring their properties and genericity in both  and  actions.

Contribution

It defines new notions of directional properties for  actions, proves their spectral invariance, and investigates the relationships and genericity of these properties.

Findings

Directional ergodicity implies directional weak mixing for weakly mixing actions.

Directional properties are spectral invariants for  actions.

Analysis of non-ergodic and non-weakly mixing directions.

Abstract

We define notions of direction $L$ ergodicity, weak mixing, and mixing for a measure preserving $\mathbb Z^d$ action $T$ on a Lebesgue probability space $(X,\mu)$, where $L\subseteq\mathbb R^d$ is a linear subspace. For $\mathbb R^d$ actions these notions clearly correspond to the same properties for the restriction of $T$ to $L$. For $\mathbb Z^d$ actions $T$ we define them by using the restriction of the unit suspension $\widetilde T$ to the direction $L$ and to the subspace of $L^2(\widetilde X,\widetilde \mu)$ perpendicular to the suspension rotation factor. We show that for $\mathbb Z^d$ actions these properties are spectral invariants, as they clearly are for $\mathbb R^d$ actions. We show that for weak mixing actions $T$ in both cases, directional ergodicity implies directional weak mixing. For ergodic $\mathbb Z^d$ actions $T$ we explore the relationship between directional properties defined via unit suspensions and embeddings of $T$ in $\mathbb R^d$ actions. Genericity questions and the structure of non-ergodic and non-weakly mixing directions are also addressed.