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

Abstract

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