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

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 $\tilde{T}$ to the direction $L$ and to the subspace of $L^2(\tilde{X},\tilde{\mu })$ 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.