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 $(X,μ)$, 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 $L^2(\widetilde X,\widetilde μ)$ 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.