Directional ergodicity, weak mixing and mixing for Zd- and Rd-actions

For a measure preserving $Z^d$- or $R^d$-action $T$, on a Lebesgue probability space $(X,\mu )$, and a linear subspace $L\subseteq R^d$, we define notions of direction $L$ ergodicity, weak mixing, and strong mixing. For $R^d$-actions, it is clear that these direction $L$ properties should correspond to the same properties for the restriction of $T$ to $L$. But since an arbitrary $L\subseteq R^d$ does not necessarily correspond to a nontrivial subgroup of $Z^d$, a different approach is needed for $Z^d$-actions. In this case, we define direction $L$ ergodicity, weak mixing, and mixing in terms of the restriction of the unit suspension $\tilde{T}$ to $L$, but also restricted to the subspace of $L^2(\tilde{X},\tilde{\mu })$ perpendicular to the suspension direction. For $Z^d$-actions, we show (as is more or less clear for $R^d$) that these directional properties are spectral properties. For weak mixing $Z^d$- and $R^d$-actions, we show that directional ergodicity is equivalent to directional weak mixing. For ergodic $Z^d$-actions $T$, we explore the relationship between direction $L$ properties as defined via unit suspensions and embeddings of $T$ in $R^d$-actions. Finally, the structure of possible sets of non-ergodic and non-weakly mixing directions is determined, and genericity questions are discussed.