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 subset of 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 subset of 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 T to L , but also restricted to the subspace of L-2 ( X , mu ) perpendicular to the suspension direction. For Z(d)-actions, we show (as is more or less clear for Rd) 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. (c) 2023 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.