Directional ergodicity, weak mixing and mixing for [formula omitted]- and [formula omitted]-actions

Abstract

For a measure preserving Zd- or Rd-action T, on a Lebesgue probability space (X,μ), and a linear subspace L⊆Rd, we define notions of direction L ergodicity, weak mixing, and strong mixing. For Rd-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⊆Rd does not necessarily correspond to a nontrivial subgroup of Zd, a different approach is needed for Zd-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 L2(X˜,μ˜) perpendicular to the suspension direction. For Zd-actions, we show (as is more or less clear for Rd) that these directional properties are spectral properties. For weak mixing Zd- and Rd-actions, we show that directional ergodicity is equivalent to directional weak mixing. For ergodic Zd-actions T, we explore the relationship between direction L properties as defined via unit suspensions and embeddings of T in Rd-actions. Finally, the structure of possible sets of non-ergodic and non-weakly mixing directions is determined, and genericity questions are discussed.