Directional ergodicity and weak mixing for actions of \(\mathbb R^d\) and \(\mathbb Z^d\)

• Published in: arXiv.org 2022-11
• Main Authors: Robinson, E Arthur, Rosenblatt, Joseph, Ayşe A \c{S}ahin
• Format: Article
• Language: English
• Subjects: Ergodic processes
• ISSN: 2331-8422

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 \(\widetilde T\) to the direction \(L\) and to the subspace of \(L^2(\widetilde X,\widetilde \mu)\) 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.