2022/03/13 by E. Arthur Robinson Jr., Robinson, E. Arthur, Joseph Rosenblatt +6
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.DS
paper · pdf · doi:10.48550/arxiv.2203.06710
The new version includes additional examples, a new "Further Directions" section, and updated references
openalex publication_date 2022/03/13 · openalex created_date 2022/05/05 · arxiv created 2022/11/29 · arxiv updated 2022/11/30 · openalex updated_date 2026/07/28
We define notions of direction L ergodicity, weak mixing, and mixing for a measure preserving \mathbb Zd action T on a Lebesgue probability space (X,μ), where L⊆\mathbb Rd is a linear subspace. For \mathbb Rd actions these notions clearly correspond to the same properties for the restriction of T to L. For \mathbb Zd actions T we define them by using the restriction of the unit suspension \widetilde T to the direction L and to the subspace of L2(\widetilde X,\widetilde μ) perpendicular to the suspension rotation factor. We show that for \mathbb Zd actions these properties are spectral invariants, as they clearly are for \mathbb Rd actions. We show that for weak mixing actions T in both cases, directional ergodicity implies directional weak mixing. For ergodic \mathbb Zd actions T we explore the relationship between directional properties defined via unit suspensions and embeddings of T in \mathbb Rd actions. Genericity questions and the structure of non-ergodic and non-weakly mixing directions are also addressed.