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

2022/03/13 by Robinson, E. Arthur, Joseph D. Rosenblatt, Rosenblatt, Joseph +3
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2203.06710

openalex publication_date 2022/03/13 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

We define notions of direction L ergodicity, weak mixing, and mixing for a\nmeasure preserving mathbb Z^d action T on a Lebesgue probability space\n(X,\μ), where L\⊆ mathbb R^d is a linear subspace. For mathbb\nR^d actions these notions clearly correspond to the same properties for the\nrestriction of T to L. For mathbb Z^d actions T we define them by\nusing the restriction of the unit suspension widetilde T to the direction\nL and to the subspace of L²( widetilde X, widetilde \μ) perpendicular to\nthe suspension rotation factor. We show that for mathbb Z^d actions these\nproperties are spectral invariants, as they clearly are for mathbb R^d\nactions. We show that for weak mixing actions T in both cases, directional\nergodicity implies directional weak mixing. For ergodic mathbb Z^d actions\nT we explore the relationship between directional properties defined via unit\nsuspensions and embeddings of T in mathbb R^d actions. Genericity\nquestions and the structure of non-ergodic and non-weakly mixing directions are\nalso addressed.\n

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