Asymptotic geometry of non-mixing sequences. (English) Zbl 1039.37001

Einsiedler, Manfred; Ward, Thomas

Ergodic Theory Dyn. Syst. 23, No. 1, 75-85 (2003).

Let \(\alpha\) be a \(\mathbb Z^d\)-action by measure preserving transformations on a nontrivial probability space \((X,{\mathcal S},\mu).\) The action \(\alpha\) is said to be mixing of order \(r\) if, for any sequence \((n_1^{(j)},\cdots ,n_r^{(j)})\) of \(r\)-tuples in \(\mathbb Z^d\) with \(n_s^{(j)}-n_t^{(j)}\to \infty\) as \(j\to \infty,\) for \(s\not=t\), and for any sets \(A_1,\cdots ,A_r\in {\mathcal S}\), one has \[ \lim_{j\to \infty}\mu(\alpha^{-n_1^{(j)}}(A_1)\cap\cdots \alpha^{-n_r^{(j)}}(A_r))=\mu(A_1)\cdots \mu(A_r). \] A finite set \(\{n_1,\cdots ,n_r\}\) of integers is a mixing shape for \(\alpha\) if \[ \lim_{k\to \infty}\mu(\alpha^{-kn_1}(A_1)\cap\cdots \alpha^{-kn_r}(A_r))=\mu(A_1)\cdots \mu(A_r). \] The authors consider algebraic dynamical systems, i.e., \(X\) is a compact metrizable abelian group, \(\mu\) is a Haar measure and \(\alpha\) acts by automorphisms. They prove a special case of the conjecture: An algebraic dynamical system for which all shapes of cardinality \(r\) are mixing is mixing of order \(r\). They prove the validity of the conjecture when \(\alpha\) is a \(\mathbb Z^2\)-action, and \(r=3.\) Their proof makes use of non-Archimedean norms in function fields of positive characteristic which allows them to exhibit an asymptotic shape in nonmixing sequences for algebraic \(\mathbb Z^2\)-actions. This gives a relationship between the order of mixing and the convex hull of the defining polynomial.

Reviewer: Karma Dajani (Utrecht)

MSC:

37A25	Ergodicity, mixing, rates of mixing
37A05	Dynamical aspects of measure-preserving transformations

Keywords:

algebraic dynamical systems; mixing of order \(r\); mixing shapes