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

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.

### MSC:

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