Some problems about joint ergodicity and uniform distribution

Joint ergodicity is concerned with studying the ergodic behavior of several measure-preserving transformations on the diagonal of product space. In this paper, we introduce notions of joint ergodicity, weak mixing and strong mixing for arbitrary sequences $\{T\sb{(1),n}\}\sbsp{n=0}{\infty}$, $\{T\sb{(2),n}\}\sbsp{n = 0}{\infty},\cdots,\{T\sb{(s),n}\}\sbsp{n=0}{\infty}$ of measure-preserving transformations of a probability space $(X,{\cal B},\mu).$ These notions generalize those of the setup--$T\sb1,T\sb2,\cdots,T\sb{s}$ are jointly ergodic (jointly weakly mixing, jointly strongly mixing) in the usual sense. Some criterions for verifying the joint ergodicity, joint weak mixing or joint strong mixing of given sequences are provided.

We study sequences of affine transformations of compact abelian groups. Sequences of affine transformations are jointly weakly mixing if and only if the sequences of epimorphisms lying below are jointly ergodic. A sufficient condition for given several affine transformations being jointly ergodic is given. Specially, we study skew products and compound skew products of finite dimensional tori. All of them are non-commutative. We obtain some necessary and sufficient conditions about their joint ergodicity.

The behavior of sequence of partial sums of epimorphisms and affine transformations of compact abelian groups is studied. If $\sigma$ is an epimorphism, we define$$\{\sigma\sb{k} (n)= {\sum\limits\sbsp{j=0}{n-1}}\ \sigma\sb{k-1}(j)\}\sbsp{n=0}{\infty}$$the sequence of k-degree partial sums of $\sigma$ by induction, where $\{\sigma\sb1(n)= \sum\sbsp{j=0}{n-1}\ \sigma\sp{j}\}\sbsp{n=0}{\infty}$. We prove that sequence of 1-degree partial sums of an epimorphism $\sigma$ of the finite dimensional torus is ergodic if and only if the matrix of $\sigma$ has no non-trivial root of the unity as its eigenvalue. If $\sigma$ is an ergodic epimorphism of the compact abelian group G, then for any positive integer $k, \{\sigma\sb{k}(n)x\}\sbsp{n=0}{\infty}$ is uniformly distributed for almost all $x \in G.$ If $\sigma\sb1,\sigma\sb2,\cdots,\sigma\sb{s}$ are commuting jointly ergodic epimorphisms of the compact abelian group G, then for any positive integers $k\sb1,k\sb2,\cdots,k\sb{s},$ $\{(\sigma\sb{k\sb1}(n)x,\sigma\sb{k\sb2}(n)x, \cdots,\sigma\sb{k\sb{s}}(n)x)\}\sbsp{n=0}{\infty}$ is uniformly distributed in the product $G \times G \times\cdots\times G$ for almost all $x \in G$. An interesting application of above results is the following theorem: almost all points $\{x\sb{n}\}\sbsp{n=1}{\infty}$ in the infinite dimensional torus have the properties that $\{\sum\sbsp{j=1}{n}x\sb{j}\}\sbsp{n=1}{\infty}$ and $\{\sum\sbsp{j=1}{n}jx\sb{j}\}\sbsp{n=1}{\infty}$ are uniformly distributed mod 1.