We prove that mixing on rank-one transformations is equivalent to "the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums". In particular, all polynomial staircase transformations are mixing.

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by $dₙ\left(x\right):={\sum }_{i=1}^n{1}_E\left(2^{i-1}x\right)\left(mod2\right)$, where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if $N^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that $N^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges a.e. and the limit equals 1/3 or 2/3 depending on x.

In ergodic theory, certain sequences of averages $A_{k}f$ may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence $A_{m_k}f$ of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are $A_{k}f\left(x\right)=1/\left(2^k\right){\sum }_{j=4^k+1}^{4^k+2^k}f\left(T^{j}x\right)$, then the subsequence $A_{k²}f$ will not be pointwise good even on $L^{\infty }$, but the subsequence $A_{2^k}f$ will be pointwise good on L¹. Understanding when the hyperexponential...

In this article some properties of Markovian mean ergodic operators are studied. As an application of the tools developed, and using the admissibility feature, a “reduction of order” technique for multiparameter admissible superadditive processes is obtained. This technique is later utilized to obtain a.e. convergence of averages $n^{-2}{\sum }_{i,j=0}^{n-1}{f}_{(i,j)}$ as well as their weighted version.