Ergodic averages of commuting transformations with distinct degree polynomial iterates

We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)... f_\ell(T_\ell^{p_\ell(n)}x)$, where $p_1,...,p_\ell$ are integer polynomials with distinct degrees, and $T_1,...,T_\ell$ are commuting, invertible measure preserving transformations, acting on the same probability space. This establishes several cases of a conjecture of Bergelson and Leibman, that complement the case of linear polynomials, recently established by Tao. Furthermore, we show that, unlike the case of linear polynomials, for polynomials of distinct degrees, the corresponding characteristic factors are mixtures of inverse limits of nilsystems. We use this particular structure, together with some equidistribution results on nilmanifolds, to give an application to multiple recurrence and a corresponding one to combinatorics.Comment: 44 pages, small correction in the proof of Lemma 7.5, appeared in the Proceedings of the London Mathematical Societ

Random sequences and pointwise convergence of multiple ergodic averages

We prove pointwise convergence, as N → ∞, for the multiple ergodic averages (1/N) Σn=1N f(Tnx) · g(San x), where T and S are commuting measure preserving transformations, and an is a random version of the sequence [nc] for some appropriate c \u3e 1. We also prove similar mean convergence results for averages of the form (1/N) Σ n=1N (Tan x) · g(S an x), as well as pointwise results when T and S are powers of the same transformations. The deterministic versions of these results, where one replaces an with [nc], remain open, and we hope that our method will indicate a fruitful way to approach these problems as well

Powers of Sequences and Convergence of Ergodic Averages

A sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure preserving system (X, B, µ, T) and any bounded measurable function f, the averages 1 PN N n=1 f(T snx) converge in the L2 norm. We construct a sequence (sn) that is good for the mean ergodic theorem, but the sequence (s2 n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (sk n) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form 1 PN N n=1 f1(T snx)f2(T 2snx)... fℓ(T ℓsnx). W