Convergence of diagonal ergodic averages

Abstract

Tao has recently proved that ifT1,…,Tlare commuting, invertible, measure-preserving transformations on a dynamical system, then for anyL∞functionsf1,…,fl, the average (1/N)∑n=0N−1∏i≤lfi∘Tinconverges in theL2norm. Tao’s proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence ‘backwards’. In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao’s argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.

Ergodic averages of commuting transformations with distinct degree polynomial iterates

“…Unlike previous arguments in [4,11,13,22,31,32], where one finds ways to sidestep the problem of giving precise algebraic descriptions of the factor systems that control the limiting behaviour of special cases of the averages (2), a distinctive feature of the proof of Theorem 1.1 is that we give such descriptions. † Furthermore, we did not find it advantageous to work within a suitable extension of our system in order to simplify our study (like the 'pleasant' or 'magic' extensions that were introduced in [4] and in [22], respectively).…”

“…When all the polynomials in (1) are linear, after a series of partial results [11,16,20,29,33] that were obtained using ergodic theory, convergence was established in [31] using a finitary argument. Subsequently, motivated by ideas from [31], several other proofs of this 'linear' result were found using non-standard analysis [32], and then ergodic theory [4,22]. Proofs of convergence for general polynomial iterates have been given only under very strong ergodicity assumptions [5,27].…”

“…Unlike previous arguments in [11,13,31,32,2,22], where one finds ways to sidestep the problem of giving precise algebraic descriptions of the factor systems that control the limiting behavior of special cases of the averages (2), a distinctive feature of the proof of Theorem 1.1 is that we give such descriptions.…”

“…We take ℓ ≥ 2, assume that the results holds for ℓ − 1 transformations, and we are going to prove that it holds for ℓ transformations. Without loss of generality we can assume that deg( (32) converge to 0 in L 2 (µ). Therefore we can restrict ourselves to the case where the function f 1 is measurable with respect to Z k 0 ,T 1 .…”

On the norm convergence of non-conventional ergodic averages

“…This, in turn, is easily seen to be equivalent to a purely finitary result about the behaviour of certain sequences of averages of 1-bounded functions on (Z/NZ) d for very large N, and the bulk of Tao's work then goes into proving this last result. Interestingly, Towsner has shown in [17] how the asymptotic behaviour of these purely finitary averages can be re-interpreted back into an ergodic-theoretic assertion by building a suitable 'proxy' probability-preserving system from these averages themselves, using constructions from nonstandard analysis. Tao's method of analysis can be extended to the case of individual actions T i of a higher-rank r and an arbitrary Følner sequence in Z r , but with the base-point shifts a N all zero, quite straightforwardly, but seems to require more work in order to be extended to a proof for the above base-point-uniform version.…”