Norm convergence of multiple ergodic averages for commuting transformations

By Terence Tao. Ergodic Theory and Dynamical Systems, 2007

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Abstract Let T1,…,Tl:X→X be commuting measure-preserving transformations on a probability space $(X, \mathcal {X}, \mu )$. We show that the multiple ergodic averages $\bfrac {1}{N} \sum _{n=0}^{N-1} f_1(T_1^n x) \cdots f_l(T_l^n x)$ are convergent in $L^2(X,\mathcal {X},\mu )$ as $N \to \infty $ for all $f_1,\ldots ,f_l \in L^\infty (X,\mathcal {X},\mu )$; this was previously established for l=2 by Conze and Lesigne [J. P. Conze and E. Lesigne. Théorèmes ergodique por les mesures diagonales. Bull. Soc. Math. France112 (1984), 143–175] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiTj−1 by Frantzikinakis and Kra [N. Frantzikinakis and B. Kra. Convergence of multiple ergodic averages for some commuting transformations. Ergod. Th. & Dynam. Sys.25 (2005), 799–809] (with the l=3 case of this result established earlier by Zhang [Q. Zhang. On the convergence of the averages $\bfrac {1}{N} \sum _{n=1}^N f_1(R^n x) f_2(S^n x) f_3(T^n x)$. Mh. Math.122 (1996), 275–300]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l=2 case of our arguments is a finitary analogue of those by Conze and Lesigne.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6aa88ff71cc07cb27abfd197243c5904adf9727c

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Citations in the corpus

Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces (2007, methodology): “The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from prima-facie ineffective proofs in the fixed point theory of such mappings.”
Some open problems on multiple ergodic averages (2011, background): “, T` are powers of the same transformation ([32, 35]), when the polynomials are linear [42] (with alternate proofs given in [43, 3, 29]), and when the polynomials have distinct degrees [15].”
Multiple recurrence for two commuting transformations (2009, background): “Ever since Tao [17] proved the norm convergence of multiple ergodic averages with several commuting transformations of the form 1 N N∑ n=1 T n1 f1 · · · T n d fd , (1) several other proofs via different approaches have been given by Austin [1], Host [12] and Towsner [18].”
Multiple recurrence and convergence for hardy sequences of polynomial growth (2009, background): “Nevertheless, we now have several different proofs of convergence in the case where the sequences are linear ([26], [38], [33], [35],[1], [25]), and the case where all the sequences are integer polynomials was treated in [27] and [29].”
Ergodic averages of commuting transformations with distinct degree polynomial iterates (2009, background): “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.”
On the norm convergence of non-conventional ergodic averages (2008, unknown): “1 |IN | ∑ n∈IN+aN d∏ i=1 fi ◦ T n i converge to some limit in L2(µ) that does not depend on the choice of (aN )N≥1 or (IN )N≥1.”
Norm convergence of nilpotent ergodic averages (2011, methodology): “Reducible functions will play a similar role than the one played by basic antiuniform functions in [19].”
Effective metastability of Halpern iterates in CAT(0) spaces (2011, methodology): “This paper applies techniques from mathematical logic to extract an explicit uniform rate of metastability (in the sense of Tao [49, 50]) from a recent proof due to Saejung [42] of a strong convergence theorem for Halpern iterations in the context of CAT(0) spaces.”
Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence (2010, methodology): “Note that the multiple ergodic average for commuting transformations was obtained by Tao [26] using finitary ergodic method, see [3, 18] for more traditional ergodic proofs.”
PROOF-THEORETIC METHODS IN NONLINEAR ANALYSIS (2019, methodology): “Tao [72] also established (without bound) a uniform metastable version of the Mean Ergodic Theorem in Hilbert space and used that uniformity as a base step for a generalization to commuting families of operators.”