On the norm convergence of non-conventional ergodic averages
By Tim Austin. Ergodic Theory and Dynamical Systems, 2008
91 citations recorded by the source corpus.
Abstract We offer a proof of the following non-conventional ergodic theorem: If Ti:ℤr↷(X,Σ,μ) for i=1,2,…,d are commuting probability-preserving ℤr-actions, (IN)N≥1 is a Følner sequence of subsets of ℤr, (aN)N≥1 is a base-point sequence in ℤr and f1,f2,…,fd∈L∞(μ) then the non-conventional ergodic averages \[ \frac {1}{|I_N|}\sum _{n \in I_N + a_N} \prod _{i=1}^df_i\circ T_i^n \] converge to some limit in L2(μ) that does not depend on the choice of (aN)N≥1 or (IN)N≥1. The leading case of this result, with r=1 and the standard sequence of averaging sets, was first proved by Tao, following earlier analyses of various more special cases and related results by Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler. While Tao’s proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/b4edadccf630d5ef6ff4aa1b61142448e6a94f3f
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Citations in the corpus
- Ergodic averages of commuting transformations with distinct degree polynomial iterates (2009, background): “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.”
- Some open problems on multiple ergodic averages (2011, methodology): “See also [4, 5] where techniques from [3] have been refined and extended, aiming to eventually handle the case of general polynomial iterates.”
- 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.”