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Multiple recurrence for two commuting transformations. À paraître, Ergodic Theory Dynam. Systems
"... Abstract. This paper is devoted to a study of the multiple recurrence of two commuting transformations. We derive a result which is similar but not identical to that of one single transformation established by Bergelson, Host and Kra. We will use the machinery of “magic systems ” established recentl ..."
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Abstract. This paper is devoted to a study of the multiple recurrence of two commuting transformations. We derive a result which is similar but not identical to that of one single transformation established by Bergelson, Host and Kra. We will use the machinery of “magic systems ” established recently by B. Host for the proof. hal00441767, version 1 17 Dec 2009 1.
Multiple recurrence and convergence of Hardy sequences of polynomial growth
, 2010
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Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems
 Pacific J. Math
"... Abstract. The Furstenberg recurrence theorem (or equivalently, Szemerédi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M, τ) with an automorphism α: M → M, and a nonnegative a ∈ M with τ(a)> 0, on ..."
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Abstract. The Furstenberg recurrence theorem (or equivalently, Szemerédi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M, τ) with an automorphism α: M → M, and a nonnegative a ∈ M with τ(a)> 0, one has lim infN→ ∞ 1N ∑N n=1 Re τ(aα n(a)... α(k−1)n(a))> 0; a subsequent result of Host and Kra shows that this limit exists. In particular, Re τ(aαn(a)... α(k−1)n(a))> 0 for all n in a set of positive density. From the von Neumann algebra perspective, it is thus natural to ask to what extent these results remain true when the abelian hypothesis is dropped. All three claims hold for k = 2, and we show in this paper that all three claims hold for all k when the von Neumann algebra is asymptotically abelian, and that the last two claims hold for k = 3 when the von Neumann algebra is
ERGODIC AVERAGES OF COMMUTING TRANSFORMATIONS WITH DISTINCT DEGREE POLYNOMIAL ITERATES
, 2010
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Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence
 Adv. Math
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On the computational content of convergence proofs via Banach limits
"... This paper addresses new developments in the ongoing proof mining program, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method to extract rates of metastability (i ..."
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This paper addresses new developments in the ongoing proof mining program, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method to extract rates of metastability (in the sense of Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all what is known) rely on the axiom of choice. In this paper we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X. Combined with our method of eliminating Banach limits this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.
Effective metastability of Halpern iterates in CAT(0) spaces
"... This paper provides an effective uniform rate of metastability (in the sense of Tao) on the strong convergence of Halpern iterations of nonexpansive mappings in CAT(0) spaces. The extraction of this rate from an ineffective proof due to Saejung is an instance of the general proof mining program whic ..."
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This paper provides an effective uniform rate of metastability (in the sense of Tao) on the strong convergence of Halpern iterations of nonexpansive mappings in CAT(0) spaces. The extraction of this rate from an ineffective proof due to Saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs. This methodology is applied here for the first time to a proof that uses Banach limits and hence makes a substantial reference to the axiom of choice.