文摘
Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang–Shao–Ye technique and the Host machinery of magic systems, we prove that for a system (X, µ, S, T) with commuting transformations S and T, the average $$\frac{1}{{{N^2}}}\sum\limits_{i,j = 0}^{N - 1} {{f_0}} \left( {{S^i}x} \right){f_1}\left( {{T^j}x} \right){f_2}\left( {{S^i}{T^j}x} \right)$$ converges a.e. as N goes to infinity for any f0, f1, f2 ∈ L∞(µ).References[1]I. Assani, Pointwise convergence of ergodic averages along cubes, J. Analyse Math. 110 (2010), 241–269.MathSciNetCrossRefMATHGoogle Scholar[2]T. Austin, On the norm convergence of nonconventional ergodic averages, Ergodic Theory and Dynamical Systems 30 (2010), 321–338.MathSciNetCrossRefMATHGoogle Scholar[3]V. Bergelson, The multifarious Poincaré Recurrence Theorem, in Descriptive Set Theory and Dynamical Systems, (M. Foreman, A. S. Kechris, A. Louveau and B. Weiss. eds.), Cambridge University Press, New York, 2000, pp. 31–57.CrossRefGoogle Scholar[4]J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math. 404 (1990), 140–161.MathSciNetMATHGoogle Scholar[5]Q. Chu, Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems 31 (2011), 771–792.MathSciNetCrossRefMATHGoogle Scholar[6]Q. Chu and N. Frantzikinakis, Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations, Ergodic Theory Dynam. Systems 32 (2012), 877–897.MathSciNetCrossRefMATHGoogle Scholar[7]C. Demeter and C Thiele, On the two dimensional bilinear Hilbert transform, Amer. J. Math. 132.1 (2010), 201–256.MathSciNetCrossRefMATHGoogle Scholar[8]S. Donoso and W. Sun, Dynamical cubes and a criteria for systems having product extensions, J. Mod. Dyn. 9 (2015), 365–405.MathSciNetCrossRefMATHGoogle Scholar[9]B. Host, Ergodic seminorms for commuting transformations and applications, Studia Math. 195 (2009), 31–49.MathSciNetCrossRefMATHGoogle Scholar[10]B. Host and B. Kra, Averaging along cubes, in Dynamical Systems and Related Topics, (M. Brin, B. Hasselblatt and Y. Pesin eds.) Cambridge University Press, Cambridge, 2004.Google Scholar[11]B. Host and B. Kra, Nonconventional averages and nilmanifolds, Ann. of Math. (2) 161 (2005), no. 1, 398–488.MathSciNetCrossRefMATHGoogle Scholar[12]W. Huang, S. Shao and X. Ye, Strictly ergodic models and the convergence of nonconventional pointwise ergodic averages, arXiv:1312.7213.[13]W. Huang, S. Shao and X. Ye, Pointwise convergence of multiple ergodic averages and strictly ergodic models, arXiv:1406.5930.[14]R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech. 19 (1969/1970) 717–729.MathSciNetMATHGoogle Scholar[15]W. Krieger, On unique ergodicity, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, University of California Press, Berkeley, Calif., 1972, pp. 327–346.Google Scholar[16]A. Leibman, Lower bounds for ergodic averages, Ergodic Theory Dynam. Systems 22 (2002), 863–872.MathSciNetCrossRefMATHGoogle Scholar[17]T. Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory and Dynamical Systems 28 (2008), 657–688.MathSciNetCrossRefMATHGoogle Scholar[18]B. Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc. (N.S.) 13 (1985), 143–146.MathSciNetCrossRefMATHGoogle ScholarCopyright information© Hebrew University of Jerusalem 2016Authors and AffiliationsSebastián Donoso12Email authorWenbo Sun31.Centro de Modelamiento Matemático and Departamento de Ingeniería MatemáticaUniversidad de ChileSantiagoChile2.Université Paris-EstLaboratoire d’analyse et de mathématiques appliquéesMarne la Vallée Cedex 2France3.Department of MathematicsNorthwestern UniversityEvanstonUSA About this article CrossMark Print ISSN 0021-2172 Online ISSN 1565-8511 Publisher Name The Hebrew University Magnes Press About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s11856-016-1423-5_A pointwise cubic average for two ", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s11856-016-1423-5_A pointwise cubic average for two ", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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