no. 5
Analysis of two step nilsequences
[Analyse des nilsuites d’ordre deux]
Host, Bernard1 ; Kra, Bryna2
1 Université Paris-Est Laboratoire d’analyse et de mathématiques appliquées UMR CNRS 8050, 5 bd Descartes 77454 Marne la Vallée Cedex 2 (France)
2 Department of Mathematics Northwestern University 2033 Sheridan Road, Evanston IL 60208-2730 (USA)
Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1407-1453.

Les nilsuites sont apparues dans l’étude des moyennes ergodiques multiples associées à la démonstration par Furstenberg du théorème de Szemerédi. Depuis, elles ont aussi joué un rôle dans des questions de combinatoire additive. Les nilsuites sont une généralisation des suites presque périodiques et nous déterminons quelles parties de la théorie des suites presque périodiques peuvent s’étendre aux nilsuites d’ordre deux. Nous établissons les propriétés de base de ces suites et donnons une classification.

Nilsequences arose in the study of the multiple ergodic averages associated to Furstenberg’s proof of Szemerédi’s Theorem and have since played a role in problems in additive combinatorics. Nilsequences are a generalization of almost periodic sequences and we study which portions of the classical theory for almost periodic sequences can be generalized for two step nilsequences. We state and prove basic properties for two step nilsequences and give a classification scheme for them.

DOI : 10.5802/aif.2389
Classification : 37A45, 37B05, 42A75, 43A85
Keywords: Nilsequence, nilmanifold, almost periodic sequence
Mot clés : nilsuite, nilvariété, suite presque-périodique

Host, Bernard 1 ; Kra, Bryna 2

1 Université Paris-Est Laboratoire d’analyse et de mathématiques appliquées UMR CNRS 8050, 5 bd Descartes 77454 Marne la Vallée Cedex 2 (France)
2 Department of Mathematics Northwestern University 2033 Sheridan Road, Evanston IL 60208-2730 (USA) 
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Host, Bernard; Kra, Bryna. Analysis of two step nilsequences. Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1407-1453. doi : 10.5802/aif.2389. https://aif.centre-mersenne.org/articles/10.5802/aif.2389/

[1] Auslander, L.; Green, L.; Hahn, F. Flows on homogeneous spaces, Ann. Math. Studies, 53, Princeton Univ. Press, 1963 | Zbl

[2] Bergelson, V.; Host, B.; Kra, B. Multiple recurrence and nilsequences, Inventiones Math., Volume 160 (2005), pp. 261-303 (with an Appendix by I. Ruzsa) | DOI | MR | Zbl

[3] Bergelson, V.; Leibman, A. Distribution of values of bounded generalized polynomials, Acta Math., Volume 198 (2007), pp. 155-230 | DOI | MR

[4] Corwin, L.; Greenleaf, F.P. Representations of nilpotent Lie groups and their applications, Cambridge University Press, 1990 | MR | Zbl

[5] Furstenberg, H. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, Journal d’Analyse Math., Volume 31 (1977), pp. 204-256 | DOI | Zbl

[6] Green, B.; Tao, T. Linear equations in the primes (To appear, Annals of Math)

[7] Green, B.; Tao, T. Quadratic uniformity of the Möbius function (To appear, Annales de l’Institut Fourier) | Numdam | Zbl

[8] Host, B.; Kra, B. Uniformity seminorms on ℓ ∞ (ℤ) and applications (submitted)

[9] Host, B.; Maass, A. Nilsystèmes d’ordre deux et parallélépipèdes (To appear, Bull. Math. Soc. France)

[10] Leibman, A. Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold, Ergod. Th. & Dynam. Sys., Volume 25 (2005), p. 201-113 | DOI | MR | Zbl

[11] Lesigne, E. Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques, Ergod. Th. & Dynam. Sys., Volume 11 (1991), pp. 379-391 | DOI | MR | Zbl

[12] Malcev, A. On a class of homogeneous spaces, Amer. Math. Soc. Transl., Volume 9 (1962), pp. 276-307

[13] Montgomery, D.; Zippin, L. Topological Transformation Groups, Interscience Publishers, 1955 | MR | Zbl

[14] Parry, W. Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math., Volume 91 (1969), pp. 757-771 | DOI | MR | Zbl

[15] Parry, W. Dynamical systems on nilmanifolds, Bull. London Math. Soc., Volume 2 (1970), pp. 37-40 | DOI | MR | Zbl

[16] Rudolph, D. J.; Petersen, K.; Salama, I. Eigenfunctions of T×S and the Conze-Lesigne algebra, Ergodic Theory and its Connections with Harmonic Analysis (1995), pp. 369-432 | MR | Zbl

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