Analysis of two step nilsequences

Bernard Host[1]; Bryna Kra[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)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 5, page 1407-1453
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Host, Bernard, and Kra, Bryna. "Analysis of two step nilsequences." Annales de l’institut Fourier 58.5 (2008): 1407-1453. <http://eudml.org/doc/10353>.

@article{Host2008,
abstract = {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.},
affiliation = {Université Paris-Est Laboratoire d’analyse et de mathématiques appliquées UMR CNRS 8050, 5 bd Descartes 77454 Marne la Vallée Cedex 2 (France); Department of Mathematics Northwestern University 2033 Sheridan Road, Evanston IL 60208-2730 (USA)},
author = {Host, Bernard, Kra, Bryna},
journal = {Annales de l’institut Fourier},
keywords = {Nilsequence; nilmanifold; almost periodic sequence; nilsequence},
language = {eng},
number = {5},
pages = {1407-1453},
publisher = {Association des Annales de l’institut Fourier},
title = {Analysis of two step nilsequences},
url = {http://eudml.org/doc/10353},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Host, Bernard
AU - Kra, Bryna
TI - Analysis of two step nilsequences
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 5
SP - 1407
EP - 1453
AB - 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.
LA - eng
KW - Nilsequence; nilmanifold; almost periodic sequence; nilsequence
UR - http://eudml.org/doc/10353
ER -

References

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  10. A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold, Ergod. Th. & Dynam. Sys. 25 (2005), 201-113 Zbl1080.37003MR2122919
  11. E. Lesigne, Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques, Ergod. Th. & Dynam. Sys. 11 (1991), 379-391 Zbl0709.28012MR1116647
  12. A. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Transl. 9 (1962), 276-307 
  13. D. Montgomery, L. Zippin, Topological Transformation Groups, (1955), Interscience Publishers Zbl0068.01904MR73104
  14. W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757-771 Zbl0183.51503MR260975
  15. W. Parry, Dynamical systems on nilmanifolds, Bull. London Math. Soc. 2 (1970), 37-40 Zbl0194.05601MR267558
  16. D. J. Rudolph, Eigenfunctions of T × S and the Conze-Lesigne algebra, Ergodic Theory and its Connections with Harmonic Analysis (1995), 369-432, Cambridge University Press Zbl0877.28012MR1325712

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