Hostname: page-component-6d856f89d9-jrqft Total loading time: 0 Render date: 2024-07-16T04:02:03.349Z Has data issue: false hasContentIssue false
  • English
  • Français

On the ergodicity of Weyl sum cocycles

Published online by Cambridge University Press:  01 December 2007

GERNOT GRESCHONIG ,
MAHESH NERURKAR  and
DALIBOR VOLNÝ
GERNOT GRESCHONIG
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstraße 15, A-1090 Vienna, Austria (email: [email protected])
MAHESH NERURKAR
Affiliation:
Department of Mathematical Sciences, Rutgers University – Camden, Armitage Hall, 311 North 5th Street, Camden, NJ 08102, USA (email: [email protected])
DALIBOR VOLNÝ
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Avenue de l’Université, F-76801 Saint Etienne du Rouvray, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We present the quadratic Weyl sums with θ,x∈[0,1) as cocycles over a measure-preserving transformation on the two-dimensional torus. We show then that these cocycles are not coboundaries for every irrational θ∈[0,1), and that for a dense Gδ set of θ∈[0,1) the corresponding skew product is ergodic. For each of those θ, there exists a dense Gδ set of full measure of x∈[0,1) for which the sequence , n=1,2,… , is dense in .

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007 , pp. 1851 - 1863
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Aaronson, J., Lemańczyk, M. and Volný, D.. A cut salad of cocycles. Fund. Math. 157(2–3) (1998), 99–119.CrossRefGoogle Scholar
[3]Fayad, B.. On the ergodicity of the Weyl sums cocycle. Acta Arith. 125(4) (2006), 305–316.CrossRefGoogle Scholar
[4]Forrest, A. H.. The limit points of Weyl sums and other continuous cocycles. J. London Math. Soc. (2) 54(3) (1996), 440–452.CrossRefGoogle Scholar
[5]Forrest, A. H.. Symmetric cocycles and classical exponential sums. Colloq. Math. 84/85(1) (2000), 125–145.CrossRefGoogle Scholar
[6]Furstenberg, H.. Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961), 573–601.CrossRefGoogle Scholar
[7]Hardy, G. H. and Littlewood, J. E.. The trigonometric series associated with the elliptic θ-functions. Acta Math. 37 (1914), 193–239.CrossRefGoogle Scholar
[8]Schmidt, K.. Cocycles on Ergodic Transformation Groups (Macmillan Lectures in Mathematics, 1). Macmillan Company of India, Ltd., Delhi, 1977.Google Scholar
[9]Volný, D.. Completely squashable smooth ergodic cocycles over irrational rotations. Topol. Methods Nonlinear Anal. 22(2) (2003), 331–344.CrossRefGoogle Scholar
[10]Volný, D. and Nerurkar, M.. On the ergodicity of Weyl sum cocycles. Preprint.Google Scholar