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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])
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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

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