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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces


Authors: M. Lemanczyk, F. Parreau and D. Volný
Journal: Trans. Amer. Math. Soc. 348 (1996), 4919-4938
MSC (1991): Primary 28D05, 47A10
MathSciNet review: 1389783
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Abstract: Given an irrational rotation, in the space of real bounded variation functions it is proved that there are ergodic cocycles whose small perturbations remain ergodic; in fact, the set of ergodic cocycles has nonempty dense interior.

Given a pseudo-homogeneous Banach space and an irrational rotation, we study the set of elements satisfying the mean ergodic theorem. Once such a space is not homogeneous, we prove it is not reflexive and not separable. In ``natural" cases, up to $L^1$-cohomology, the only elements satisfying the mean ergodic theorem are those from the closure of trigonometric polynomials.

For pseudo-homogeneous spaces admitting a Koksma's inequality ergodicity of the corresponding cylinder flows can be deduced from spectral properties of some circle extensions. In particular this is the case of Lebesgue spectrum (in the orthocomplement of the space of eigenfunctions) for the circle extension.


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Additional Information

M. Lemanczyk
Affiliation: Department of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
Email: mlem@mat.uni.torun.pl

F. Parreau
Affiliation: Laboratoire d’Analyse, Géométrie et Applications, URA CNRS 742, Université Paris-Nord, Av. J.-B. Clément, 93430 Villetaneuse, France
Email: parreau@math.univ-paris13.fr

D. Volný
Affiliation: Mathematical Institute, Charles University, Sokolovská 83, 186 00 Praha 8, Czech Republic
Email: dvolny@karlin.mff.cuni.cz

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01799-0
PII: S 0002-9947(96)01799-0
Received by editor(s): July 3, 1995
Additional Notes: Research of the first author was partially supported by KBN grant 2 P301 031 07 (1994)
Research of the third author was supported by grant GAUK 368 of Charles University
Article copyright: © Copyright 1996 American Mathematical Society