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ArticleOriginal scientific text

Title

Generic smooth cocycles of degree zero over irrational rotations

Authors 1

Affiliations

  1. Institute of Mathematics, Technical University of Wrocław, 50-370 Wrocław, Poland

Abstract

If a rotation α of has unbounded partial quotients then "most" of its skew-product diffeomorphic extensions to the 2-torus × defined by C1 cocycles of topological degree zero enjoy nontrivial ergodic properties. In fact they admit a cyclic approximation with speed o(1/n) and have nondiscrete (simple) spectrum. Similar results are obtained for Cr cocycles if α admits a sufficiently good approximation by rationals. For a.e. α and generic C1 cocycles the speed can be improved to o(1/(nlogn)). For generic α and generic Cr cocycles (r = 1,...,∞) the spectral measure of the skew product has a continuous component and Hausdorff dimension zero.

Keywords

ENG
Anzai skew product, weakly mixing cocycle, cyclic approximation, simple spectrum

Bibliography

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Studia Mathematica
1995/115/3
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Pages:
241-250
Main language of publication
English
Received
1994-07-04
Accepted
1995-04-11
Published
1995
Exact and natural sciences