Constructions of smooth and analytic cocycles over irrational circle rotations

Dalibor Volný

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 4, page 745-764
  • ISSN: 0010-2628

Abstract

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We define a class of step cocycles (which are coboundaries) for irrational rotations of the unit circle and give conditions for their approximation by smooth and real analytic coboundaries. The transfer functions of the approximating (smooth and real analytic) coboundaries are close (in the supremum norm) to the transfer functions of the original ones. This result makes it possible to construct smooth and real analytic cocycles which are ergodic, ergodic and squashable (see [Aaronson, Lemańczyk, Volný]), of type I I I 0 , or which are coboundaries with nonintegrable transfer functions. The cocycles are constructed as sums of coboundaries.

How to cite

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Volný, Dalibor. "Constructions of smooth and analytic cocycles over irrational circle rotations." Commentationes Mathematicae Universitatis Carolinae 36.4 (1995): 745-764. <http://eudml.org/doc/247726>.

@article{Volný1995,
abstract = {We define a class of step cocycles (which are coboundaries) for irrational rotations of the unit circle and give conditions for their approximation by smooth and real analytic coboundaries. The transfer functions of the approximating (smooth and real analytic) coboundaries are close (in the supremum norm) to the transfer functions of the original ones. This result makes it possible to construct smooth and real analytic cocycles which are ergodic, ergodic and squashable (see [Aaronson, Lemańczyk, Volný]), of type $III_0$, or which are coboundaries with nonintegrable transfer functions. The cocycles are constructed as sums of coboundaries.},
author = {Volný, Dalibor},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {smooth cocycle; real analytic cocycle; transfer function; type $III_0$; ergodic and squashable; distributions of a cocycle; ergodicity; skew product; coboundary; distribution of a cocycle; transfer functions; real analytic cocycle},
language = {eng},
number = {4},
pages = {745-764},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Constructions of smooth and analytic cocycles over irrational circle rotations},
url = {http://eudml.org/doc/247726},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Volný, Dalibor
TI - Constructions of smooth and analytic cocycles over irrational circle rotations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 4
SP - 745
EP - 764
AB - We define a class of step cocycles (which are coboundaries) for irrational rotations of the unit circle and give conditions for their approximation by smooth and real analytic coboundaries. The transfer functions of the approximating (smooth and real analytic) coboundaries are close (in the supremum norm) to the transfer functions of the original ones. This result makes it possible to construct smooth and real analytic cocycles which are ergodic, ergodic and squashable (see [Aaronson, Lemańczyk, Volný]), of type $III_0$, or which are coboundaries with nonintegrable transfer functions. The cocycles are constructed as sums of coboundaries.
LA - eng
KW - smooth cocycle; real analytic cocycle; transfer function; type $III_0$; ergodic and squashable; distributions of a cocycle; ergodicity; skew product; coboundary; distribution of a cocycle; transfer functions; real analytic cocycle
UR - http://eudml.org/doc/247726
ER -

References

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  13. Lemańczyk M., Personal communication, . 
  14. Liardet P., Volný D., Sums of continuous and differentiable functions in dynamical systems, preprint. MR1459847
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