# Constructions of smooth and analytic cocycles over irrational circle rotations

Commentationes Mathematicae Universitatis Carolinae (1995)

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

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topVolný, 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

top- Aaronson J., Lemańczyk M., Volný D., Salad of cocycles, preprint.
- Baggett L.W., Medina H.A., Merrill K.D., On functions that are trivial cocycles for a set of irrationals, II, to appear. Zbl0876.28024MR1285971
- Baggett L.W., Merrill K.D., Smooth cocycles for an irrational rotation, Israel J. Math. 79 (1992), 281-288. (1992) Zbl0769.28013MR1248918
- Billingsley P., Convergence of Probability Measures, Wiley New York (1968). (1968) Zbl0172.21201MR0233396
- Hamachi T., Type $II{I}_{0}$ cocycles with unbounded gaps, Commentationes Math. Univ. Carolinae 36.4 (1995), 713-720. (1995) MR1378692
- Herman M.R., Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, IHES Publications Math. 49 (1979), 5-234. (1979) Zbl0448.58019MR0538680
- Katok, Constructions in Ergodic Theory, manuscript. Zbl1130.37304
- Khinchine, Continued Fractions, P. Noordhoff, Ltd. Groningen (1963). (1963) MR0161834
- Kuipers L., Niederreiter H., Uniform Distribution of Sequences, Wiley New York (1974). (1974) Zbl0281.10001MR0419394
- Kwiatkowski J., Lemańczyk M., Rudolph D., On weak isomorphism of measure preserving diffeomorphisms, Israel J. Math. 80 (1992), 33-64. (1992) MR1248926
- Kwiatkowski J., Lemańczyk M., Rudolph D., A class of cocycles having an analytic modification, Israel J. Math. 87 (1994), 337-360. (1994) MR1286834
- Lemańczyk M., Analytic nonregular cocycles over irrational rotations, Commentationes Math. Univ. Carolinae 36.4 (1995), 727-735. (1995) MR1378694
- Lemańczyk M., Personal communication, .
- Liardet P., Volný D., Sums of continuous and differentiable functions in dynamical systems, preprint. MR1459847
- Parry W., Tuncel S., Classification Problems in Ergodic Theory, London Math. Society Lecture Notes 67, Cambridge University Press Cambridge (1982). (1982) Zbl0487.28014MR0666871
- Schmidt K., Cocycles of Ergodic Transformation Groups, Macmillan Lectures in Math. vol. 1, Macmillan Company of India (1977). (1977) MR0578731
- Stewart M., Irregularities of uniform distribution, Acta Math. Acad. Sci. Hungar. 37 (1981), 185-221. (1981) Zbl0475.10040MR0616890
- Volný D., On limit theorems and category for dynamical systems, Yokohama Math. J. 38 (1990), 29-35. (1990) MR1093661

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