Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
smooth cocycle; real analytic cocycle; transfer function; type $III_0$; ergodic and squashable; distributions of a cocycle
smooth cocycle; real analytic cocycle; transfer function; type $III_0$; ergodic and squashable; distributions of a cocycle
Summary:
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.
References:
[1] Aaronson J., Lemańczyk M., Volný D.: Salad of cocycles. preprint.
[2] Baggett L.W., Medina H.A., Merrill K.D.: On functions that are trivial cocycles for a set of irrationals, II. to appear. MR 1285971 | Zbl 0876.28024
[3] Baggett L.W., Merrill K.D.: Smooth cocycles for an irrational rotation. Israel J. Math. 79 (1992), 281-288. MR 1248918 | Zbl 0769.28013
[4] Billingsley P.: Convergence of Probability Measures. Wiley New York (1968). MR 0233396 | Zbl 0172.21201
[5] Hamachi T.: Type $III_0$ cocycles with unbounded gaps. Commentationes Math. Univ. Carolinae 36.4 (1995), 713-720. MR 1378692
[6] Herman M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. IHES Publications Math. 49 (1979), 5-234. MR 0538680 | Zbl 0448.58019
[7] Katok: Constructions in Ergodic Theory. manuscript. Zbl 1130.37304
[8] Khinchine: Continued Fractions. P. Noordhoff, Ltd. Groningen (1963). MR 0161834
[9] Kuipers L., Niederreiter H.: Uniform Distribution of Sequences. Wiley New York (1974). MR 0419394 | Zbl 0281.10001
[10] Kwiatkowski J., Lemańczyk M., Rudolph D.: On weak isomorphism of measure preserving diffeomorphisms. Israel J. Math. 80 (1992), 33-64. MR 1248926
[11] Kwiatkowski J., Lemańczyk M., Rudolph D.: A class of cocycles having an analytic modification. Israel J. Math. 87 (1994), 337-360. MR 1286834
[12] Lemańczyk M.: Analytic nonregular cocycles over irrational rotations. Commentationes Math. Univ. Carolinae 36.4 (1995), 727-735. MR 1378694
[13] Lemańczyk M.: Personal communication.
[14] Liardet P., Volný D.: Sums of continuous and differentiable functions in dynamical systems. preprint. MR 1459847
[15] Parry W., Tuncel S.: Classification Problems in Ergodic Theory. London Math. Society Lecture Notes 67, Cambridge University Press Cambridge (1982). MR 0666871 | Zbl 0487.28014
[16] Schmidt K.: Cocycles of Ergodic Transformation Groups. Macmillan Lectures in Math. vol. 1, Macmillan Company of India (1977). MR 0578731
[17] Stewart M.: Irregularities of uniform distribution. Acta Math. Acad. Sci. Hungar. 37 (1981), 185-221. MR 0616890 | Zbl 0475.10040
[18] Volný D.: On limit theorems and category for dynamical systems. Yokohama Math. J. 38 (1990), 29-35. MR 1093661
Search
Partner of
