Sur l'absence de mélange pour des flots spéciaux au-dessus d'une rotation irrationnelle

M. Lemańczyk

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 1, page 29-41
  • ISSN: 0010-1354

Abstract

top
We prove the absence of mixing for special flows built over (1) an irrational rotation and under a function whose Fourier coefficients are of order O(1/|n|), and (2) an irrational rotation (satisfying a diophantine condition) and under a function having a finite number of singularities of a logarithmic type. These results generalize two theorems of Kochergin.

How to cite

top

Lemańczyk, M.. "Sur l'absence de mélange pour des flots spéciaux au-dessus d'une rotation irrationnelle." Colloquium Mathematicae 84/85.1 (2000): 29-41. <http://eudml.org/doc/210806>.

@article{Lemańczyk2000,
author = {Lemańczyk, M.},
journal = {Colloquium Mathematicae},
keywords = {mixing; flow built under a function},
language = {fre},
number = {1},
pages = {29-41},
title = {Sur l'absence de mélange pour des flots spéciaux au-dessus d'une rotation irrationnelle},
url = {http://eudml.org/doc/210806},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Lemańczyk, M.
TI - Sur l'absence de mélange pour des flots spéciaux au-dessus d'une rotation irrationnelle
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 1
SP - 29
EP - 41
LA - fre
KW - mixing; flow built under a function
UR - http://eudml.org/doc/210806
ER -

References

top
  1. [1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Math. Surveys Monogr. 50, Amer. Math. Soc., Providence, 1997. 
  2. [2] J. Aaronson, M. Lemańczyk, C. Mauduit and H. Nakada, Koksma's inequality and group extensions of Kronecker transformations, in: Algorithms, Fractals and Dynamics, Y. Takahashi (ed.), Plenum Press, 1995, 27-50. Zbl0878.28009
  3. [3] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, New York, 1982. 
  4. [4] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon, Oxford, 1960. Zbl0086.25803
  5. [5] K. M. Khanin and Ya. G. Sinai, Mixing of some classes of special flows over a circle rotation, Funktsional. Anal. i Prilozhen. 26 (1992), no. 3, 1-21 (in Russian). 
  6. [6] A. Ya. Khintchin, Continued Fractions, Univ. of Chicago Press, 1960. 
  7. [7] A. V. Kočergin [A. V. Kochergin], On the absence of mixing in special flows over the rotation of a circle and in flows on a two-dimensional torus, Dokl. Akad. Nauk SSSR 205 (1972), 515-518 (in Russian); English transl.: Soviet Math. Dokl. 13 (1972), 949-952. 
  8. [8] A. V. Kočergin [A. V. Kochergin], Time change for flows and mixing, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1275-1298 (in Russian). Zbl0286.28013
  9. [9] A. V. Kočergin [A. V. Kochergin], On the mixing of special flows over interval exchange maps and in smooth flows on surfaces, Mat. Sb. 96 (1975), 471-502 (in Russian). Zbl0321.28012
  10. [10] A. V. Kočergin [A. V. Kochergin], Nonsingular saddle points and the absence of mixing, Mat. Zametki 19 (1976), 453-468 (in Russian); English transl.: Math. Notes 19 (1976), 277-286. Zbl0344.28008
  11. [11] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, 1974. Zbl0281.10001
  12. [12] M. Lemańczyk and C. Mauduit, Ergodicity of a class of cocycles over irrational rotations, J. London Math. Soc. 49 (1994), 124-132. Zbl0801.28009
  13. [13] M. Lemańczyk, F. Parreau and D. Volný, Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces, Trans. Amer. Math. Soc. 348 (1996), 4919-4938. Zbl0876.28021
  14. [14] V. V. Ryzhikov, The absence of mixing in special flows over rearrangements of segments, Math. Notes 55 (1994), 648-650. Zbl0849.28009
  15. [15] K. Schmidt, Cocycles of Ergodic Transformation Groups, Lecture Notes in Math. 1, Mac Millan of India, 1977. Zbl0421.28017

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.