A cut salad of cocycles

Jon Aaronson; Mariusz Lemańczyk; Dalibor Volný

Fundamenta Mathematicae (1998)

  • Volume: 157, Issue: 2-3, page 99-119
  • ISSN: 0016-2736

Abstract

top
We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed.

How to cite

top

Aaronson, Jon, Lemańczyk, Mariusz, and Volný, Dalibor. "A cut salad of cocycles." Fundamenta Mathematicae 157.2-3 (1998): 99-119. <http://eudml.org/doc/212297>.

@article{Aaronson1998,
abstract = {We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed.},
author = {Aaronson, Jon, Lemańczyk, Mariusz, Volný, Dalibor},
journal = {Fundamenta Mathematicae},
keywords = {skew product; group extension; cocycle; ergodic measure-preserving transformation; squashable; non-coalescent; Maharam transformation},
language = {eng},
number = {2-3},
pages = {99-119},
title = {A cut salad of cocycles},
url = {http://eudml.org/doc/212297},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Aaronson, Jon
AU - Lemańczyk, Mariusz
AU - Volný, Dalibor
TI - A cut salad of cocycles
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 2-3
SP - 99
EP - 119
AB - We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed.
LA - eng
KW - skew product; group extension; cocycle; ergodic measure-preserving transformation; squashable; non-coalescent; Maharam transformation
UR - http://eudml.org/doc/212297
ER -

References

top
  1. [A1] J. Aaronson, The asymptotic distributional behaviour of transformations preserving infinite measures, J. Anal. Math. 39 (1981), 203-234. Zbl0499.28013
  2. [A2] J. Aaronson, The intrinsic normalising constants of transformations preserving infinite measures, ibid. 49 (1987), 239-270. Zbl0644.28013
  3. [A-L-M-N] J. Aaronson, M. Lemańczyk, C. Mauduit and H. Nakada, Koksma's inequality and group extensions of Kronecker transformations, in: Algorithms, Fractals and Dynamics (Okayama and Kyoto, 1992), Y. Takahashi (ed.), Plenum, New York, 1995, 27-50. Zbl0878.28009
  4. [A-L-V] J. Aaronson, M. Lemańczyk and D. Volný, A salad of cocycles, preprint, internet: http://www.math.tau.ac.il/~aaro, 1995. 
  5. [D] A. Danilenko, Comparison of cocycles of measured equivalence relations and lifting problems, Ergodic Theory Dynam. Systems 18 (1998), 125-151. Zbl0919.28015
  6. [F-M] J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, I, Trans. Amer. Math. Soc. 234 (1977), 289-324. 
  7. [G-L-S] P. Gabriel, M. Lemańczyk and K. Schmidt, Extensions of cocycles for hyperfinite actions and applications, Monatsh. Math. 123 (1997), 209-228. Zbl0887.28008
  8. [G-S] V. I. Golodets and S. D. Sinel'shchikov, Locally compact groups appearing as ranges of cocycles of ergodic ℤ-actions, Ergodic Theory Dynam. Systems 5 (1985), 45-57. 
  9. [H] P. Halmos, Lectures on Ergodic Theory, Chelsea, New York, 1953. 
  10. [H-P] F. Hahn and W. Parry, Some characteristic properties of dynamical systems with quasi-discrete spectrum, Math. Systems Theory 2 (1968), 179-190. Zbl0167.32902
  11. [H-O-O] T. Hamachi, Y. Oka and M. Osikawa, A classification of ergodic non-singular transformation groups, Mem. Fac. Sci. Kyushu Univ. Ser. A 28 (1974), 113-133. Zbl0293.28011
  12. [K-W] Y. Katznelson and B. Weiss, Commuting measure preserving transformations, Israel J. Math. 12 (1972), 161-172. Zbl0239.28014
  13. [K] W. Krieger, On ergodic flows and isomorphism of factors, Math. Ann. 223 (1976), 19-70. Zbl0332.46045
  14. [L-L-T] M. Lemańczyk, P. Liardet and J-P. Thouvenot, Coalescence of circle extensions of measure preserving transformations, Ergodic Theory Dynam. Systems 12 (1992), 769-789. Zbl0785.58030
  15. [L-V] P. Liardet and D. Volný, Sums of continuous and differentiable functions in dynamical systems, Israel J. Math. 98 (1997), 29-60. Zbl0881.28012
  16. [L] D. Lind, Locally compact measure preserving flows, Adv. Math. 15 (1975), 175-193. Zbl0293.28012
  17. [M] D. Maharam, Incompressible transformations, Fund. Math. 56 (1964), 35-50. Zbl0133.00304
  18. [M-S] C. Moore and K. Schmidt, Coboundaries and homomorphisms for non-singular actions and a problem of H. Helson, Proc. London Math. Soc. 40 (1980), 443-475. Zbl0428.28014
  19. [O] D. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems, Yale Math. Monographs 5, Yale Univ. Press, New Haven, 1974. 
  20. [O-W] D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987), 1-142. 
  21. [R] D. Rudolph, Classifying the isometric extensions of a Bernoulli shift, ibid. 34 (1978), 36-60. Zbl0415.28012
  22. [S] K. Schmidt, Cocycles of Ergodic Transformation Groups, Lecture Notes in Math. 1, Mac Millan of India, 1977. Zbl0421.28017
  23. [Z] R. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Funct. Anal. 27 (1978), 350-372. Zbl0391.28011

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.