Semisimplicity, joinings and group extensions

A. Del Junco; M. Lemańczyk; M. Mentzen

Studia Mathematica (1995)

  • Volume: 112, Issue: 2, page 141-164
  • ISSN: 0039-3223

Abstract

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We present a theory of self-joinings for semisimple maps and their group extensions which is a unification of the following three cases studied so far: (iii) Gaussian-Kronecker automorphisms: [Th], [Ju-Th]. (ii) MSJ and simple automorphisms: [Ru], [Ve], [Ju-Ru]. (iii) Group extension of discrete spectrum automorphisms: [Le-Me], [Le], [Me].

How to cite

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Del Junco, A., Lemańczyk, M., and Mentzen, M.. "Semisimplicity, joinings and group extensions." Studia Mathematica 112.2 (1995): 141-164. <http://eudml.org/doc/216143>.

@article{DelJunco1995,
abstract = {We present a theory of self-joinings for semisimple maps and their group extensions which is a unification of the following three cases studied so far: (iii) Gaussian-Kronecker automorphisms: [Th], [Ju-Th]. (ii) MSJ and simple automorphisms: [Ru], [Ve], [Ju-Ru]. (iii) Group extension of discrete spectrum automorphisms: [Le-Me], [Le], [Me].},
author = {Del Junco, A., Lemańczyk, M., Mentzen, M.},
journal = {Studia Mathematica},
keywords = {semisimple maps; ergodic automorphism; minimal self-joinings; ergodic group extensions},
language = {eng},
number = {2},
pages = {141-164},
title = {Semisimplicity, joinings and group extensions},
url = {http://eudml.org/doc/216143},
volume = {112},
year = {1995},
}

TY - JOUR
AU - Del Junco, A.
AU - Lemańczyk, M.
AU - Mentzen, M.
TI - Semisimplicity, joinings and group extensions
JO - Studia Mathematica
PY - 1995
VL - 112
IS - 2
SP - 141
EP - 164
AB - We present a theory of self-joinings for semisimple maps and their group extensions which is a unification of the following three cases studied so far: (iii) Gaussian-Kronecker automorphisms: [Th], [Ju-Th]. (ii) MSJ and simple automorphisms: [Ru], [Ve], [Ju-Ru]. (iii) Group extension of discrete spectrum automorphisms: [Le-Me], [Le], [Me].
LA - eng
KW - semisimple maps; ergodic automorphism; minimal self-joinings; ergodic group extensions
UR - http://eudml.org/doc/216143
ER -

References

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  1. [An] H. Anzai, Ergodic skew product transformations on the torus, Osaka J. Math. 3 (1951), 83-99. Zbl0043.11203
  2. [Fi-Ru] A. Fieldsteel and D. Rudolph, An ergodic transformation with trivial Kakutani centralizer, Ergodic Theory Dynamical Systems 12 (1992), 459-478. Zbl0763.58016
  3. [Fu] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981. Zbl0459.28023
  4. [Gl-Ho-Ru] E. Glasner, B. Host and D. Rudolph, Simple systems and their higher order self-joinings, Israel J. Math. 78 (1992), 131-142. Zbl0779.28010
  5. [Ju-Ru] A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynamical Systems 7 (1987), 531-557. Zbl0646.60010
  6. [Ju-Th] A. del Junco and J.-P. Thouvenot, The theory for Gaussian-Kronecker automorphisms, preprint. 
  7. [Ke-Ne1] H. B. Keynes and D. Newton, Choquet Theory and Ergodic Measures for Compact Group Extensions, Lecture Notes in Math. 318, Springer, 1973. Zbl0256.28014
  8. [Ke-Ne2] H. B. Keynes and D. Newton, The structure of ergodic measures for compact group extensions, Israel J. Math. 18 (1974), 363-389. Zbl0299.54033
  9. [Ke-Ne3] H. B. Keynes and D. Newton, Ergodic measures for nonabelian compact group extensions, Compositio Math. 32 (1976), 53-70. Zbl0318.28006
  10. [Le] M. Lemańczyk, Ergodic abelian group extensions of rotations, preprint, Toruń 1990. 
  11. [Le] M. Lemańczyk and M. K. Mentzen, Compact subgroups in the centralizer of natural factors of an ergodic group extension of a rotation determine all factors, Ergodic Theory Dynamical Systems 10 (1990), 763-776. Zbl0725.54030
  12. [Me] M. K. Mentzen, Ergodic properties of group extensions of dynamical systems with discrete spectra, Studia Math. 101 (1991), 19-31. Zbl0809.28015
  13. [Ne] D. Newton, On canonical factors of ergodic dynamical systems, J. London Math. Soc. 19 (1979), 129-136. Zbl0425.28012
  14. [Ne1] D. Newton, Coalescence and spectrum of automorphisms of a Lebesgue space, Z. Wahrsch. Verw. Gebiete 19 (1971), 117-122. Zbl0201.38401
  15. [Ru] D. Rudolph, An example of measure preserving map with minimal self-joinings, and applications, J. Analyse Math. 35 (1979), 97-122. Zbl0446.28018
  16. [Ru1] D. Rudolph, Classifying the isometric extensions of a Bernoulli shift, ibid. 34 (1978), 36-60. Zbl0415.28012
  17. [Th] J.-P. Thouvenot, The metrical structure of some Gaussian processes, in: Proc. Ergodic Theory and Related Topics II, Georgenthal 1986, 195-198. 
  18. [Ve] W. A. Veech, A criterion for a process to be prime, Monatsh. Math. 94 (1982), 335-341. Zbl0499.28016
  19. [Zi] R. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373-409. Zbl0334.28015
  20. [Zi1] R. Zimmer, Ergodic actions with generalized discrete spectrum, ibid., 555-588. Zbl0349.28011

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