Simple systems are disjoint from Gaussian systems

Andrés del Junco; Mariusz Lemańczyk

Studia Mathematica (1999)

  • Volume: 133, Issue: 3, page 249-256
  • ISSN: 0039-3223

Abstract

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We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less, divisibility up to distal extensions. The theory of Kronecker Gaussians also plays a crucial role.

How to cite

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del Junco, Andrés, and Lemańczyk, Mariusz. "Simple systems are disjoint from Gaussian systems." Studia Mathematica 133.3 (1999): 249-256. <http://eudml.org/doc/216616>.

@article{delJunco1999,
abstract = {We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less, divisibility up to distal extensions. The theory of Kronecker Gaussians also plays a crucial role.},
author = {del Junco, Andrés, Lemańczyk, Mariusz},
journal = {Studia Mathematica},
keywords = {infinite divisibility; ergodic self joinings; Gaussian dynamical system; discrete spectrum; weakly mixing; time one map},
language = {eng},
number = {3},
pages = {249-256},
title = {Simple systems are disjoint from Gaussian systems},
url = {http://eudml.org/doc/216616},
volume = {133},
year = {1999},
}

TY - JOUR
AU - del Junco, Andrés
AU - Lemańczyk, Mariusz
TI - Simple systems are disjoint from Gaussian systems
JO - Studia Mathematica
PY - 1999
VL - 133
IS - 3
SP - 249
EP - 256
AB - We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less, divisibility up to distal extensions. The theory of Kronecker Gaussians also plays a crucial role.
LA - eng
KW - infinite divisibility; ergodic self joinings; Gaussian dynamical system; discrete spectrum; weakly mixing; time one map
UR - http://eudml.org/doc/216616
ER -

References

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  1. [C,F,S] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, Berlin, 1982. 
  2. [G,H,R] E. Glasner, B. Host and D. Rudolph, Simple systems and their higher order self-joinings, Israel J. Math. 78 (1992), 131-142. Zbl0779.28010
  3. [I,L,R,S] A. Iwanik, M. Lemańczyk, T. de la Rue et J. de Sam Lazaro, Quelques remarques sur les facteurs des systèmes dynamiques gaussiens, Studia Math. 125 (1997), 247-254. 
  4. [J,L,M] A. del Junco, M. Lemańczyk and M. K. Mentzen, Semisimplicity, joinings and group extensions, ibid. 112 (1995), 141-164. Zbl0814.28007
  5. [J,R] A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems 7 (1987), 531-557. Zbl0646.60010
  6. [L,P,Th] M. Lemańczyk, F. Parreau and J.-P. Thouvenot, Gaussian automorphisms whose ergodic self-joinings are Gaussian, preprint. Zbl0977.37003
  7. [Th1] J.-P. Thouvenot, Some properties and applications of joinings in ergodic theory, in: Ergodic Theory and its Connections with Harmonic Analysis (Proc. Alexandria Conference), K. E. Petersen and I. Salama (eds.), London Math. Soc. Lecture Note Ser. 205, Cambridge Univ. Press, Cambridge, 1995, 207-235. Zbl0848.28009
  8. [Th2] J.-P. Thouvenot, Utilisation des processus gaussiens en théorie ergodique, preprint. 
  9. [Th3] J.-P. Thouvenot, The metrical structure of some Gaussian processes, in: Proc. Conf. on Ergodic Theory and Related Topics II (Georgenthal, 1986), Teubner Texte Math. 94, Teubner, Leipzig, 1987, 195-198. 
  10. [V] W. A. Veech, A criterion for a process to be prime, Monatsh. Math. 94 (1982), 335-341. Zbl0499.28016

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