# 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

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topdel 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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- [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
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