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.
@article{bwmeta1.element.bwnjournal-article-smv133i3p249bwm,
author = {Andr\'es del Junco and Mariusz Lema\'nczyk},
title = {Simple systems are disjoint from Gaussian systems},
journal = {Studia Mathematica},
volume = {133},
year = {1999},
pages = {249-256},
zbl = {0931.37001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv133i3p249bwm}
}
del Junco, Andrés; Lemańczyk, Mariusz. Simple systems are disjoint from Gaussian systems. Studia Mathematica, Tome 133 (1999) pp. 249-256. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv133i3p249bwm/
[00000] [C,F,S] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, Berlin, 1982.
[00001] [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. | Zbl 0779.28010
[00002] [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.
[00003] [J,L,M] A. del Junco, M. Lemańczyk and M. K. Mentzen, Semisimplicity, joinings and group extensions, ibid. 112 (1995), 141-164. | Zbl 0814.28007
[00004] [J,R] A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems 7 (1987), 531-557. | Zbl 0646.60010
[00005] [L,P,Th] M. Lemańczyk, F. Parreau and J.-P. Thouvenot, Gaussian automorphisms whose ergodic self-joinings are Gaussian, preprint. | Zbl 0977.37003
[00006] [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. | Zbl 0848.28009
[00007] [Th2] J.-P. Thouvenot, Utilisation des processus gaussiens en théorie ergodique, preprint.
[00008] [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.
[00009] [V] W. A. Veech, A criterion for a process to be prime, Monatsh. Math. 94 (1982), 335-341. | Zbl 0499.28016