A topological analog of Halmos' conjugacy lemma

Abstract

Abstract. Using a new lemma indicated by the title, and a recent measure preserving version of Lusin's Theorem, we prove the following theorem: Any isomorphism-invariant measure theoretic property which is "typical" for automorphisms of a kebesgue space is also "typical" for Lebesgue measure preserving homeomorphisms of the unit cube I n, n > 2. We also prove a partial converse of this theorem. Taken together, these results clarify the relationship between pairs of theorems proved by several authors, which established the "typicality" of specific properties (such as ergodicity and weak mixing) separately in the measurable and continuous cases.The determination of typical, or generic, properties of measure preserving transformations has proceeded at different paces but with eventually the same results in the measurable (i) and continuous (ii) contexts. For example, it is known for both (i) automorphisms of a Lebesgue space, and (ii) measure preserving homeomorphisms of a compact manifold, that ergodicity, weak mixing and zero entropy are typical. [By "typical" we mean dense G a in an appropriate space.) In this paper we explain the striking similarity of the "typicality" results which have been established separately in the above two contexts, in the following pairs: ergodicity [5,13], weak mixing [6,9], and other properties in [8,9] and [6,3]. We do this by proving the following result [Theorem 6) under the assumption I that the underlying manifold has the fixed point property: Any isomorphism-invariant measure theoretic property which is typical for automorphisms o/ a Lehesgue space is typical for measure preserving homeomorphisms. We also prove [Theorem 5) a partial converse of this result.As a basic tool in the comparison of the measurable and continuous contexts, we use a measure preserving version of Lusin's Theorem developed in a previous paper [3]. That result (Theorem 1 of this paper) is in turn based on a similar theorem due to J.C.

Propriétés dynamiques génériques des homéomorphismes conservatifs

“…On commence, dans cette partie, par donner une nouvelle preuve de la généricité du non-mélange fort ; dans la partie suivante 6.5, nous établissons une nouvelle preuve de la généricité du mélange faible. Ces deux résultats 6 sont des applications « historiques » du théorème de transfert : ces généricités ont été établies dans Auto(X, µ) par respectivement P. Halmos [28] et V. Rokhlin [59] dans les années 40 7,8 . Ces preuves ont été reprises par P. Halmos lui-même dans sont livre fondateur [26] au chapitre « Category » ; celles-ci se transferent directement à l'espace Homeo(X, µ).…”

“…6 Que l'on a déjà obtenues au chapitre 4. 7 Il est quelque peu anachronique de parler d'application historique lorsqu'on applique le théorème de transfert, prouvé dans les années 70, à des résultats datant des années 40.…”

“…À défaut d'un réel analogue topologique du lemme d'Halmos 3 , il montra en 1978 une variante dans l'ensemble des automorphismes mais pour la topologie forte : l'ensemble des homéomorphismes est inclus dans l'adhérence (pour la topologie forte) de la classe de conjugaison (dans l'ensemble des automorphismes) d'un automorphisme apériodique. Ceci lui permit d'établir son très beau théorème de transfert [6], [7], qui affirme que toute propriété ergodique générique parmi les automorphismes pour la topologie faible l'est aussi parmi les homéomorphismes pour la topologie forte. L'étude de la dynamique générique des homéomorphismes conservatifs de variétés non compactes a été amorcée l'année suivante, lorsque V. Prasad a montré que l'ergodicité est générique dans l'espace R n muni de la mesure de Lebesgue [54].…”

Properties generic for Lebesgue space automorphisms are generic for measure-preserving manifold homeomorphisms

“…2.6. Genericity results in G and M. In this section we relate the genericity results stated in §1.3 for G to those for M. In particular, we show that any measure-theoretic property which is generic in G is also generic in M. The results in this section were obtained first in [8] using a technique that applied only to manifolds with the fixed point property and subsequently generalized in [9] to arbitrary compact manifolds. We begin with a result which treats the two spaces G and M in a similar fashion.…”

“…The restriction here to the approximation only of homeomorphisms in M[X, µ] is not serious, as this version of the above theorem is in fact sufficient to prove the main results of the next section. Thus this proof could be used to prove the results of the next section for the special case where X is the unit cube, as was done in [8].…”