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

Abstract

First we describe the work of the first author leading to the conclusion that any property generic (in the weak topology) for measure-preserving bijections of a Lebesgue probability space is also generic (in the compact-open topology) for homeomorphisms of a compact manifold preserving a fixed measure. Then we describe the work of both authors in extending this result to non-compact manifolds, with modifications based on the ends of the manifold. These results can be thought of as generalizations of the original work which established genericity for the specific property of ergodicity (Oxtoby and Ulam, 1941) and subsequent work for other properties such as weak mixing (Katok and Stepin, 1970). The techniques used to obtain the titled theorem are also applied to related areas, such as fixed point theorems and chaos theory, and some new results are obtained.

Propriétés dynamiques génériques des homéomorphismes conservatifs

“…-Enfin, dans le cas dissipatif (c'est-à-dire si on considère des systèmes dynamiques qui ne préservent pas nécessairement une mesure fixée), il est très facile de voir qu'il existe un ouvert dense d'homéomorphismes (et donc a fortiori un ouvert dense de difféomorphismes C r pour tout r) qui possèdent des orbites périodiques attractives, et ne peuvent donc pas être topologiquement transitifs. Les résultats de généricité obtenus dans les différents cadres sont en général logiquement indépendants les uns des autres 9 . Il est néanmoins toujours enrichissant de comparer ces résultats, ne serait-ce que pour relativiser la portée de chacun d'entre eux.…”

Genericity in topological dynamics

“…Section 7 proves part (3) of theorem 1.2. Section 8 discusses parts (4) and (5) of that theorem, and part (6) is proved in section 9. Finally, in section 10 we outline some extensions and open questions.…”

“…Since then the status of many other properties in this category have been established. A parallel program has been carried out for volume preserving homeomorphisms of manifolds, and a certain unification has been achieved between these two categories [7,5]. Similar questions have been studied in the smooth category, where there are many open questions.…”

Generic properties of extensions

“…Since then, much research has been done in finding "typical" properties of measure-preserving dynamical systems. See, e.g., [28], [26], [1], [27], [2], [5], [3], [12], [4], [33], [21].…”