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Approximation des ensembles \(\omega\)-limites des difféomorphismes par des orbites pério-diques. (Approximation of \(\omega\)-limit sets of diffeomorphisms by periodic orbits). (French) Zbl 1024.37011
This paper is devoted to the approximation of \(\omega\)-limit sets of diffeomorphisms by periodic orbits. Indeed, let \({\mathfrak M}\) be a compact manifold, \(Q\) the set of its \(C^1\)-diffeomorphisms (possibly symplectic or volume preserving). The author proves that there exists a dense \(G_\delta {\mathcal S}\) of \(\text{Diff}^1({\mathfrak M})^Q\) (here by \(\text{Diff}^1({\mathfrak M})\) is denoted a set of \(C^1\)-diffeomorphisms of \({\mathfrak M}\)), such that for any \(f\in{\mathcal S}\), every \(\omega\)-limit set of \(f\) is the limit (in the sense of Hausdorff topology) of a sequence of periodic orbits. As a result the author obtains interesting properties on the structure of \(\omega\)-limit sets. Moreover, the author defines a new notion of attractors and describes them precisely in several cases.

MSC:
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C27 Periodic orbits of vector fields and flows
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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