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Recurrence and genericity. (Récurrence et généricité.) (French. English summary) Zbl 1071.37015
The central technical result is the following version of a \(C^1\)-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds (Théorème 1.2): If \(f\) is a diffeomorphism of a compact manifold \(M\) such that all periodic orbits are hyperbolic, then for any points \(x\) and \(y\) on the same pseudo-orbit, \(f\) can be \(C^1\)-approximated by a diffeomorphism with respect to that \(x\) and \(y\) lie on the same orbit. We mention a few of the numerous consequences derived. For a residual subset of \(\text{Diff}^1(M)\), any points \(x\) and \(y\) are on the same pseudo-orbit if and only if any two respective neighborhoods meet under suitable iteration of one. For a residual subset of \(\text{Diff}^1(M)\), the chain recurrent set coincides with the nonwandering set. If \(M\) is connected and \(f\) is in a suitable residual subset of \(\text{Diff}^1(M)\), then \(f\) is transitive if the nonwandering set is all of \(M\). Also, a conjecture of Hurley is confirmed in the \(C^1\)-topology. Moreover, all of the results have counterparts for volume preserving diffeomorphisms.
The \(C^1\)-connecting lemma goes back to S. Hayashi. The formulation presented here is proved along the lines of work by M.-C. Arnaud and is elaborated in an appendix.

MSC:
37C20 Generic properties, structural stability of dynamical systems
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
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