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.


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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