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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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