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An “orbit closing lemma”. (Un lemme de fermeture d’orbites: Le “orbit closing lemma”.) (French) Zbl 0867.58040
Summary: Let $$f$$ be a diffeomorphism (resp. symplectic diffeomorphism, resp. volume preserving diffeomorphism) of a Riemannian manifold $$(M,d)$$. Let $$\Sigma(f)$$ be the set of points $$x\in M$$ such that for every neighbourhood $$U$$ of $$f$$ in the $$C^1$$ topology and every $$\varepsilon > 0$$, there exist $$g\in U$$ and $$y\in M$$ such that:
(i) $$y$$ is a periodic point of $$g$$ with period $$m$$;
(ii) $$g=f$$ in $$M\setminus \bigcup_{0\leq k\leq m} B_\varepsilon (f^kx)$$;
(iii) $$\forall i\in [0,m]$$, $$d(g^iy,f^ix)<\varepsilon$$.
Then $$\Sigma(f)$$ is a countable intersection of open subsets of the set $$R(f)$$ of recurrent points of $$f$$ and is dense in $$R(f)$$.

##### MSC:
 37B99 Topological dynamics
##### Keywords:
orbit closing lemma