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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
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