an:00972684
Zbl 0867.58040
Arnaud, Marie-Claude
An ``orbit closing lemma''
FR
C. R. Acad. Sci., Paris, S??r. I 323, No. 11, 1175-1178 (1996).
00038499
1996
j
37B99
orbit closing lemma
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 \(R(f)\).