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A simpler proof of $$C^ 1$$ closing lemma. (English) Zbl 0616.58024
The closing lemma states that a recurrent point of a dynamical system is a periodic point for some perturbation of the dynamical system. For $$C^ 0$$ perturbations, the result is easy. However, for $$C^ 1$$ perturbations the result is very difficult. It was first proved by C. Pugh [Am. J. Math. 89, 956-1009 (1967; Zbl 0167.218)] for diffeomorhisms and by C. Pugh and C. Robinson [Ergodic Theory Dyn. Syst. 3, 261-313 (1983; Zbl 0548.58012)] for flows. The $$C^ r$$ case for $$r\geq 2$$ is still open. A new proof of the $$C^ 1$$ closing lemma for flows is presented in the paper under review.
Reviewer: C.Chicone

MSC:
 37C75 Stability theory for smooth dynamical systems