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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

37C75 Stability theory for smooth dynamical systems