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An improvement of the \(C^ 1\) closing lemma for endomorphisms. (English) Zbl 0824.58013
From the abstract: “L. Wen proved the \(C^ 1\) closing lemma for endomorphisms with finitely many singularities. The arguments and tools of Wen are available for endomorphisms which have infinitely many singularities but at most finitely many ones in the nonwandering sets. By refining the argument of Wen we prove the \(C^ 1\) closing lemma for endomorphisms with finitely many singularities in the nonwandering sets. By using this lemma we can slightly improve the characterization of \(C^ 1\) absolutely \(\Omega\)-stable endomorphisms. That is, for an endomorphism \(f\) with finitely many singularities in the nonwandering set, \(f\) is \(C^ 1\) absolutely \(\Omega\)-stable if and only if \(f\) has a neighbourhood \(\mathcal U\) such that every \(g\) in \(\mathcal U\) satisfies weak Axiom \(A\)”.

MSC:
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
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