A counter-example to a \(C^ 2\) closing lemma. (English) Zbl 0642.58036

Gutierrez, C.

Ergodic Theory Dyn. Syst. 7, 509-530 (1987).

Let M be a compact manifold containing a two dimensional punctured torus. Given \(p\in M\) and an integer \(r\geq 2\), there exists \(X\in {\mathcal X}^{\infty}(M)\) having non-trivial recurrent orbits and such that, for some neighbourhood \({\mathcal U}\) of \(X|_{(M\setminus \{p\})}\) in \({\mathcal X}^ r(M\setminus \{p\})\) (under the Whitney \(C^ r\) topology), no \(Y\in {\mathcal U}\) has closed orbits.

Reviewer: J.So

MSC:

37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37J40	Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion

Keywords:

C\({}^ r\)-closing lemma; non-wandering point; periodic point; stability conjecture; \(C^ r\)-connecting lemma; \(C^ 1\) ergodic closing lemma; bifurcation theory

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