Gutierrez, C. A counter-example to a \(C^ 2\) closing lemma. (English) Zbl 0642.58036 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 Cited in 11 Documents 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 PDF BibTeX XML Cite \textit{C. Gutierrez}, Ergodic Theory Dyn. Syst. 7, 509--530 (1987; Zbl 0642.58036) Full Text: DOI References: [1] de Melo, Geometric Theory of Dynamical Systems (1982) [2] Newhouse, Publ. IHES. 57 pp 5– (1983) · Zbl 0518.58031 · doi:10.1007/BF02698773 [3] Lang, Introduction to Diophantine Approximations (1966) [4] Hirsh, Invariant Manifolds (1977) · doi:10.1007/BFb0092042 [5] Herman, Pub. Math. none pp 5– (none) [6] Gutierrez, Ergod. Th. & Dynam. Sys. 6 pp 45– (1986) [7] Gutierrez, Ergod. Th. & Dynam. Sys. 6 pp 17– (1986) [8] Denjoy, J. Mathematique 9 pp 333– (1932) [9] Coexeter, Introduction to Geometry (1961) [10] DOI: 10.1007/BF01389816 · Zbl 0247.58007 · doi:10.1007/BF01389816 [11] Sotomayor, Publ. Math. IHES. 44 pp none– (1974) [12] Slater, Proc. Camb. Phil. Soc. 63 pp 1115– (1967) [13] DOI: 10.2307/2373414 · Zbl 0167.21804 · doi:10.2307/2373414 [14] DOI: 10.1016/0022-0396(75)90054-6 · Zbl 0309.34043 · doi:10.1016/0022-0396(75)90054-6 [15] DOI: 10.1016/0022-0396(82)90002-X · Zbl 0506.58029 · doi:10.1016/0022-0396(82)90002-X [16] DOI: 10.1016/0040-9383(65)90018-2 · Zbl 0107.07103 · doi:10.1016/0040-9383(65)90018-2 [17] Palis, Global Analysis 14 pp 223– (1970) · doi:10.1090/pspum/014/0267603 [18] DOI: 10.2307/2007021 · Zbl 0511.58029 · doi:10.2307/2007021 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.