%0 Generic %1 gao2022proof %A Gao, Chang %D 2022 %K article %T A proof of the $C^r$ closing lemma and stability conjecture %U http://arxiv.org/abs/2206.02974 %X This paper presents a proof of the $C^r$ closing lemma and stability conjecture for flows and diffeomorphisms on locally compact Riemannian manifolds. The dimension of the perturbation is reduced by extending the flow box of the periodic orbit so as to control the $C^r$ size of Lie derivatives. Density of the periodic points are hence preserved by homeomorphisms, due to the topological invariance of structurally stable differential systems under perturbations. Necessity of hyperbolicity of nonwandering set is proved by contradicting the nonhyperbolic tendency of periodic points.

@misc{gao2022proof, abstract = {This paper presents a proof of the $C^r$ closing lemma and stability conjecture for flows and diffeomorphisms on locally compact Riemannian manifolds. The dimension of the perturbation is reduced by extending the flow box of the periodic orbit so as to control the $C^r$ size of Lie derivatives. Density of the periodic points are hence preserved by homeomorphisms, due to the topological invariance of structurally stable differential systems under perturbations. Necessity of hyperbolicity of nonwandering set is proved by contradicting the nonhyperbolic tendency of periodic points.}, added-at = {2022-06-08T15:39:38.000+0200}, author = {Gao, Chang}, biburl = {https://www.bibsonomy.org/bibtex/29c0a86ed27d82988b0f02a4f7be629c0/lacerdagabo}, description = {A proof of the $C^r$ closing lemma and stability conjecture}, interhash = {5eed28be880debf2d5ece387e9a5c44a}, intrahash = {9c0a86ed27d82988b0f02a4f7be629c0}, keywords = {article}, note = {cite arxiv:2206.02974Comment: 11 pages}, timestamp = {2022-06-08T15:39:38.000+0200}, title = {A proof of the $C^r$ closing lemma and stability conjecture}, url = {http://arxiv.org/abs/2206.02974}, year = 2022 }