We prove that a Cl-generic volume-preserving dynamical system （diffeomor- phism or flow） has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, as in ...We prove that a Cl-generic volume-preserving dynamical system （diffeomor- phism or flow） has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, as in [10, 27], we prove that the Cl-robustness, within the volume-preserving context, of the expansiveness property and the weak specifica- tion property, imply that the dynamical system （diffeomorphism or flow） is Anosov.展开更多
In this paper,we define a generalized Lipschitz shadowing property for flows and prove that a flowΦgenerated by a C1vector field X on a closed Riemannian manifold M has this generalized Lipschitz shadowing property i...In this paper,we define a generalized Lipschitz shadowing property for flows and prove that a flowΦgenerated by a C1vector field X on a closed Riemannian manifold M has this generalized Lipschitz shadowing property if and only if it is structurally stable.展开更多
Let M be a closed smooth manifold M, and let f ： M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set A of f. We show that if f has the C^1-stably average shadowing property on A, then A...Let M be a closed smooth manifold M, and let f ： M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set A of f. We show that if f has the C^1-stably average shadowing property on A, then A admits a dominated splitting.展开更多
We prove that any C1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the v...We prove that any C1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.展开更多
Let f ： Md →Md（d 〉 2） be a diffeomorphism on a compact C^∞ manifold on M. If a diffeomorphism f belongs to the Cl-interior of the set of all diffeomorphisms having the barycenter property, then f is Ω-stable. M...Let f ： Md →Md（d 〉 2） be a diffeomorphism on a compact C^∞ manifold on M. If a diffeomorphism f belongs to the Cl-interior of the set of all diffeomorphisms having the barycenter property, then f is Ω-stable. Moreover, if a generic diffeomorphism f has the barycenter property, then f is 12-stable. We also apply our results to volume preserving diffeomorphisms.展开更多
基金partially supported by National Funds through FCT-"Fundacao para a Ciencia e a Tecnologia",(PEst-OE/MAT/UI0212/2011)supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry,ICT&Future Planning(No.2014R1A1A1A05002124)supported by National Natural Science Foundation of China(No.11301018 and 11371046)
文摘We prove that a Cl-generic volume-preserving dynamical system （diffeomor- phism or flow） has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, as in [10, 27], we prove that the Cl-robustness, within the volume-preserving context, of the expansiveness property and the weak specifica- tion property, imply that the dynamical system （diffeomorphism or flow） is Anosov.
基金supported by National Natural Science Foundation of China(12071018)Fundamental Research Funds for the Central Universitiessupported by the National Research Foundation of Korea(NRF)funded by the Korea government(MIST)(2020R1F1A1A01051370)。
文摘In this paper,we define a generalized Lipschitz shadowing property for flows and prove that a flowΦgenerated by a C1vector field X on a closed Riemannian manifold M has this generalized Lipschitz shadowing property if and only if it is structurally stable.
基金supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology(Grant No.2011-0007649)supported by National Natural Science Foundation of China(Grant No.11026041)
文摘Let M be a closed smooth manifold M, and let f ： M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set A of f. We show that if f has the C^1-stably average shadowing property on A, then A admits a dominated splitting.
基金supported by National Funds through FCT-"Fundao para a Ciênciae a Tecnologia"(Grant No.PEst-OE/MAT/UI0212/2011)supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology,Korea(Grant No.2011-0007649)
文摘We prove that any C1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.
基金Supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Science,ICT&Future Planning(Grant No.2014R1A1A1A05002124)
文摘Let f ： Md →Md（d 〉 2） be a diffeomorphism on a compact C^∞ manifold on M. If a diffeomorphism f belongs to the Cl-interior of the set of all diffeomorphisms having the barycenter property, then f is Ω-stable. Moreover, if a generic diffeomorphism f has the barycenter property, then f is 12-stable. We also apply our results to volume preserving diffeomorphisms.
