In this paper, we give a partial answer to the problem proposed by Lan Wen. Roughly speaking, we prove that for a fixed i, f has C^1 persistently no small angles if and only if f has a dominated splitting of index i o...In this paper, we give a partial answer to the problem proposed by Lan Wen. Roughly speaking, we prove that for a fixed i, f has C^1 persistently no small angles if and only if f has a dominated splitting of index i on the C^1 i-preperiodic set P＊^1（f）. To prove this, we mainly use some important conceptions and techniques developed by Christian Bonatti. In the last section, we also give a characterization of the finest dominated splitting for linear cocvcles.展开更多
In this paper, we solve the problem proposed by Lan Wen for the case of dimM = 3. Roughly speaking, we prove that for fixed i, f has C1 persistently no small angles of index i if and only if f has a dominated splittin...In this paper, we solve the problem proposed by Lan Wen for the case of dimM = 3. Roughly speaking, we prove that for fixed i, f has C1 persistently no small angles of index i if and only if f has a dominated splitting of index i on the C1 i-preperiodic set P＊i（f）.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
We study bi-Lyapunov stable homoclinic classes for a C^(1)generic flow on a closed Rieman-nian manifold and prove that such a homoclinic class contains no singularity.This enables a parallel study of bi-Lyapunov stabl...We study bi-Lyapunov stable homoclinic classes for a C^(1)generic flow on a closed Rieman-nian manifold and prove that such a homoclinic class contains no singularity.This enables a parallel study of bi-Lyapunov stable dynamics for flows and for diffeomorphisms.For example,we can then show tha t a bi-Lyapunov st able homoclinic class for a C^(1)generic flow is hyperbolic if and only if all periodic orbits in the class have the same stable index.展开更多
It seems that in Mane＇s proof of the C^1 Ω-stability conjecture containing in the famous paper which published in I. H. E. S. （1988）, there exists a deficiency in the main lemma which says that for f ∈F^1 （M） t...It seems that in Mane＇s proof of the C^1 Ω-stability conjecture containing in the famous paper which published in I. H. E. S. （1988）, there exists a deficiency in the main lemma which says that for f ∈F^1 （M） there exists a dominated splitting TMPi（f） =Ei^s the direlf sum of E and F Fi^u（O 〈 i 〈 dim M） such that if Ei^s is contracting, then Fi^u is expanding. In the first part of the paper, we give a proof to fill up this deficiency. In the last part of the paper, we, under a weak assumption, prove a result that seems to be useful in the study of dynamics in some other stability context.展开更多
文摘In this paper, we give a partial answer to the problem proposed by Lan Wen. Roughly speaking, we prove that for a fixed i, f has C^1 persistently no small angles if and only if f has a dominated splitting of index i on the C^1 i-preperiodic set P＊^1（f）. To prove this, we mainly use some important conceptions and techniques developed by Christian Bonatti. In the last section, we also give a characterization of the finest dominated splitting for linear cocvcles.
基金Liu and Liang are supported by NNSFC (#10671006)
文摘In this paper, we solve the problem proposed by Lan Wen for the case of dimM = 3. Roughly speaking, we prove that for fixed i, f has C1 persistently no small angles of index i if and only if f has a dominated splitting of index i on the C1 i-preperiodic set P＊i（f）.
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.
文摘We study bi-Lyapunov stable homoclinic classes for a C^(1)generic flow on a closed Rieman-nian manifold and prove that such a homoclinic class contains no singularity.This enables a parallel study of bi-Lyapunov stable dynamics for flows and for diffeomorphisms.For example,we can then show tha t a bi-Lyapunov st able homoclinic class for a C^(1)generic flow is hyperbolic if and only if all periodic orbits in the class have the same stable index.
基金The first author is supported by NSFC (No. 10171004) Ministry of Education Special Funds for Excellent Doctoral ThesisThe second author is supported by Ministry of Education Special Funds for Excellent Doctoral Thesis
文摘It seems that in Mane＇s proof of the C^1 Ω-stability conjecture containing in the famous paper which published in I. H. E. S. （1988）, there exists a deficiency in the main lemma which says that for f ∈F^1 （M） there exists a dominated splitting TMPi（f） =Ei^s the direlf sum of E and F Fi^u（O 〈 i 〈 dim M） such that if Ei^s is contracting, then Fi^u is expanding. In the first part of the paper, we give a proof to fill up this deficiency. In the last part of the paper, we, under a weak assumption, prove a result that seems to be useful in the study of dynamics in some other stability context.
