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
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
基金国家重点基础研究发展规划(973)(the National Grand Fundamental Research 973 Program of China under Grant No．2002CB312103)河南省自然科学基金(the Natural Science Foundation of Henan Province of China under Grant No．0611051900)。
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.
