It is proved that for any C^1 self-covering map f of a compact connected Riemann manifoldwithout boundary, if f satisfies both axiom A and the no-cycle property, and Ω(f) has (s-c-u.)structure, then the structure of ...It is proved that for any C^1 self-covering map f of a compact connected Riemann manifoldwithout boundary, if f satisfies both axiom A and the no-cycle property, and Ω(f) has (s-c-u.)structure, then the structure of its orbit space is topologically stable (semi-stable) underC^0 perturbation and structurally stable under C^1 perturbation. It seems very difficult to theauthors to extend the results of Robbinson and Nitecki to the case of self-maps. Such exten-sions were expected to be solved by Z. Nitecki. This paper may be regarded as a step for-ward in this direction.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It is proved that for any C^1 self-covering map f of a compact connected Riemann manifoldwithout boundary, if f satisfies both axiom A and the no-cycle property, and Ω(f) has (s-c-u.)structure, then the structure of its orbit space is topologically stable (semi-stable) underC^0 perturbation and structurally stable under C^1 perturbation. It seems very difficult to theauthors to extend the results of Robbinson and Nitecki to the case of self-maps. Such exten-sions were expected to be solved by Z. Nitecki. This paper may be regarded as a step for-ward in this direction.
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.
