Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (2): 552-558.doi: 10.1016/S0252-9602(12)60037-X

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DIFFEOMORPHISMS WITH VARIOUS C1 STABLE PROPERTIES

 TIAN Xue-Ting, SUN Wen-Xiang   

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China|Departamento de Matem´atica, Universidade Federal de Alagoas, Macei´o 57072-090, Brazil LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
  • Received:2009-08-05 Revised:2011-02-27 Online:2012-03-20 Published:2012-03-20
  • Supported by:

    Tian is the corresponding author and supported by CAPES(Brazil); Sun is supported by National Natural Science Foundation (10671006, 10831003) and National Basic Research Program of China (973 Program) (2006CB805903).

Abstract:

Let M be a smooth compact manifold and ∧ be a compact invariant set. In this article, we prove that, for every robustly transitive set ∧, f| satisfies a C1-generic-stable shadowable property (resp., C1-generic-stable transitive specification property or C1-generic-stable barycenter property) if and only if ∧ is a hyperbolic basic set. In partic-ular, f| satisfies a C1-stable shadowable property (resp., C1-stable transitive specification property or C1-stable barycenter property) if and only if ∧ is a hyperbolic basic set. Similar results are valid for volume-preserving case.

Key words: Specification property, hyperbolic basic set, topologically transitive, shad-owing property

CLC Number: 

  • 37A25
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