• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

2004 Fiscal Year Final Research Report Summary

Studies of the Palis Conjecture and new directions of dynamical systems

Research Project

Project/Area Number 14540199
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionThe University of Tokyo

Principal Investigator

HAYASHI Shuhei  The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (20247208)

Project Period (FY) 2002 – 2004
Keywordshyperbolicity / homoclinic bifurcations / ergodic closing lemma / partial hyperbolicity / dominated splitting / connecting lemma / Lyapunov exponents / ergodic measures
Research Abstract

In this research, I studied the following two subjects ; 1.solving the Palis Conjecture in higher dimension, and 2.finding a new direction of dynamical systems for the research after the solution of the Palis Conjecture.
For the first one, I talked results in several international conferences, and the solution has been announced in a paprer in preparation. But, the proof has not been admitted to be correct by authorities. The whole proof can be divided into three parts. When a diffeomorphism can not be approximated by one exhibiting a homodinic bifurcations in the Palis Conjecture, we have process of proving that it is Axiom A as follows :
(1)Constructing a dominated splitting on the support of every ergodic measure using an extension of the ergodic closing lemma ;
(2)In the dominated splitting on the support of the ergodic measure above, the subspace corresponding the zero Lyapunov exponent is zero ; that is, it is a hyperbolic ergodic measure
(3)The closure of the supports of the ergodic measures is, in fact, a hyperbolic set, and by the uniformity of the hyperbolicity (in a neighborhood of the diffeomorphisim), it can be extended to the nonwandering set.
In this period of the research, I wrote a paper for (1) as a preprint, and for (2) and (3), I am preparing the paper now.
For the second subject, I don't have results yet. But I am interested in the "complexity" and computer science that Smale proposed, and considered a possibility to applications in the theory of dynamical systems. Not only the international conference "New directions in Dynamical Systems" held in Kyoto, I participated in several workshops and the articles I wrote in Japanese journals for non-specialists were come from this research.

  • Research Products

    (4 results)

All 2003

All Journal Article (4 results)

  • [Journal Article] トポロジーからカオスへ2003

    • Author(s)
      林 修平
    • Journal Title

      数理科学 480

      Pages: 19-26

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] スメール2003

    • Author(s)
      林 修平
    • Journal Title

      数学セミナー 503

      Pages: 74-81

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] From Topology to Chaos2003

    • Author(s)
      Shuhei Hayashi
    • Journal Title

      Suuri Kagaku 6

      Pages: 19-26

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Smale2003

    • Author(s)
      Shuhei Hayashi
    • Journal Title

      Sugaku Seminar 8

      Pages: 74-81

    • Description
      「研究成果報告書概要(欧文)」より

URL: 

Published: 2006-07-11  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi