Morse Decomposition of Semiflows on Topological Spaces


This paper studies Morse decompositions of discrete and continuous-time semiflows on compact Hausdorff topological spaces. We extend two classical results which are well-known facts for flows on compact metric spaces: the characterization of the Morse decompositions through increasing sequences of attractors and the existence of Lyapunov functions.


Semiflows Topological spaces Morse decompositions Lyapunov functions 

Ams Subject Classification 2000

Primary: 37B35 37B25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Braga Barros C. J. and San Martin L. A. B.: Chain transitive sets for flows on flag bundles. Forum Math., to appear.Google Scholar
  2. 2.
    Choi S.K., Chu C.-K., Park J.S. (2002). Cahin recurrent sets for flows on non- spaces. J. Dyn. Diff. Eq. 14, 597–612MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Colonius F., Kliemann W. (2000). The Dynamics of Control. Birkhäuser, BostonGoogle Scholar
  4. 4.
    Conley C. (1978). Isolated invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math., Vol. 38. American Mathematical Society, Providence, RI.Google Scholar
  5. 5.
    Hirsch M., Smith H., Zhao X. (2001). Chain transitivity, attractivity and strong repellors for semidynamical systems. J. Dyn. Diff. Eq. 13, 107–131MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hurley M. (1995). Chain recurrence, semiflows, and gradients. J. Dyn. Diff. Eq. 7, 437–456MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hurley M. (1998). Lyapunov functions and attractors in arbitrary metric spaces. Proc. Am. Math. Soc. 126, 245–256MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Folland G. (1984). Real Analysis. Willey, New YorkMATHGoogle Scholar
  9. 9.
    Freedman H., So I., Joseph W.-H. (1989). Persistence in discrete semidynamical systems. SIAM J. Math. Anal. 20, 930–938MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kuratowski K. (1966). Topology, Vol 1. Academic Press, New YorkGoogle Scholar
  11. 11.
    Lima E.L. (1977). Espaços Métricos. IMPA, Rio de JaneiroMATHGoogle Scholar
  12. 12.
    Norton D.E. (1995). The fundamental theorem of dynamical systems. Comment. Math., Univ. Carolinae 36(3): 585–597MATHMathSciNetGoogle Scholar
  13. 13.
    Patrão M., and San Martin L. A. B. Semiflows on Topological Spaces: Chain Transitivity and Semigroups. J. Dyn. Diff. EqDOI: 10.1007/s10884-006-9032-3.Google Scholar
  14. 14.
    Patrão M., and San Martin L. A. B. Morse Decomposition of semiflows on fiber bundles. Preprint.Google Scholar
  15. 15.
    Rybakowski K. P. (1987). The Homotopy Index and Partial Differential Equations. Universitext Springer-Verlag, BerlinMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Estadual de CampinasCampinas-Brasil

Personalised recommendations