Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T09:28:35.901Z Has data issue: false hasContentIssue false

Extending flows from isolated invariant sets

Published online by Cambridge University Press:  14 October 2010

W. L. Bloch
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA

Abstract

Let M be a closed n-dimensional manifold, and let U be a n-dimensional isolating block such that U is smoothly embedded in M. Let φ be a smooth semi-flow on U and let Λ contained in U, be isolated and invariant under φ Then there exists a semi-flow φ′ on M which extends φ such that φ′ is Morse-Smale outside of U, and no new recurrence is introduced in U. The theorem is true for any finite number of pairwise-disjoint Ui. Furthermore, if Λ is hyperbolic, topologically transitive and is the closure of periodic orbits, then φ′ is an Axiom A flow and is Ω-stable. In dimensions two and three, we have the stronger result that φ′ is structurally stable. Also, as a corollary, we give sufficient conditions for the flow φ′ to be nonsingular. One application of the corollary permits the formation of allowable knots and links in three-manifolds such that there exists a structurally stable nonsingular Morse-Smale flow φ′ which contains the specified knots and links in Ω(φ′) Moreover, the knots and links can be specified to be any combination of attractors, repellers or saddles.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Asimov, D.. Round handles and nonsingular Morse-Smale flows. Ann. of Math. 102 (1975), 4154.CrossRefGoogle Scholar
[2]Bowen, R.. One-dimensional hyperbolic sets for flows. J. Differential Equations 12 (1972), 173179.CrossRefGoogle Scholar
[3]Bowen, R.. On Axiom A Diffeomorphisms. CBMS no. 35. Amer. Math. Soc: Providence, 1977.Google Scholar
[4]Christy, J.. Dissertation. University of California: Berkeley, 1984.Google Scholar
[5]Conley, C. and Easton, R.. Isolated invariant sets and isolating blocks. Trans. Amer. Math. Soc. 158 (1971), 3561.CrossRefGoogle Scholar
[6]Franks, J.. Homology and Dynamical Systems. CBMS no. 49. Amer. Math. Soc: Providence, 1982.CrossRefGoogle Scholar
[7]Hirsch, M. W.. Differential Topology. Springer-Verlag: New York, 1976.CrossRefGoogle Scholar
[8]Hirsch, M. W. and Pugh, C.. Stable manifolds and hyperbolic sets. In: S. S. Chern and S. Smale, eds. Global Analysis, Proc. Symp. Pure Math. no. 14, pp 133–163. Amer. Math. Soc: Providence, 1970.Google Scholar
[9]Hirsch, M. W., Pugh, C. and Shub, M.. Invariant Manifolds. Lecture Notes in Math. no. 583. Springer-Verlag: New York, 1977.CrossRefGoogle Scholar
[10]Kennedy, J. and Yorke, J.. Preprint. 1992.Google Scholar
[11]Morgan, J.. Nonsingular Morse-Smale flows on 3-dimensional manifolds. Topology 18 (1979), 4153.CrossRefGoogle Scholar
[12]Palis, J. and Melo, W. de. Geometric Theory of Dynamical Systems. Springer-Verlag: New York, 1982.CrossRefGoogle Scholar
[13]Peixoto, M.. Structural stability on 2-dimensional manifolds. Topology 1 (1962), 101120.CrossRefGoogle Scholar
[14]Pugh, C. and Shub, M.. The Ω-stability theorem for flows. Invent. Math. 11 (1970), 150158.CrossRefGoogle Scholar
[15]Robinson, C.. Structural stability of vector fields. Ann. of Math. 99 (1974), 154175.CrossRefGoogle Scholar
[16]Rolfsen, D.. Knots and Links. Publish or Perish: Houston, 1990.Google Scholar
[17]Siegel, C.. Note on differential equations on the torus. Ann. of Math. 46 (1945), 423428.CrossRefGoogle Scholar
[18]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747817.CrossRefGoogle Scholar
[19]Williams, R. F.. Expanding attractors. Inst. Hautes Études Sci. Publ. Math. 43 (1973), 169203.CrossRefGoogle Scholar