Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T09:22:56.025Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcations of meromorphic vector fields on the Riemann sphere

Published online by Cambridge University Press:  14 October 2010

Jesús Muciño-Raymundo  and
Carlos Valero-Valdés
Jesús Muciño-Raymundo
Affiliation:
Institute de Matemáticas, UNAM, Coyoacán, México 04510 D.F. Mexico
Carlos Valero-Valdés
Affiliation:
Institute de Matemáticas, UNAM, Coyoacán, México 04510 D.F. Mexico
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {Xθ} be a family of rotated singular real foliations in the Riemann sphere which is the result of the rotation of a meromorphic vector field X with zeros and poles of multiplicity one. We prove that the set of bifurcation values, in the circle {θ}, is for each family a set with at most a finite number of accumulation points. A condition which implies a finite number of bifurcation values is given. We also show that the property of having an infinite set of bifurcation values defines open but not dense sets in the space of meromorphic vector fields with fixed degree.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 6 , December 1995 , pp. 1211 - 1222
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]Douady, A. and Hubbard, J.. On the density of Strebel differentials. Invent. Math. 30 (1975), 175–179.CrossRefGoogle Scholar
[2]Duff, G. F. D.. Limit-cycles and rotated vector fields. Ann. Math. (2) 57 (1953), 15–31.CrossRefGoogle Scholar
[3]Gluck, H.. Dynamical behavior of geodesic fields. Springer Lecture Notes in Mathematics 819, pp 190–215. Springer: Berlin, 1980.Google Scholar
[4]Griffiths, Ph. and Harris, J.. Principles of Algebraic Geometry. Wiley Interscience: New York, 1988.Google Scholar
[5]Kerchoff, S., Masur, H. and Smillie, J.. Ergodicity of billard flows and quadratic differentials. Ann. Math. (2), 124 (1986), 293–311.CrossRefGoogle Scholar
[6]Masur, H. and Smillie, J.. Haussdorff dimension of sets of nonergodic measured foliations. Ann. Math. (2), 134 (1991), 455–543.CrossRefGoogle Scholar
[7]Perko, L.. Differential Equations and Dynamical Systems. Springer: Berlin-Heidelberg, 1991.CrossRefGoogle Scholar
[8]Plante, J.. Foliations with measure preserving holonomy. Ann. Math. (2) 102 (1975), 327–361.CrossRefGoogle Scholar
[9]Spivak, M.. A Comprehensive Introduction to Differential Geometry Vol. II. Publish or Perish: Houston, Texas, 1970.Google Scholar
[10]Strebel, K.. Quadratic Differentials. Springer: Berlin-Heidelberg, 1984.CrossRefGoogle Scholar