More counterexamples to Coleman’s conjecture

Pixton, Dennis

doi:10.1090/S0002-9939-1981-0603618-4

More counterexamples to Coleman’s conjecture
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by Dennis Pixton PDF

Proc. Amer. Math. Soc. 82 (1981), 145-148 Request permission

Abstract:

For any $m,n \geqslant 2$ we construct a smooth vector field with a topologically hyperbolic equilibrium of type $(m,n)$ which is not locally topologically conjugate to a linear vector field. This refutes Coleman’s conjecture in all cases not covered by previous work of Neumann, Walker, and Wilson.

References

Hyperbolic stationary points

Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5
Dean A. Neumann, Topologically hyperbolic equilibria in dynamical systems, J. Differential Equations 37 (1980), no. 1, 49–59. MR 583338, DOI 10.1016/0022-0396(80)90087-X
Russell B. Walker, Conjugacies of topologically hyperbolic fixed points: a necessary condition on foliations, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 446–457. MR 591203
F. Wesley Wilson Jr., A uniform continuity condition which is equivalent to Coleman’s conjecture, J. Differential Equations 36 (1980), no. 1, 12–19. MR 571123, DOI 10.1016/0022-0396(80)90071-6

Coleman’s Conjecture on topological hyperbolicity