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
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C. Coleman, Hyperbolic stationary points, Reports of the First Internat. Congr. Nonlinear Oscillations (Kiev, 1969), Vol. 2, Qualitative Methods in the Theory of Nonlinear Oscillations (Ju. A. Mitropol’skiǐ and A. N. Śarkovskiǐ, eds.), Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1970, pp. 222-227.
- 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, Global Theory of Dynamical Systems (Proc. Northwestern, 1979), edited by Z. Nitecki and C. Robinson, Lecture Notes in Math., vol. 819, Springer, New York, 1980, pp. 458-470.
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 145-148
- MSC: Primary 58F25; Secondary 34D30, 58F15
- DOI: https://doi.org/10.1090/S0002-9939-1981-0603618-4
- MathSciNet review: 603618