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More counterexamples to Coleman's conjecture

Author: Dennis Pixton
Journal: Proc. Amer. Math. Soc. 82 (1981), 145-148
MSC: Primary 58F25; Secondary 34D30, 58F15
MathSciNet review: 603618
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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.

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  • [C] 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'skii and A. N. Śarkovskii, eds.), Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1970, pp. 222-227.
  • [H] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR 0448362
  • [N] Dean A. Neumann, Topologically hyperbolic equilibria in dynamical systems, J. Differential Equations 37 (1980), no. 1, 49–59. MR 583338,
  • [Wa] 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
  • [Wi1] 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,
  • [Wi2] -, 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.

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Keywords: Hyperbolic equilibrium, topological hyperbolicity, topological conjugacy
Article copyright: © Copyright 1981 American Mathematical Society

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