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A property of vector fields without singularity in {\mathcal G}^1(M)

Published online by Cambridge University Press:  26 March 2001

HIROYOSHI TOYOSHIBA
HIROYOSHI TOYOSHIBA
Affiliation:
Department of Mathematics, School of Education, Waseda University, Tokyo 169-8050, Japan
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Abstract

We prove the following property: if X in \mathcal G^1(M) has no singularity and x \in \Sigma(X), then \overline{\operatorname{orbit}(x)} \cap \overline{\operatorname{per}(X)} \not = \emptyset. In addition, if we assume \overline{\operatorname{per}_i(X)} \cap \overline{\operatorname{per}_j(X)} = \emptyset for i \not = j, then \overline{\operatorname{per}(X)} = \bigcup_{i=0}^{n-1} \overline{\operatorname{per}_i(X)} is a hyperbolic set. Moreover, we shall give a proof of the \Omega-stability conjecture for flows.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 21 , Issue 1 , February 2001 , pp. 303 - 314
Copyright
© 2001 Cambridge University Press

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