Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generalized Poincaré stability criterion

Authors: Carmen Chicone and R. C. Swanson
Journal: Proc. Amer. Math. Soc. 81 (1981), 495-500
MSC: Primary 58F15; Secondary 58F10
MathSciNet review: 597670
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Phi _t^\char93 \eta = {\Phi ^{ - t}} \circ \eta \circ {\phi ^t}$ define a semigroup on the Banach space $ \Gamma (M,E)$ of continuous sections of $ E$ over $ M$. It is known that $ ({\Phi ^t},{\phi ^t})$ is hyperbolic iff $ \Phi _\char93 ^t$ has spectrum off the unit circle for $ t \ne 0$. We prove that a third equivalent condition is that the (unbounded!) infinitesimal generator $ L$ of $ \{ \Phi_t^\char93 \}$ have its spectrum disjoint from the imaginary axis. In two dimensions this property coincides with the Poincaré stability criterion for a periodic orbit of a planar dynamical system.

References [Enhancements On Off] (What's this?)

  • [1] C. Chicone and R. C. Swanson, The spectrum of the adjoint representation and the hyperbolicity of dynamical systems, J. Differential Equations 36 (1980), 28-40. MR 571125 (81h:58047)
  • [2] D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math. 95 (1972), 66-82. MR 0288785 (44:5981)
  • [3] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., Vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 0089373 (19:664d)
  • [4] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin and New York, 1977. MR 0501173 (58:18595)
  • [5] R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977), 351-370. MR 0482849 (58:2894)
  • [6] J. Mather, Characterization of Anosov diffeomorphisms, Indag. Math. 30 (1968), 479-483. MR 0248879 (40:2129)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F15, 58F10

Retrieve articles in all journals with MSC: 58F15, 58F10

Additional Information

Keywords: Hyperbolic flow, spectrum, adjoint representation
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society