A generalized Poincaré stability criterion
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- by Carmen Chicone and R. C. Swanson PDF
- Proc. Amer. Math. Soc. 81 (1981), 495-500 Request permission
Abstract:
Let $\Phi _t^\# \eta = {\Phi ^{ - t}} \circ \eta \circ {\phi ^t}$ 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 $\Phi _\# ^t$ 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 $\{ \Phi _t^\# \}$ 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
- Carmen Chicone and R. C. Swanson, The spectrum of the adjoint representation and the hyperbolicity of dynamical systems, J. Differential Equations 36 (1980), no. 1, 28–39. MR 571125, DOI 10.1016/0022-0396(80)90073-X
- D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math. (2) 95 (1972), 66–82. MR 288785, DOI 10.2307/1970854
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
- Ricardo Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977), 351–370. MR 482849, DOI 10.1090/S0002-9947-1977-0482849-4
- John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math. 30 (1968), 479–483. MR 0248879
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 81 (1981), 495-500
- MSC: Primary 58F15; Secondary 58F10
- DOI: https://doi.org/10.1090/S0002-9939-1981-0597670-2
- MathSciNet review: 597670