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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Essential versus $ \char93 $-spectrum for smooth diffeomorphisms


Author: Russell B. Walker
Journal: Proc. Amer. Math. Soc. 93 (1985), 532-538
MSC: Primary 58F15; Secondary 58F19
MathSciNet review: 774018
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Abstract: J. Robbin conjectures in his 1972 survey article (Bull. Amer. Math. Soc. 78, 923-952) that the "essential" and $ \char93 $-spectra are identical for all $ {C^1}$-diffeomorphisms. If so, the stability conjecture of S. Smale follows. A $ \char93 $-spectrum may be attached to any orbit or invariant set and is a generalization of the set of eigenvalues of $ Tf$ at a fixed point. The essential spectrum is the closure of this spectrum, restricted to the periodic set of $ f$. So Robbin's conjecture meant that the periodic orbits carry the growth rate behavior of their closure. A counterexample is constructed and other conjectures made.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0774018-5
PII: S 0002-9939(1985)0774018-5
Keywords: $ \char93 $-spectrum, adjoint operator, essential spectrum, hyperbolicity
Article copyright: © Copyright 1985 American Mathematical Society