Essential versus -spectrum for smooth diffeomorphisms

Author:
Russell B. Walker

Journal:
Proc. Amer. Math. Soc. **93** (1985), 532-538

MSC:
Primary 58F15; Secondary 58F19

DOI:
https://doi.org/10.1090/S0002-9939-1985-0774018-5

MathSciNet review:
774018

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Abstract | References | Similar Articles | Additional Information

Abstract: J. Robbin conjectures in his 1972 survey article (Bull. Amer. Math. Soc. **78**, 923-952) that the "essential" and -spectra are identical for *all* -diffeomorphisms. If so, the stability conjecture of S. Smale follows. A -spectrum may be attached to any orbit or invariant set and is a generalization of the set of eigenvalues of at a fixed point. The essential spectrum is the closure of this spectrum, restricted to the periodic set of . 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:
https://doi.org/10.1090/S0002-9939-1985-0774018-5

Keywords:
-spectrum,
adjoint operator,
essential spectrum,
hyperbolicity

Article copyright:
© Copyright 1985
American Mathematical Society