Essential versus #-spectrum for smooth diffeomorphisms

Walker, Russell B.

doi:10.1090/S0002-9939-1985-0774018-5

Essential versus $\#$-spectrum for smooth diffeomorphisms
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by Russell B. Walker

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

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

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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 ${C^1}$-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 $Tf$ 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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