Essential versus $\#$-spectrum for smooth diffeomorphisms

Abstract

J. Robbin conjectures in his 1972 survey article (Bull. Amer. Math. Soc. 78, 923-952) that the and #-spectra are identical for all C'-diffeomor- phisms. 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 7/ at a fixed point. The 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 con- structed and other conjectures made. A counterexample to the 1972 conjecture of J. Robbin (Ro), that the and #-spectra are identical for all C^diffeomorphisms, is constructed in this note. The #-spectrum of a C^diffeomorphism,/: Mm -» Mm, is the total of the operator on bounded C°-vector fields,/*: C°(M, TM) -» C°(M, TM) given by f*(i) = Tf-li\of. The # -spectrum may be naturally restricted to orbits of / or to any invariant set of /. In the terminology of Robbin's survey paper (Ro), the essential spectrum of /is the closure of the #-spectrum restricted to the periodic set of/, per/. In 1970 J. Franks showed that when/is structurally stable (see (Sml and PM) for definitions and background), the contains no complex number of modulus one (Fr). It follows from the combined work of J. Kupka (Ku), S. Smale (Sm2), M. M. Peixoto (Pe), and C. Pugh (Pul, Pu2), that structural stability implies weak Axiom A; that is, the #-spectrum over per/contains no complex numbers of modulus one (per/is hyperbolic), and per/ equals the nonwandering set of/, s(/). So, were the known to equal the #-spectrum over per/, then from the structural stability of/, it would follow that fi(/) is hyperbolic and per/ = s(/). In the standard terminology of dynamical systems, from Robbin's conjecture, S. Smale's 1967 conjecture that structural stability implies Axiom A would follow (Sml).