Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1985-0774018-5
MathSciNet review: 774018
Full-text PDF

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 $ \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.


References [Enhancements On Off] (What's this?)

  • [CS] C. Chicone and R. C. Swanson, Spectral theory for linearizations of dynamical systems, J. Differential Equations 40 (1981), 155-167. MR 619131 (82h:58039)
  • [Do] H. R. Dowson, Spectral theory of linear operators, Academic Press, London, 1978. MR 511427 (80c:47022)
  • [Fr] J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 92 (1970), 907-918. MR 0283812 (44:1042)
  • [HPS] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin, 1977. MR 0501173 (58:18595)
  • [HS] M. Hirsch and S. Smale, Differential equations, dynamical systems and linear algebra, Academic Press, New York, 1974. MR 0486784 (58:6484)
  • [Jo] R. Johnson, Analyticity of spectral subbundles, J. Differential Equations 35 (1980), 366-387. MR 563387 (81c:58056)
  • [Ku] J. Kupka, Contributions a la theorie des champs generiques, Contributions to Differential Equations, Vol. 2, 1963, pp. 457-484. MR 0165536 (29:2818a)
  • [PM] J. Palis and W. de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York, 1982. MR 669541 (84a:58004)
  • [Pe] M. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214-227. MR 0209602 (35:499)
  • [Pu1] C. Pugh, The closing lemma, Amer. J. Math. 89 (1967), 956-1009. MR 0226669 (37:2256)
  • [Pu2] -, An improved closing lemma and general density theorem, Amer. J. Math. 89 (1967), 1010-1021. MR 0226670 (37:2257)
  • [Ro] J. Robbin, Topological conjugacy and structural stability for discrete dynamical systems, Bull. Amer. Math. Soc. 78 (1972), 923-952. MR 0312529 (47:1086)
  • [SS] R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), 320-358. MR 0501182 (58:18604)
  • [Se] J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc. 203 (1975), 359-390. MR 0368080 (51:4322)
  • [Sm1] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 0228014 (37:3598)
  • [Sm2] -, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1963), 97-116. MR 0165537 (29:2818b)
  • [Sw] R. Swanson, The spectrum of vector bundle flows with invariant subbundles, Proc. Amer. Math. Soc. 83 (1981), 143-145. MR 620000 (83a:58078)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F15, 58F19

Retrieve articles in all journals with MSC: 58F15, 58F19


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0774018-5
Keywords: $ \char93 $-spectrum, adjoint operator, essential spectrum, hyperbolicity
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society