Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T20:44:24.900Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterisation of quasi-Anosov diffeomorphisms

Published online by Cambridge University Press:  17 April 2009

Graham Couper
Graham Couper
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, California, USA; Department of Mathematics, University of Newcastle, Newcastle, New South Wales.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ƒ be a C1 diffeomorphism of a compact C∞ boundary–less manifold, and let ƒ# be the operator on the bounded or continuous sections of the tangent bundle (with supremum norm) defined by ƒ#η = Tƒ о η о ƒ−1. The main result of this paper is that ƒ is quasi-Anosov if and only if 1 – f# is injective and has closed range.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 17 , Issue 3 , December 1977 , pp. 321 - 334
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Dunford, Nelson and Schwartz, Jacob T., Linear operators, Part I (Pure and Applied Mathematics, 7. Interscience, New York, London, 1958).Google Scholar
[2]Franks, John, “Necessary conditions for stability of diffeomorphisms”. Trans. Amer. Math. Soc. 158 (1971), 301–308.CrossRefGoogle Scholar
[3]Franks, John, “Differentially Ω-stable diffeomorphisms”, Topology 11 (1972), 107–113.CrossRefGoogle Scholar
[4]Franks, John, “Absolutely structurally stable diffeomorphisms”, Proc. Amer. Math. Soc. 37 (1973), 293–296.CrossRefGoogle Scholar
[5]Franks, John and Robinson, Clark, “A quasi–Anosov diffeomorphism that is not Anosov”, Trans. Amer. Math. Soc. 223 (1976), 267–278.CrossRefGoogle Scholar
[6]Guckenheimer, John, “Absolutely Ω-stable diffeomorphisms”, Topology 11 (1972), 195–197.CrossRefGoogle Scholar
[7]Mane, Ricardo, “Persistent manifolds are normally hyperbolic”, Bull. Amer. Math. Soc. 80 (1974), 90–91.CrossRefGoogle Scholar
[8]Mañé, R., “Quasi-Anosov diffeomorrhisms”, Dynamical systems, 27–29 (Proc. Sympos. University of Warwick, 1973/74. Lecture Notes in Mathematics, 468. Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[9]Mañé, Ricardo, “On infinitesimal and absolute stability of diffeomorphisms”, Dynamical systems, 151–161 (Proc. Sympos. University of Warwick, 1973/74. Lecture Notes in Mathematics, 468. Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[10]Mañé, Ricardo, “Expanding diffeomorphisms”, Dynamical systems, 162–174 (Proc. Sympos. University of Warwick, 1973/74. Lecture Motes in Mathematics, 468. Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[11]Mather, John N., “Characterization of Anosov diffeomorphisms”, K. Nederl. Akad. Wetensah. Proc. Ser. A 71 = Indag. Math. 30 (1968), 479–483.CrossRefGoogle Scholar
[12]Robbin, J.W., “A structural stability theorem”, Ann. of Math. (2) 94 (1971), 447–493.CrossRefGoogle Scholar
[13]Robinson, R. Clark, “Structural stability of C 1 -diffeomorphisms”, J. Differential Equations 22 (1976), 28–73.CrossRefGoogle Scholar