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C1-stably shadowable chain components

Published online by Cambridge University Press:  01 June 2008

KAZUHIRO SAKAI
KAZUHIRO SAKAI*
Affiliation:
Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505, Japan (email: kazsakai@cc.utsunomiya-u.ac.jp)
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Abstract

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Let p be a hyperbolic periodic saddle of a diffeomorphism f on a closed $C^\infty $ manifold M, and let Cf(p) be the chain component of f containing p. In this paper, we show that if Cf(p) is C1-stably shadowable, then (i) Cf(p) is the homoclinic class of p and admits a dominated splitting $E \oplus F$ (where the dimension of E is equal to that of the stable eigenspace of p); (ii) the Cf(p)-germ of f is expansive if and only if Cf(p) is hyperbolic; and (iii) when M is a surface, Cf(p) is locally maximal if and only if Cf (p)is hyperbolic.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008 , pp. 987 - 1029
Copyright
Copyright © Cambridge University Press 2008

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