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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Persistent manifolds are normally hyperbolic


Author: Ricardo Mañé
Journal: Trans. Amer. Math. Soc. 246 (1978), 261-283
MSC: Primary 58F15
MathSciNet review: 515539
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Abstract: Let M be a smooth manifold, $ f:\,M\,\mid$   a$ \,{{C}^{1}}$ diffeomorphism and $ V \subset M\,{\text{a}}\,{{\text{C}}^1}$ compact submanifold without boundary invariant under f (i.e. $ f\left( V \right)\, = \,V$). We say that V is a persistent manifold for f if there exists a compact neighborhood U of V such that $ { \cap _{n\, \in \,{\textbf{z}}}}\,{f^n}\left( U \right)\, = \,V$, and for all diffeomorphisms $ g:\,M\,\mid $ near to f in the $ {C^1}$ topology the set $ {V_g}\, = \,{ \cap _{n\, \in \,{\textbf{z}}}}{g^n}\left( U \right)$ is a $ {C^1}$ submanifold without boundary $ {C^1}$ near to V. Several authors studied sufficient conditions for persistence of invariant manifolds. Hirsch, Pugh and Shub proved that normally hyperbolic manifolds are persistent, where normally hyperbolic means that there exist a Tf-invariant splitting $ TM/V\, = \,{N^s}V\, \oplus \,{N^u}V\, \oplus \,TV$ and constants $ K\, > \,0$, $ 0\, < \,\lambda \, < \,1$ such that:

\begin{displaymath}\begin{gathered}\left\Vert {{{\left( {Tf} \right)}^n}/N_x^sV}... ... \right)}}V} \right\Vert\, \leq \,K{\lambda ^n} \end{gathered} \end{displaymath}

for all $ n\, > \,0$, $ x\, \in \,V$. In this paper we prove the converse result, namely that persistent manifolds are normally hyperbolic.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0515539-0
PII: S 0002-9947(1978)0515539-0
Article copyright: © Copyright 1978 American Mathematical Society