Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F15

Retrieve articles in all journals with MSC: 58F15

Additional Information

Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society