Persistent manifolds are normally hyperbolic

Let M be a smooth manifold, f : M ∣ a C 1 f:\,M\,\mid \text {a}\,{{C}^{1}} diffeomorphism and V ⊂ M a C 1 V \subset M\,{\text {a}}\,{{\text {C}}^1} compact submanifold without boundary invariant under f (i.e. f ( V ) = V 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 ∩ n ∈ z f n ( U ) = V { \cap _{n\, \in \,{\textbf {z}}}}\,{f^n}\left ( U \right )\, = \,V , and for all diffeomorphisms g : M ∣ g:\,M\,\mid near to f in the C 1 {C^1} topology the set V g = ∩ n ∈ z g n ( U ) {V_g}\, = \,{ \cap _{n\, \in \,{\textbf {z}}}}{g^n}\left ( U \right ) is a C 1 {C^1} submanifold without boundary C 1 {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 T M / V = N s V ⊕ N u V ⊕ T V TM/V\, = \,{N^s}V\, \oplus \,{N^u}V\, \oplus \,TV and constants K > 0 K\, > \,0 , 0 > λ > 1 0\, > \,\lambda \, > \,1 such that: \[ ‖ ( T f ) n / N x s V ‖ ≤ K λ n , ‖ ( T f ) − n / N x u V ‖ ≤ K λ n , ‖ ( T f ) n / N x s V ‖ ⋅ ‖ ( T f ) − n / T f n ( x ) V ‖ ≤ K λ n \begin {gathered} \left \| {{{\left ( {Tf} \right )}^n}/N_x^sV} \right \|\, \leq \,K{\lambda ^n},\,\left \| {{{\left ( {Tf} \right )}^{ - n}}/N_x^uV} \right \|\, \leq \,K{\lambda ^n}, \left \| {{{\left ( {Tf} \right )}^n}/N_x^sV} \right \|\, \cdot \,\left \| {{{\left ( {Tf} \right )}^{ - n}}/{T_{{f^n}\left ( x \right )}}V} \right \|\, \leq \,K{\lambda ^n} \end {gathered} \] for all n > 0 n\, > \,0 , x ∈ V x\, \in \,V . In this paper we prove the converse result, namely that persistent manifolds are normally hyperbolic.