Stable manifolds for hyperbolic sets

turns out that they STABLE MANIFOLDS AND HYPERBOLIC SETS ' MORRIS W. I-HIRSCH AND CHARLES c. PUGH - 0. Introduction. Let U be an open set in a smooth manifold M and f '. U —» M a C‘ map. A fixed point x of f is hyperbolic if the derivative ’I;f: M, , —g M, is an isomorphism and its spectrum is separated by the unit circle. If 'I'= 7}; this means that M, has a unique splitting E, x E, under Tsuch that T|E, is expanding and T|Ez is contracting. That is, for suitable equivalent norms on E 1 and E2, m3qllT_'lE1ll- qTlE2q l 1-. The classical stable manifold theory says that this convenient behavior of T; f is reflected in the behavior off in a neighborhood Vof x: there is a submanifold W‘ of M tangent to E, at x such that, - W’n V= ye I/lim (f|V) qy = x, _ llq® there is also a submanifold W” tangent to E, such that W‘ n V= ye V|lim (f|V) “'y = x. See for example Kelley [1, Appendix, 15 and 14, which contains further references. _ We call W‘ and W‘ local stable and unstable manifolds off at x, respectively. It enjoy the same diflerentiability as j; and if f is C‘ they depend continuously on f in the C‘ topologies. ' '. _ For technical reasons we allow M to be an infinite dimensional manifold modelled on a Banach space. _ i ' _ The notion of hyperbolic fixed point can be generalized to that of a hyperbolic set A c: U. ‘This means that f (A) = A, and TAM (the tangent bundle of M over A) has an invariant splitting E1 GB B, such that 'If|E, is expanding and '. l]'|E1 is contracting. (For this theory M is assumed finite dimensional and A compact, although generalizations are possible. ) In Smale’s theory of Q-stability, and related topics 12, 13, “generalized stable manifold theoremq plays a key role: there is a neighborhood Vol‘ A, and submanifolds W’ (x), Wq (x) tangent to E, (x) and E, (x) respectively for each x e A, such that W‘ (x) = ye Vllim dmvry. (flV) ‘x) ‘= 0. lqII) Wq (x) = ye Vl1imd ( (f|V) q‘y. (flV) q'fc) _= 0- l“Q l3J