Stable manifolds for hyperbolic sets

Abstract

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, m3q{llT_'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