Vol. 96, No. 1, 1981

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Invariant manifolds for regular points

Jorge Lewowicz

Vol. 96 (1981), No. 1, 163–173

In this article we prove, for a differentiable vector field or a diffeomorphism on a smooth manifold, that the set of points such that the semitrajectories issuing from them approach a particular semitrajectory at a given exponential rate, constitute a differentiable submanifold, provided the differential of the flow has a certain similar behavior on that trajectory. (See Theorem 1 below, for a precise statement). In particular, the stable manifold theorem for hyperbolic sets ([3], [6, XI]) follows as a corollary.

Although we only consider the C1-case, the same methods, which are essentially classical ([2, Ch. XIII]), could be applied to obtain higher differentiability properties.

Since I have not seen in the literature this type of results for points which are neither equilibrium nor periodic points, and on account of [6, XI–8], I thought that their publication might not be entirely devoid of interest.

Mathematical Subject Classification
Primary: 58F15
Secondary: 58F10
Published: 1 September 1981
Jorge Lewowicz