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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.
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