Abstract. Let M be a smooth submanifold of R^n. We say that M admits a local (real-analytic) automorphism around a point p if there exist an open neighborhood U of p and a real-analytic mapping F from U to R^n, whose differential at p is invertible, such that F(M) is contained in M. We are interested in the properties (mainly the topological ones) of the orbit of p under such transformations. Already interesting is the case of smooth real curves in the plane.