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We study local biholomorphisms with finite orbits in some neighborhood of theorigin since they are intimately related to holomorphic foliations with closedleaves. We describe the structure of the set of periodic points in dimension 2.As a consequence we show that given a local biholomorphism $F$, in dimension 2with finite orbits, there exists an analytic curve passing through the originand contained in the fixed point set of some non-trivial iterate of $F.$ As anapplication we obtain that at least one eigenvalue of the linear part of $F$ atthe origin is a root of unity. Moreover, we show that such a result is sharp byexhibiting examples of local biholomorphisms, with finite orbits, such thatexactly one of the eigenvalues is a root of unity. These examples are subtlesince we show they can not be embedded in one parameter groups.

A fixed point curve theorem for finite orbits local diffeomorphisms