A fixed point curve theorem for finite orbits local diffeomorphisms

TL;DR

This paper investigates local biholomorphisms with finite orbits near the origin, revealing structural properties of their periodic points, eigenvalue conditions, and providing examples that demonstrate the sharpness of these results.

Contribution

It establishes a fixed point curve theorem for finite orbits of local biholomorphisms and characterizes eigenvalues, including constructing examples that show the limits of these properties.

Findings

Existence of an analytic curve in the fixed point set for such biholomorphisms.

At least one eigenvalue of the linear part is a root of unity.

Examples showing eigenvalue conditions are sharp and cannot be extended to one-parameter groups.

Abstract

We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. 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 2 with finite orbits, there exists an analytic curve passing through the origin and contained in the fixed point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of $F$ at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of local biholomorphisms, with finite orbits, such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they can not be embedded in one parameter groups.