Characteristic directions of two-dimensional biholomorphisms

Abstract

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

A flower theorem in dimension two

“…is arbitrarily uniformly close to u and we have from (10) that this finite product is arbitrarily uniformly close to 1 if |x m y n | is small enough. Therefore, we can assume that ε is small enough so that |u(x, y) − 1| < 1/2 for all (x, y) ∈ U k .…”

“…In dimension two, no complete description of the dynamics of F is known. Some partial analogs of Leau-Fatou flower theorem have been obtained, guaranteeing either the existence of one-dimensional stable manifolds [4,6,1,10] or of twodimensional ones [7,14]. With no extra assumptions on F , the most general result is due to Abate [1], who showed that F always supports some stable dynamics: either F has a curve of fixed points or there exist one-dimensional stable manifolds of F with 0 in their boundary.…”

Birational properties of tangent to the identity germs without non‐degenerate singular directions

“…This allows to apply Hakim's result to get a parabolic curve for the lifted germs, transversal to the exceptional divisor of the modification, so that it descends to a parabolic curve for $$f$$. Several authors addressed the problem of finding stable manifolds for (possibly non‐isolated) $$\text{two}$$‐dimensional tangent to the identity germs, and the picture is quite complete now, see, for example, [1, 2, 10, 15, 18, 23, 24, 26, 29, 32, 38, 39]. The description of parabolic manifolds has been recently instrumental for the construction of examples of wandering domains, see [5, 6].…”

“…Besides being only formal, the surface $$S$$ is also singular, and [23] cannot be applied to $$f|_S$$ even if $$S$$ were convergent. When working on the model $$X_{\pi _0}$$, the strict transform $$\widetilde{S}$$ of $$S$$ is smooth and invariant by the lift $$\widetilde{f}$$ of $$f$$.…”

“…A direct computation shows that $$\widetilde{f}|_{\widetilde{S}}$$ has only two characteristic directions, corresponding to the tangent space of the exceptional divisor $$\pi _0^{-1}(0)$$. In general, $$\pi _0^{-1}(0)$$ could provide the only separatrices of $$\widetilde{f}$$, and constructing parabolic manifolds would require other techniques (similar to [23]).…”