Characteristic directions of two-dimensional biholomorphisms

TL;DR

This paper investigates the dynamics of two-dimensional biholomorphisms near tangent fixed points, establishing the existence of invariant curves or manifolds aligned with characteristic directions, and identifying conditions for parabolic curves.

Contribution

It proves the existence of either fixed point curves or parabolic manifolds associated with each characteristic direction of tangent to the identity diffeomorphisms in a72, advancing understanding of local complex dynamics.

Findings

Existence of analytic fixed point curves tangent to characteristic directions.

Presence of k parabolic manifolds tangent to the same directions.

At least one parabolic manifold contains a parabolic curve.

Abstract

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $\mathbb{C}^2$ 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.