Spiral Domains and Lavaurs-Type Renormalization for Parabolic Germs of ${\mathbb{C}}^2$

Abstract

We study the local dynamics of holomorphic germs $P:{\mathbb{C}}^2\to {\mathbb{C}}^2$ tangent to the identity whose 2-jet at the origin is $(J_0^{2}P)(z,w)=(z-z^2,w+w^2+b{z}^2)$. We prove the existence of parabolic domains for all values of the parameter $b$, showing in particular that for $b>1/4$ there are spiral domains, i.e. parabolic domains whose orbits converge to the origin without being tangent to any fixed direction. We then establish a Lavaurs-type renormalization theorem for a class of non-skew-product maps, extending earlier results known in the skew-product case. As applications, we obtain new topological invariants for such germs and construct a Fatou component with both rank-one and rank-zero limit maps. We also give an example of a polynomial self-map of ${\mathbb{C}}^3$ with an elliptic fixed point admitting a wandering domain with non-contractible limit set.