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 $(J_0^2P)(z,w)= (z-z^2,w+w^2+bz^2)$. 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.