Spiralling domains in dimension 2

In this talk, I will present a joint work in progress with Xavier Buff. We study the dynamics of polynomial endomorphisms of C^2 which are tangent to the identity at a fixed point. Our goal is to show the existence of such maps for which the immediate basin of attraction of the fixed point has an infinite number of distinct invariant connected components, where the orbits converge to the fixed point without being tangent to any direction.