Dynamics of skew-products tangent to the identity

Abstract

We study the local dynamics of generic skew-products tangent to the identity, i.e. maps of the form $P(z,w)=(p(z),q(z,w))$ with $\mathrm{d}{P}_0=\text{Id}$. More precisely, we focus on maps with non-degenerate second differential at the origin; such maps have local normal form $P(z,w)=(z-z^2+O(z^3),w+w^2+b{z}^2+O(\Vert (z,w){\Vert }^3))$. We prove the existence of parabolic domains, and prove that inside these parabolic domains the orbits converge non-tangentially if and only if $b\in (1/4,+\infty )$. Furthermore, we prove the existence of a type of parabolic implosion, in which the renormalization limits are different from previously known cases. This has a number of consequences: under a diophantine condition on coefficients of $P$, we prove the existence of wandering domains with rank 1 limit maps. We also give explicit examples of quadratic skew-products with countably many grand orbits of wandering domains, and we give an explicit example of a skew-product map with a Fatou component exhibiting historic behaviour. Finally, we construct various topological invariants, which allow us to answer a question of Abate.