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}=\textup{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}+bz^{2}+O(\|(z,w)\|^{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.