Dynamics of skew-products tangent to the identity

TL;DR

This paper investigates the local dynamics of generic skew-products tangent to the identity, revealing conditions for parabolic domains, parabolic implosion phenomena, and the existence of wandering domains and historic behavior.

Contribution

It introduces new results on parabolic domains, parabolic implosion, and wandering domains for skew-products tangent to the identity, with explicit examples and topological invariants.

Findings

Existence of parabolic domains with non-tangential convergence conditions.

Identification of a new type of parabolic implosion with distinct renormalization limits.

Construction of explicit examples of wandering domains and Fatou components with historic behavior.

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 $dP_0=\mathrm{id}$. 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 $b \in (\frac{1}{4},+\infty)$. 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.