Fatou components of elliptic polynomial skew products

Abstract

We investigate the description of Fatou components for polynomial skew products in two complex variables. The non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov [Fatou theory in two dimensions. PhD Thesis, University of Michigan, 2004], and the geometrically attracting case was studied in Peters and Vivas [Polynomial skew products with wandering Fatou-disks. Math. Z.283(1–2) (2016), 349–366] and Peters and Smit [Fatou components of attracting skew products. Preprint, 2015, http://arxiv.org/abs/1508.06605]. In Astorg et al [A two-dimensional polynomial mapping with a wandering Fatou component. Ann. of Math. (2), 184 (2016), 263–313] it was proven that wandering domains can exist near a parabolic invariant fiber. In this paper we study the remaining case, namely the dynamics near an elliptic invariant fiber. We prove that the two-dimensional Fatou components near the elliptic invariant fiber correspond exactly to the Fatou components of the restriction to the fiber, under the assumption that the multiplier at the elliptic invariant fiber satisfies the Brjuno condition and that the restriction polynomial has no critical points on the Julia set. We also show the description does not hold when the Brjuno condition is dropped. Our main tool is the construction of expanding metrics on nearby fibers, and one of the key steps in this construction is given by a local description of the dynamics near a parabolic periodic cycle.