Fatou Components of Attracting Skew-Products

Abstract

We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004, 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). In 2014, it was shown in Astorg et al. (Ann Math, arXiv:1411.1188, 2014) that wandering domains can exist near a parabolic invariant fiber. In Peters and Vivas (Math Z, arXiv:1408.0498, 2014, the geometrically attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew-products; this class contains the maps studied in Peters and Vivas (Math Z, arXiv:1408.0498, 2014. Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds.

Parametrization of unstable manifolds and Fatou disks for parabolic skew-products

“…Fatou components on the attracting skew-product case have been explored also by Peters and Smit [6]. We prove in this article that, in the parabolic case, a similar construction can be done as in the case of geometrically attracting.…”

Polynomial skew-products in dimension 2: Bulging and Wandering Fatou components

“…The geometrically attracting case have been further investigated by Peters and Smit in [24]. They focused their investigation on polynomial skew-products such that the action on the invariant attracting fiber is subhyperbolic, that is the polynomial does not have parabolic periodic points and all critical points lying on the Julia set are pre-periodic.…”

“…Proposition 1 (Peters-Smit, [24]) Let F : C 2 → C 2 be a polynomial skew-product of the form (1). Assume that the origin is an attracting, not superattracting, fixed point for g with corresponding basin B g , and the polynomial f 0 (z) := f (z, 0) is subhyperbolic.…”

“…Fix a neighbourhood U of the post-critical set of f 0 . Then by [24,Proposition 15], there exists a set E ⊂ C of full mesure in a neighbourhood of the origin such that for all w 0 ∈ E there exists a constant C = C(w 0 ,U) such that for all n ∈ N we have Card z :…”

“…Each critical point x contained in the Julia set is eventually mapped into a repelling periodic point, and up to considering an iterate of F we may assume that it is eventually mapped into a repelling fixed point with multiplier µ, with |µ| > 1. The main tool to control the orbits of the critical points of F is obtained using a linearization map of the unstable manifold of the repelling fixed point, given by a map Φ : C → C satisfying Φ(µt) = f k 0 • Φ(t) for some k ∈ N. Thanks to [24,Proposition 10], there exist C > 1 and 0 < γ < 1 so that…”