Polynomial skew-products with wandering Fatou-disks

Abstract

Until recently, little was known about the existence of wandering Fatou components for rational maps in more than one complex variables. In 2014, examples of wandering Fatou components were constructed in Astorg et al. [1] for polynomial skew-products with an invariant parabolic fiber. In 2004 Lilov already proved the non-existence of wandering Fatou components for polynomial skew-products in the basin of an invariant super-attracting fiber. In the current work we investigate in how far his methods carry over to the geometrically attracting case. Lilov proved a stronger statement, namely that the orbit of any horizontal disk in the super-attracting basin must eventually intersect a fattened Fatou components. Here we give explicit constructions to show that this stronger result is false in the geometrically attracting case. However, we also prove that the constructed disks do not give rise to wandering Fatou components. Our construction therefore leaves open the existence of wandering Fatou components in the geometrically attracting case, showing however that the situation is significantly more complicated than in the super-attracting case.

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

“…As explained in the introduction, we prove a theorem similar to the one in [7]. That is, we prove that there exists some skew product parabolic maps that have wandering Fatou disks.…”

“…After we prove this parametrization theorem, we use a similar strategy to the one in [7] to prove that in the parabolic case, forward orbits of one-dimensional disks D that lie above A do not necessarily intersect fattened Fatou components. Therefore, Lilov's theorem is false in general for the parabolic case.…”

“…More precisely, we prove the following: Theorem (Peters [7]) There exist skew-product polynomial maps of the form F (t, z) = (αt, p(z) + q(t)), where α < 1 and p and q are polynomials and a vertical holomorphic disk D ⊂ {t = t 0 } whose forward orbit accumulates at a point (0, z 0 ), where z 0 is a repelling fixed point in the Julia set of p.…”

Fatou Components of Attracting Skew-Products

“…The geometrically attracting case was recently studied by Vivas and the first author in [9]. In order to state results from that paper, let us first recall in greater detail what Lilov proved for the super-attracting case.…”

“…In [9] 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 [9]. 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.…”