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 [math.DS], 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.
Highlights
Sullivan’s No Wandering Domains Theorem [10] states that every Fatou component of a rational function f : C → Cis either periodic or pre-periodic
We investigate whether polynomial skew-products can have wandering domains in the basin of an attracting fixed point of g
A sufficiently large iterate of the map is expanding on the Julia set, which immediately implies the non-existence of wandering Fatou components
Summary
Sullivan’s No Wandering Domains Theorem [10] states that every Fatou component of a rational function f : C → Cis either periodic or pre-periodic. The easiest class is given by the hyperbolic polynomials, for which the forward orbits of all critical points stay bounded away from the Julia set In this case, a sufficiently large iterate of the map is expanding on the Julia set, which immediately implies the non-existence of wandering Fatou components. A sufficiently large iterate of the map is expanding on the Julia set, which immediately implies the non-existence of wandering Fatou components It was pointed out by Lilov [7] that an attracting or super-attracting skew-product acting hyperbolically on the invariant fiber has no wandering Fatou components. We will follow the same line of argument in this paper, the main difference being that instead of having no critical points enter the disk U , we obtain sub-linear estimates on the degree of the inverse branches This will turn out to be sufficient to conclude the non-existence of wandering domains.
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