Singular perturbations of Blaschke products and connectivity of Fatou components

Abstract

The goal of this paper is to study the family of singular perturbations of Blaschke products given by B a,λ (z) = z 3 z−a 1−az + λ z 2 . We focus on the study of these rational maps for parameters a in the punctured disk D * and |λ| small. We prove that, under certain conditions, all Fatou components of a singularly perturbed Blaschke product B a,λ have finite connectivity but there are components of arbitrarily large connectivity within its dynamical plane. Under the same conditions we prove that the Julia set is the union of countably many Cantor sets of quasicircles and uncountably many point components.

Rational maps with Fatou components of arbitrarily large connectivity

“…Notice that if c − (a, λ) ∈ U c (a, λ), where U c (a, λ) is a preimage of A 0 (a, λ), then U c (a, λ) has connectivity 3 (see Lemma 2.4). If U c (a, λ) surrounds z = 0, the subsequent preimages of U c (a, λ) which also surround z = 0 have greater connectivity (see [7]). Because of this, when we talk about the sets A ∆ (a, λ) we refer to them as multiply connected preimages of A 0 surrounding z = 0.…”

“…In [7] we introduced a family of singularly perturbed Blaschke products (see Equation (3)) whose maps have, under certain dynamical conditions, Fatou components of arbitrarily large finite connectivity (see Theorem 1.1). We also provided numerical evidence showing that parameters satisfying these conditions actually exist.…”

“…However, the extra critical point c − allows the maps B a,λ to have richer dynamics than the maps Q λ,3,2 . The main theorem in [7] describes the connectivity of the Fatou components of B a,λ in the case that |λ| is small enough and c − (a, λ) belongs to the basin of attraction of z = ∞, A a,λ (∞). We denote the immediate basin of attraction of ∞, i.e.…”

“…It follows again from the Riemann-Hurwitz formula that U c has connectivity 3. Moreover, it can be proven that if U c surrounds z = 0, then it has iterated preimages of arbitrarily large connectivity (see [7,Proposition 3.1], c.f. Theorem 1.1).…”

“…It shows that, if λ ∈ D * C(a) , then there is a straight annulus inside A 0 . Within the paper [7] this is presented as one of the conditions to obtain the constant C(a) in the proof of Theorem 2.5. Even if we could avoid using it in this paper, it helps us understand how the annulus A 0 shrinks when λ tends to 0.…”