Tongues in degree 4 Blaschke products

Abstract

The goal of this paper is to investigate the family of Blasche products B a (z) = z 3 z−a 1−āz , which is a rational family of perturbations of the doubling map. We focus on the tonguelike sets which appear in its parameter plane. We first study their basic topological properties and afterwords we investigate how bifurcations take place in a neighborhood of their tips. Finally we see how the period one tongue extends beyond its natural domain of definition.

Newton-Like Components in the Chebyshev–Halley Family of Degree n Polynomials

“…In the proof of Proposition 4.6 in [6], the authors show that for a ∈ C with |a| large enough (indeed |a| > 16), we have B a (c + ) ∈ A * Ba (∞). A similar proof was previously done in [9,Lemma 2.6] for a family that includes B a (but without providing an explicit bound). Here we present an easier proof, only for real values of the parameter a.…”

“…The map B a (z) = z 3 z−a 1−az is a rational map of degree 4 studied in [8,9], and [6]. In [6,Section 4] it is proven that for a ∈ C, |a| > 15.133, c + ∈ A Ba (∞).…”

Singular perturbations of Blaschke products and connectivity of Fatou components

“…We consider the family of Blaschke products of the form B a (z) = z 3 (z − a)/(1 − az) where z ∈ C and a ∈ D * . The dynamics of these maps was studied in [5] and [6] for a ∈ C. As all Blaschke products, they leave the unit circle S 1 invariant. Moreover, z = 0 and z = ∞ are superattracting fixed points of local degree 3 and, therefore, are critical points of multiplicity 2.…”

On Some Advances in Dynamics of One Variable Complex Functions

“…For the Blaschke products B a (z) = z 3 z−a 1−az of degree 4, the dynamical properties are investigated in [15]. The concept of tongues for B a is introduced and focus on the tongue-like sets.…”