Singular perturbations of Blaschke Products and connectivity of Fatou components

2017/02/03 by Canela, Jordi
#30D05 (Secondary) #37F10 #37F45 (Primary) #37F50 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1702.01074

Abstract

The goal of this paper is to study the family of singular perturbations of Blaschke products given by B_a,λ(z)=z³\fracz-a1-az+\fracλz². We focus on the study of these rational maps for parameters a in the punctured disk \mathbbD^* 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.

Related

• Rational maps with Fatou components of arbitrarily large connectivity
• Achievable connectivities of Fatou components for a family of singular perturbations
• Singular perturbations with multiple poles of the simple polynomials
• A note on the characterization of finite Blaschke products
• On the parabolic Fatou domains