Rational maps with Fatou components of arbitrarily large connectivity

2017/04/03 by Canela, Jordi
#30D05 #37F10 #37F45 #37F50 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.00544

Abstract

We study the family of singular perturbations of Blaschke products B_a,λ(z)=z³\fracz-a1-az+\fracλz². We analyse how the connectivity of the Fatou components varies as we move continuously the parameter λ. We prove that all possible escaping configurations of the critical point c_-(a,λ) take place within the parameter space. In particular, we prove that there are maps B_a,λ which have Fatou components of arbitrarily large finite connectivity within their dynamical planes.

Related

• Singular perturbations of Blaschke Products and connectivity of Fatou components
• Achievable connectivities of Fatou components for a family of singular perturbations
• Singular perturbations with multiple poles of the simple polynomials
• Wild recurrent critical points
• Boundary dynamics in unbounded Fatou components