Dynamical Systems seminar - Achievable connectivities of Fatou components in singular perturbations

Speaker: Jordi Canela, (Universitat Jaume I) 

Abstract: In this talk we will consider the dynamical system given by the iteration of a rational map Q over the Riemann Sphere. The dynamics of Q split the Riemann Sphere into two totally invariant sets. The Fatou set consists of all points z such that the family of iterates of Q is normal, or equivalently equicontinuous, in some open neighbourhood of z. The Fatou set is open and corresponds to the set of points with stable dynamics. Its complement, the Julia set, is closed and corresponds to the set of points which present chaotic behaviour. 

 Fatou components, connected components of the Fatou set, are mapped amongst themselves under iteration of Q. A periodic Fatou component can only have connectivity 1, 2, or infinity. Despite that, preperiodic Fatou components can have arbitrarily large finite connectivity. There exist explicit examples of rational maps with Fatou components of any prescribed connectivity. However, the degree of these maps grows as the required connectivity increases. 

We study a family of singular perturbations of rational maps with a single free critical point. Under certain conditions, the dynamical planes of these singular perturbations contain Fatou components of arbitrarily large finite connectivity. In this talk we will analyze the dynamical conditions under which these Fatou components of arbitrarily large connectivity appear.