On a Family of Degree 4 Blaschke Products

Resumen

This PhD thesis belongs to the area of discrete dynamical systems in the complex plane, i.e. the iteration of analytic functions in one complex variable. Given a rational map f from the Riemann sphere onto itself, we consider the dynamical system given by its iterates. The Riemann sphere splits into two totally f-invariant subsets: the Fatou set, which is defined to be the set of points z where the family {f^n} is normal in some neighborhood of z, and its complement, the Julia set. The dynamics of the points in the Fatou set are stable in the sense of normality or equicontinuity whereas the dynamics in the Julia set present chaotic behavior. This thesis focuses on the study of the family of Blaschke products Ba(z)=z^3(z-a)/(1-\bar{a}z), where a and z are complex numbers. We study its parameter and its dynamical planes using intensive use of quasiconformal surgery techinques, which allow us to build rational maps with prescribed dynamics using quasiregular maps as models. The thesis is structured as follows. In Chapter 1 we give an overview on the preliminary results used throughout the thesis. In Chapter 2 we give an introduction to quasiconformal surgery. In Chapter 3 we give an overview of the dynamical plane of the Blaschke products Ba. We begin by studying their basic properties. Afterwards we show that the maps Ba cannot have doubly connected rotation domains (Herman rings) (Proposition 3.2.3) and prove a criterion of connectivity of the Julia set of Ba (Theorem 3.2.1). In Chapter 4 we introduce the family Mb of cubic polynomials with a superattracting fixed point. Then we show how to build polynomials Mb from Blaschke products Ba , obtaining a map ? from a subset of the parameter plane of the Ba to the parameter plane of the polynomials Mb. We also prove that the map ? is continuous and restricts to a homeomorphism on every disjoint hyperbolic component. In Chapter 5 we study the parameter plane of the Blaschke products Ba. We first describe the symmetries in the parameter plane. Then we classify the different hyperbolic dynamics which may take place and the sets of parameters for which they may happen. Afterwards we build a polynomial-like map for all non-escaping parameters contained in swapping regions which, under certain conditions, may relate the dynamics of Ba with the one of the antipolynomials pc(z) =\bar{z}^2+c (Theorem 5.3.4). Finally we parametrize all disjoint hyperbolic components whose disjoint cycles are bounded and do not lie on the unit circle (Theorem 5.4.2). In Chapter 6 we study the tongues of the Blaschke products Ba. We first prove some of their topological properties such as their connectivity modulo symmetry, their simple connectivity and the existence of a unique tip for every tongue (Theorem 6.2.1). Then we show how bifurcations take place along curves in a neighborhood of every tongue (Theorem 6.3.2). Finally we study how tongues extend in the annulus of parameters a such that 1<|a|<2. In Chapter 7 we study how the degree 4 Blaschke products Ba generalize to degree m+2 families of rational maps for m>2.