Antonio Garijo

A GENERALIZED VERSION OF THE MCMULLEN DOMAIN

We study the family of complex maps given by Fλ(z) = zn + λ/zn + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components.

Singular perturbations in the quadratic family with multiple poles

We consider the quadratic family of complex maps given by , where c is the centre of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding bounded super-attracting cycle by adding one pole to each point in the cycle. When c = − 1, the Julia set of q − 1 is the well-known basilica and the perturbed map is given by , where are integers, and λ is a complex parameter such that |λ| is very small. We focus on the topological characteristics of the Julia and Fatou sets of f λ that arise when the parameter λ becomes non-zero. We give sufficient conditions on the order of the poles so that for small λ, the Julia sets consist of the union of homeomorphic copies of the unperturbed Julia set, countably many Cantor sets of concentric closed curves, and Cantor sets of point components that accumulate on them.

On a Family of Rational Perturbations of the Doubling Map

The goal of this paper is to investigate the parameter plane of a rational family of perturbations of the doubling map given by the Blaschke products . First we study the basic properties of these maps such as the connectivity of the Julia set as a function of the parameter a. We use techniques of quasiconformal surgery to explore the relation between certain members of the family and the degree 4 polynomials . In parameter space, we classify the different hyperbolic components according to the critical orbits and we show how to parametrize those of disjoint type.

Capture Zones of the Family of Functions λzmexp(z)

We consider the family of entire transcendental maps given by Fλ,m(z)=λzmexp(z) where m≥2. All functions Fλ,m have a superattracting fixed point at z=0, and a critical point at z = -m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behavior, i.e. λ values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then nonlocally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle.

On McMullen-like mappings

We introduce a generalization of the McMullen family f (z) = z n +=z d . In 1988 C. McMullen showed that the Julia set of f is a Cantor set of circles if and only if 1=n+1=d < 1 and the simple critical values of f belong to the trap door. We generalize this behavior and we define a McMullen-like mapping as a rational map f associated to a hyperbolic postcritically finite polynomial P and a pole dataD where we encode, basically, the location of every pole of f and the local degree at each pole. In the McMullen family the polynomial P is z7! z n and the pole dataD is the pole located at the origin that maps to infinity with local degree d. As in the McMullen family f , we can characterize a McMullen-like mapping using an arithmetic condition depending only on the polynomial P and the pole dataD. We prove that the arithmetic condition is necessary using the theory of Thurston’s obstructions, and sucient by quasiconformal surgery.

A note on the period function for certain planar vector fields

Consider a smooth planar autonomous differential equation having a period annulus, 𝒫. We present a new criterion to ensure that the period function has at most one critical period on 𝒫. Our result has a compact form when the differential equation is written as . It is based on a suitable representation formula of the derivative of the period function which uses the infinitesimal generator associated to the continua of periodic orbits. We apply the criterion to several particular cases of the equation , where f(z) and are holomorphic functions and h is a C 2 smooth real valuated function.

Tongues in degree 4 Blaschke products

The goal of this paper is to investigate the family of Blasche products

Singular perturbations of z n with a pole on the unit circle

B_a(z)=z^3\frac{z-a}{1-\bar{a}z}

An effective algorithm to compute Mandelbrot sets in parameter planes

, which is a rational family of perturbations of the doubling map. We focus on the tongue-like 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.

Julia sets converging to the unit disk

We consider the family of complex maps given by where n, d ≥ 1 are integers, and a and λ are complex parameters such that |a| = 1 and |λ| is sufficiently small. We focus on the topological characteristics of the Julia and Fatou sets of .