In this paper we study the dynamical behavior of the Chebyshev-Halley methods on the family of degree n polynomials z n + c. We prove that, despite increasing the degree, it is still possible to draw the parameter space by using the orbit of a single critical point. For the methods having z = ∞ as an attracting fixed point, we show how the basins of attraction of the roots become smaller as the value of n grows. We also demonstrate that, although the convergence order of the Chebyshev-Halley family is 3, there is a member of order 4 for each value of n. In the case of quadratic polynomials, we bound the set of parameters which correspond to iterative methods with stable behaviour other than the basins of attraction of the roots.

The goal of this paper is to study the family of singular perturbations of Blaschke products given by B a,λ (z) = z 3 z−a 1−az + λ z 2 . We focus on the study of these rational maps for parameters a in the punctured disk D * 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.

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 B a (z) = z 3 z−a 1−āz . 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 z 2 + c 2 + c. In parameter space, we classify the different hyperbolic components according to the critical orbits and we show how to parametrize those of disjoint type. the Julia set J (f ). The dynamics of the points in F(f ) are stable whereas the dynamics in J (f ) present chaotic behaviour. The Fatou set F(f ) is open and therefore J (f ) is closed. Moreover, if the degree of the rational map f is greater or equal than 2, then the Julia set J (f ) is not empty and is the closure of the set of repelling fixed points of f . The connected components of F(f ), called Fatou components, are mapped under f among themselves. D. Sullivan [Sul85] proved that any Fatou component of a rational map is either periodic or preperiodic. By means of the Classification Theorem (see e.g. [Mil06]), any periodic Fatou component of a rational map is either the basin of attraction of an attracting or parabolic cycle or is a simply connected rotation domain (a Siegel disk) or is a doubly connected rotation domain (a Herman Ring). Moreover, any such component is somehow related to a critical point, i.e. a point z ∈ C such that f (z) = 0. Indeed, the basin of attraction of an attracting or parabolic cycle contains, at least, a critical point whereas Siegel disks and Herman rings have critical orbits accumulating on their boundaries. For a background on the dynamics of rational maps we refer to [Mil06] and [Bea91].The aim of this paper is to study the dynamics of the degree 4 Blaschke products given bywhere a, z ∈ C. This Blaschke family restricted to S 1 is the rational analogue of the double standard family A α,β (θ) = 2θ + α + (β/π) sin(2πθ) (mod 1) of periodic perturbations of the doubling map on S 1 . Indeed, when |a| tends to infinity, the products B a tend to e 4πit z 2 uniformly on compact sets of the punctured plane C * = C \ {0}, where t ∈ R/Z denotes the argument of a. Double standard maps extend to entire transcendental self-maps of C * . Although there is no explicit simple expression for the restriction of B a to S 1 , the global dynamics are simpler than in the transcendental case. The double standard family has been studied in several papers such as [MR07, MR08], [Dez10] and [dlLSS08].For all values of a ∈ C, the points z = 0 and z = ∞ are superattracting fixed points of local degree 3. We denote by A(0) and A(∞) their basins of attraction and by A * (0) and A * (∞) their immediate basins of attraction, i.e. the connected components of the basins containing the superattracting fixed points. If |a| ≤ 1, A(0) = A * (0) = D and A(∞) = A * (∞) = C \ D and hence J (B a ) = S 1 (see Lemma 3.3). If...

We study the family of singular perturbations of Blaschke products B a,λ (z) = z 3 z−a 1−az + λ z 2 . 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.