On a family of rational perturbations of the doubling map

Abstract

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...

Singular perturbations of Blaschke products and connectivity of Fatou components

“…We consider the family of Blaschke products of the form B a (z) = z 3 (z − a)/(1 − az) where z ∈ C and a ∈ D * . The dynamics of these maps was studied in [5] and [6] for a ∈ C. As all Blaschke products, they leave the unit circle S 1 invariant. Moreover, z = 0 and z = ∞ are superattracting fixed points of local degree 3 and, therefore, are critical points of multiplicity 2.…”

Tongues in degree 4 Blaschke products

“…For all values of a ∈ C, the points z = 0 and z = ∞ are superattracting fixed points of local degree 3 (c.f. [CFG15]). We denote by A a (0) and A a (∞) their basins of attraction and by A * a (0) and A * a (∞) their immediate basins of attraction, i.e.…”

Connectivity of the Julia set for the Chebyshev-Halley family on degree n polynomials

“…Notice that a cycle of Fatou components is considered to be simply connected if all its connected components are simply connected. The proof is analogous to the one of [8,Proposition 3.6] using the symmetry in the dynamics described in Lemma 2.2.…”

“…Since we are considering the case α = 2n−1 2n−2 , we obtain α = 1 2 . Secondly, we look for values of α so that roots given in (7) coincide with roots given in (8). They are solutions of the equation…”

“…However, this is not a valid solution since we are considering the case α = 2n−1 2n−2 . Finally, we consider the value α = n n−1 discarded in Equation (8). In this case, the degree of the operator remains 2n:…”