Singular perturbations of the unicritical polynomials with two parameters

Abstract

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

Singular Perturbations with Multiple Poles of the Simple Polynomials

“…For the connectivity of the Julia sets of rational maps, the following criterion was established in [29]. We remark that Peherstorfer and Stroh proved a similar result as Lemma 2.5 in [19,Theorem 4.2], where they required that each Fatou component contains at most one critical point (counted without multiplicity).…”

“…The following theorem was proved by Douady and Hubbard in [10, Theorem 1, p. 296]. By applying Theorem 4.1 and Fatou's theorem [4, Theorem 4.1, p. 66], the following result was proved in [29].…”

Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps

“…It is known that the Cantor circle Julia sets and Sierpiński carpet Julia sets can appear in McMullen family and the generalized McMullen family (see [5,30]). For the study of singular perturbation of unicritical polynomials, one may also refer to [27], [31], [32], [15] and [16].…”

Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families

“…Some one-parameter and two-parameter families of A•G 2 have been studied by Devaney and his collaborators in several papers, among which [3] is most related to this paper. Some two-parameter families of A • G 2 are investigated in [14] and [15]. We are interested in studying classifications of the Julia sets of regularly ramified rational maps A • R Gj with 2 ≤ j ≤ 5, especially when 3 ≤ j ≤ 5.…”

“…[15, Lemma 2.9]). If a rational function f has no Herman rings and each Fatou component contains at most one critical value, then the Julia set of f is connected.Obviously, if a simply connected domain U ⊂ C * = C \ {0} does not contain any critical value, then f −1 λ (U ) consists of exactly 24 connected components and each of them is simply connected.…”