Singular Perturbations With Multiple Poles of the Simple Polynomials
Abstract.
In this article, we study the dynamics of the following family of rational maps with one parameter:
where and . This family of rational maps can be viewed as a singular perturbations of the simple polynomial . We give a characterization of the topological properties of the Julia sets of the family according to the dynamical behaviors of the orbits of the free critical points.
Key words and phrases:
Julia set; Fatou set; Jordan domain; connectivity.2010 Mathematics Subject Classification:
Primary: 37F45; Secondary: 37F10, 37F301. Introduction
For a given rational map , we are interested in the dynamical system generated by the iterates of . In this setup, the Riemann sphere can be divided into two dynamically meaningful and completely invariant subsets: the Fatou set and the Julia set. The Fatou set of is defined to be the set of points at which the family of iterates of forms a normal family, in the sense of Montel. The complement of the Fatou set is called the Julia set, which we denote by . A connected component of the Fatou set is called a Fatou component. According to Sullivan’s theorem, every Fatou component of a rational map is eventually periodic and there are five kinds of periodic Fatou components: attracting domains, superattracting domains, parabolic domains, Siegel disks and Herman rings.
The topology of the Julia sets of rational maps, such as the connectivity and local connectivity, is an interesting and important problem in complex dynamics. It was proved by Fatou that the Julia set of a polynomial is connected if and only if the polynomial has bounded critical orbits. Recently, Qiu and Yin, independently, Kozlovski and van Strien, gave a sufficient and necessary condition for the Julia set of a polynomial to be a Cantor set and hence gave an affirmative answer to Branner and Hubbard’s conjecture (see [23] and [15]). For rational maps, the Julia sets may exhibit more complex topological structures. Pilgrim and Tan proved that if the Julia set of a hyperbolic (more generally, geometrically finite) rational map is disconnected, then, with the possible exception of finitely many periodic components and their countable collection of preimages, every Julia component is either a single point or a Jordan curve [20].
For the general rational maps, it is not easy to describe the topological structures of the corresponding Julia sets. However, for some special families of rational maps, the topological properties of the Julia sets can be studied well. Recently, Devaney considered a singular perturbation of the complex polynomials by adding a pole and studied the dynamics of the rational maps with the following form
where is a polynomial with degree whose dynamics are completely understood, , and [5]. When with , , and , the family of rational maps is commonly called the McMullen maps, which has been studied extensively by Devaney and his collaborators in a series of articles (see [3, 5, 7, 8]). Specifically, it is proved in [7] that if the orbits of the critical points of are all attracted to , then the Julia set of is either a Cantor set, a Sierpiński curve, or a Cantor set of circles. In particular, the Julia set of is a Cantor set of circles if and is small. If the orbits of the free critical points of are bounded, then has no Herman rings [27] and actually, the corresponding Julia set is connected [8]. Since the McMullen family behaves extremely rich dynamics, this family has also been studied in [21] and [24]. When with , , and , , the family of maps is called the generalized McMullen maps, which also attracts many people’s interest. Some additional dynamical phenomenon happens for this family since the parameter space becomes , which is twodimensional. For a comprehensive study on the generalized McMullen map, see [2, 12, 16, 28] and the references therein. There exist also some other special families of rational maps which were studied well. For example, see [11], [14] and [25].
Note that for McMullen maps and generalized McMullen maps, the point at infinity is always a superattracting fixed point and the origin is the unique pole. If the parameter is close to the origin, then each of these maps can be seen as a perturbation of the simple polynomial . Recently, the singular perturbation with multiple poles has been considered. For example, Garijo, Marotta and Russell studied the singular perturbations of the form in [13] and focused on the topological characteristics of the Julia and Fatou sets of that arise when the parameter becomes nonzero.
In this article, we consider the following family of rational maps
(1.1) 
where and . This family can be also seen as a perturbation of the simple polynomial if is small. We would like to mention that the perturbation here is essentially different from Devaney’s family (including McMullen maps and the generalized McMullen maps) and GarijoMarottaRussell’s family since the map has multiple poles and the origin is no longer a pole.
It is easy to see that has a superattracting fixed point at . Since the degree is , the map has critical points (counted with multiplicity). Note that the local degree of is and the origin is another critical point with local degree (see (2.1)) whose critical value is . Hence, this leaves more critical points. We call the remaining critical points and the free critical points. In §2.1 we will show that these free critical points (except ) have a common critical value (see (2.3)). We call these two critical values and the free critical values. The dynamics of is determined by the orbits of these two free critical values. In this article, we will give a quite complete description of the Julia sets of the family for arbitrary parameter .
1.1. Statement of the results
In the rest of this article, we use to denote the superattracting basin of containing . Recall that and are two free critical values of .
Theorem 1.1.
For and , the Julia set of is one of the following cases:

If , then is a Cantor set;

If and , then is connected;

If , , then there are two possibilities:

If each Fatou component contains at most one free critical value, then is connected;

If and lie in the same Fatou component, then is the union of countably many Jordan curves and uncountably many points and hence disconnected.

We will also give several typical examples to show that all the types of the Julia sets stated in Theorem 1.1 actually happen. The parameters that correspond to the examples are chosen from the parameter plane of with . See Figure 1.
The Julia sets in Theorem 1.1 (3b) are called Cantor bubbles (see the picture on the right of Figure 3). This kind of Julia sets has also been found by Devaney and Marotta in [9] for the rational maps when , is sufficiently small, and .
A subset of the Riemann sphere is called a Cantor set of circles (or Cantor circles in short) if it consists of uncountably many closed Jordan curves which is homeomorphic to , where is the Cantor middle third set and is the unit circle. By definition, a Sierpiński curve is a planar set homeomorphic to the wellknown Sierpiński carpet fractal. From Whyburn [26], it is known that any planar set which is compact, connected, locally connected, nowhere dense, and has the property that any two complementary domains are bounded by disjoint simple closed curves is homeomorphic to the Sierpiński curve. It is known that the Cantor circles Julia sets and Sierpiński curves Julia sets can appear in McMullen family and the generalized McMullen family (see [7] and [28]). However, for the family , it is proved that these two kind of Julia sets are not exist.
Theorem 1.2.
For any and , the Julia set of can never be a Cantor set of circles or a Sierpiński curve. Moreover, has no Herman rings.
One can refer [22] for the comprehensive study on the rational maps whose Julia sets are Cantor circles. The first example of the Sierpiński curve as the Julia set of a rational map was given in [17, Appendix F]. For more rational maps whose Julia sets are Sierpiński curves, see [6]. For the study of nonexistence of Herman rings for rational families, see [27], [30] and the references therein.
1.2. Organization of the article
The article is organized as follows:
In §2, we introduce the family and present some basic properties of . Some useful lemmas which are necessary in the proofs of our theorems are also prepared.
In §3, we describe the Julia set of for the case that the free critical value is attracted by and show that under this assumption, is also attracted by and the Julia set of is a Cantor set.
In §4, we discuss the case that the superattracting fixed point attracts exactly one free critical value and prove that is connected.
In §5, we deal with the case that neither nor are attracted by the superattracting basin of and show is either connected or a set of Cantor bubbles.
At the end of §3 to §5, we also give typical examples to show that the Julia sets appeared in Theorem 1.1 actually happen.
In the last section, we make some comments on with . We conjecture that Theorems 1.1 and 1.2 hold also in this case. However, we cannot give a proof here. Comparing Figures 1 and 4, there is a slight difference between them: the green part is compactly contained in the yellow part if while these two parts have nonempty intersection on their boundaries if . This is the essential obstruction that we cannot use the techniques in this article to deal with the case .
Acknowledgements. The first author is supported by the NSFC (Nos. 11301165, 11371126, 11571099) and the program of CSC (2015/2016). He also wants to acknowledge the Department of Mathematics, Graduate School of the City University of New York for its hospitality during his visit in 2015/2016. The second author is supported by the NSFC (No. 11401298) and the NSF of Jiangsu Province (No. BK20140587). We would like to thank the referee for careful reading and useful suggestions.
2. Preliminaries
In this section, we prepare some preliminary results. We first give the symmetric distribution of the critical points and the symmetric dynamical behaviors of . Then we consider the topological properties of the immediate superattracting basin of . In the rest of this article, we always assume that is an integer if there is no other special instruction.
2.1. Dynamical Symmetries
As pointed out in the introduction, the rational map
(2.1) 
has a superattracting fixed point , which is also a critical point of with multiplicity . A direct calculation shows that
(2.2) 
It is easy to see that the origin is another critical point of with multiplicity . The rest critical points of have the form
However, except , there are only two critical values for these critical points. They are
(2.3) 
In this article, we call , the free critical points, and the free critical values of . The dynamics of is determined by the orbits of these two free critical values. Since the local degree of at the origin is and the local degree of is two at every free critical point , we have
(2.4) 
Recall that is the immediate superattracting basin of . Let be a subset of and . We denote . The proof of the following lemma is straightforward.
Lemma 2.1.
Let be a complex number satisfying and suppose that is a Fatou component of . Then
and is also a Fatou component of .
The basin has fold symmetry, i.e. if and only if .
Let be a Fatou component of which is different from . Then either has fold symmetry and surrounds the origin, or , , , are pairwise disjoint, where .
In this article, we need to prove that some domains are simply connected and the following formula is very useful.
Lemma 2.2 (RiemannHurwitz’s formula, [1, §5.4, pp. 8589]).
Let be a rational map defined from to itself. Assume that
(1) is a domain in with finitely many boundary components;
(2) is a component of ; and
(3) there are no critical values of on .
Then there exists an integer such that is a branched covering map from onto with degree and
where denotes the Euler characteristic and denotes the total number of the critical points of in (counted with multiplicity).
Remark.
Let be a domain in . Then if and only if is the Riemann sphere ; if and only if is simply connected; and if and only if is doubly connected (i.e. an annulus).
2.2. The topological structure of
A simply connected domain in is called a Jordan domain if its boundary is a Jordan curve.
Lemma 2.3.
Let be a simply connected domain which contains exactly one free critical value . Then the preimage is a simply connected domain containing on which the degree of the restriction of is .
Proof.
Note that the simply connected domain contains exactly one critical value and . This means that is a connected set containing . By RiemannHurwitz’s formula (Lemma 2.2), it follows that is also simply connected on which the degree of the restriction of is . ∎
Proposition 2.4.
If contains at least one free critical value, then is completely invariant and .
Proof.
For the connectivity of the Julia sets of rational maps, the following criterion was established in [29].
Lemma 2.5 ([29, Lemma 2.9]).
Suppose that is a rational function which has no Herman rings and each Fatou component contains at most one critical value. Then the Julia set of is connected.
3. Both free critical values are escaped
In this section, we consider the case where the free critical value is attracted by . However, we will prove that lies in the superattracting basin of if does. According to Sullivan’s classification theorem, the Fatou set of is equal to and the Julia set is . For and , we use to denote the open disk centered at with radius .
Lemma 3.1.
For any , let be the parameter satisfying
(3.1) 
Denote . Then we have
maps the closed disk into its interior and is contained in a geometrically attracting basin of ;
The preimage is a Jordan domain containing on which the degree of the restriction of is . In particular, the basin is completely invariant.
Proof.
(1) If satisfies (3.1), then we have
which means that . If , then we have
This means that maps the closed disk into its interior. Therefore, is contained in a fixed Fatou component of and the orbit of is contained in .
Since the critical point and the critical value are both disjoint with , it means that does not contain any critical points. By Schwarz’s Lemma, is a geometrically attracting basin which contains an attracting fixed point in with multiplier satisfying .
(2) Since is a Jordan domain containing exactly one free critical value . The first assertion holds by Lemma 2.3. Moreover, the attracting basin containing is completely invariant. ∎
Lemma 3.2.
For , let be the parameter satisfying
(3.2) 
Then maps the closed disk into its interior. In particular, the orbit of the free critical value is contained in the immediate attracting basin of .
Proof.
For simplicity, we denote . If satisfies (3.2), since and , we have
Therefore, and . If , then
Thus maps the closed disk into its interior. The proof is complete. ∎
Recall that is the immediate superattracting basin of .
Corollary 3.3.
If , then .
Proof.
In order to prove Theorem 1.1, we need the following lemma.
Lemma 3.4 ([1, Theorem 9.8.1]).
Let be a rational map with degree at least two. If all of the critical points of lie in the immediate attracting basin of a (super)attracting fixed point of , then the Julia set of is a Cantor set.
Proof of Theorem 1.1 (1).
If is attracted by the superattracting fixed point located at , then the critical value lies in the immediate superattracting basin by Corollary 3.3. According to Proposition 2.4, is completely invariant and contains all critical points. This means that the Julia set of is a Cantor set by Lemma 3.4. ∎
Example 1.
For each , let . Then it is easy to check that is a pole of , i.e. . Therefore, by Theorem 1.1 (1), is a Cantor set.
4. Only one free critical value is escaped
In this section, we consider the case where the attracting basin of attracts exactly one free critical orbit of . By Corollary 3.3, there is only one possibility: and . In order to prove Theorem 1.1 (2), we need the polynomiallike mapping theory introduced by Douady and Hubbard in [10].
Definition.
A triple is called a polynomiallike mapping of degree if and are simply connected plane domains such that , and is a holomorphic proper mapping of degree . The filled Julia set of a polynomiallike mapping is defined as
The Julia set of the polynomiallike mapping is defined as .
Two polynomiallike mappings and of degree are said to be hybrid equivalent if there exists a quasiconformal homeomorphism defined from a neighborhood of onto that of , which conjugates to and whose complex dilatation vanishes on . The following theorem was proved by Douady and Hubbard in [10, Theorem 1, p. 296].
Theorem 4.1 (The Straightening Theorem).
Every polynomiallike mapping of degree is hybrid equivalent to a polynomial with the same degree.
By applying Theorem 4.1 and Fatou’s theorem [4, Theorem 4.1, p. 66], the following result was proved in [29].
Lemma 4.2 ([29, Corollary 4.2]).
Suppose is a polynomiallike mapping of degree . Then the Julia set of is connected if and only if all critical points of are contained in the filled Julia set of .
The following lemma is very useful when one wants to prove the nonexistence of Herman rings in the holomorphic family.
Lemma 4.3 ([30, Main Theorem]).
Any rational map having at most one critical orbit in its Julia set has no Herman rings.
Proof of Theorem 1.1 (2).
The arguments will be divided into two cases: the first one is and the second one is .
(i) Suppose that . Then is completely invariant and by Proposition 2.4. We will choose some suitable domains to construct a polynomiallike mapping and prove that its Julia set is connected and quasiconformally conjugate to the Julia set of .
Since is a superattracting fixed point, we can choose a small simply connected neighborhood of such that , and is a Jordan curve which is disjoint from the forward orbit of (note that the forward orbit of is discrete except the unique possible accumulation point at ). For , let be the connected component of containing . Then we have and . Since , there must exist such that . By Lemma 2.2, it follows that both and are simply connected. Moreover, since is a Jordan curve which is disjoint from the critical orbit of , this means that and are both Jordan domains. Note that and . It follows that is a Jordan domain by Lemma 2.3.
Let . We obtain a polynomiallike mapping with degree . Since , it means that the unique critical orbit of is contained in . Therefore, the Julia set of is connected by Lemma 4.2. Since , we know that the Julia set of the polynomiallike mapping is homeomorphic to that of the rational map . Therefore, the Julia set of the rational map is connected.
We now give specific examples to show that the Julia sets discussed in the proof of Theorem 1.1 (2) happen indeed.
Example 2.
(i) Let and . Then and ;
(ii) Let and . Then and .
Proof.
(i) If , then . By (2.1), we have
This means that forms a superattracting periodic orbit with period two and hence . If and , by (3.2), we have
By Lemma 3.2, we have .
(ii) If and , we have^{1}^{1}1The number in this inequality will be used in the construction of an example in next section.
(4.1) 
By Lemma 3.1, we have if . Note that . This means that and hence . Therefore, we only need to prove that .
Since is real, we consider the restriction of on the real axis . According to (2.1), is continuous in the interval . Recall that is a free critical point of . Note that and there exists a sufficiently large such that and . There must exist a real number such that . Since all the critical values of lie in (super) attracting basins, it follows that is a repelling fixed point and hence contained in the Julia set of .
Suppose that . Then by Proposition 2.4. There exists a smooth curve connecting with such that and . Let be a new curve defined as if and if . Then is a piecewise smooth curve which is disjoint with the lowerhalf plane. Moreover, is contained in since the Fatou set of is symmetric about the real line. Note that is compact and is open, one can move slightly in (but keep the two ends fixed) such that the new curve is contained in the upperhalf plane (except two ends). Then is a Jordan curve contained in , and the bounded component of contains a repelling fixed point . Since , it follows that . Since , it means that separates . This is a contradiction since is connected by Theorem 1.1 (2). Therefore, we have . ∎
5. Both free critical values do not escape
In this section, we consider the case that and do not belong to . Since the immediate superattracting basin of is simply connected, it follows that each Fatou component of is also simply connected.
Lemma 5.1.
Suppose that . Then we have (3.1) if .
Proof.
Proof of Theorem 1.1 (3).
Since , we know that is contained in an attracting basin which is completely invariant under by Lemmas 5.1 and 3.1.
(3a) By the hypothesis, each Fatou component of contains at most one critical value and there is at most one critical orbit in the Julia set. By Lemmas 2.5 and 4.3, this means that is connected.
(3b) If and are both contained in , then the Fatou set of is equal to and the Julia set . Let be an open disk with and denote . According to Lemma 3.1 (2), the preimage is a Jordan domain. Moreover, is a polynomiallike mapping^{2}^{2}2Although and are not domains in , one can use a coordinate transformation to obtain a polynomiallike mapping since as a rational map, is holomorphic on whole . with degree and does not contain any critical points. By Theorem 4.1, is hybird equivalent to a polynomial with degree . Note that has a superattracting fixed point at with local degree and the free critical points , , are attracted by the basin . This means that has also a superattracting fixed point with local degree and the rest critical points of are escaped to . In particular, the Julia set of is disconnected.
Let be the filled Julia component of which contains the superattracting fixed point . According to [23, Main Theorem], all the Julia components of are points except the components that are iterated onto the component . We claim that is a closed Jordan disk. Indeed, by the hyperbolicity of , one can construct a polynomiallike mapping with degree such that , and the filled Julia set of is exactly . Note that is hybird equivalent to whose filled Julia set is the closed unit disk. This means that is actually a closed Jordan disk. Therefore, the Julia set of is the union of countably many Jordan curves and uncountably many points. Since is hybird equivalent to and is contained in the attracting basin , we know that the Julia set of is homeomorphic to that of . This means that consists of countably many Jordan curves and uncountably many points. ∎
Example 3.
(i) Let and . Then , and they are in different Fatou components;
(ii) Let and . Then , and both of them are contained in an immediate attracting basin of .
6. The impossible types of Julia sets
As stated in the introduction, it was known that the Cantor circles Julia sets and Sierpiński curves Julia sets can appear in McMullen family and the generalized McMullen family. We will prove in the present section that these two kind of Julia sets are not exist for the family . Moreover, we also prove that has no Herman rings.
Lemma 6.1.
The Julia set of any polynomial can never be a Sierpiński curve.
Proof.
Let be a polynomial with degree at least two. Then has a superattracting fixed point at . Moreover, the basin containing is completely invariant. Therefore, we have . If has no bounded Fatou components, then has exactly one Fatou component and cannot be a Sierpiński curve. If has a bounded Fatou component , then . This also contradicts with the definition of the Sierpiński curve. ∎
Corollary 6.2.
The Julia set of any polynomiallike mapping can never be a Sierpiński curve.
Proof of Theorem 1.2.
By Theorem 1.1, we only need to consider cases (2) and (3).
(i) The nonexistence of Cantor circles. By definition, a Cantor circles Julia set consists uncountable many Jordan curves. Therefore, is not a Cantor set of circles by Theorem 1.1 (2) and (3).
(ii) The nonexistence of Sierpiński curves. Since any Sierpiński curve is connected, we only need to consider cases (2) and (3a). Suppose that is contained in the immediate basin of . From the proof of Theorem 1.1 (2), one can construct a polynomiallike mapping such that the Julia set of is homeomorphic to that of . By Corollary 6.2, is not a Sierpiński curve.
Suppose that . By Lemmas 5.1 and 3.1, one can construct a polynomiallike mapping as in the proof of Theorem 1.1 (3b), where , and . By the choice of the disk , the Julia set of is quasiconformally homeomorphic to that of . According to Corollary 6.2, is not a Sierpiński curve.
(iii) The nonexistence of Herman rings. The Julia set of a rational map having a Herman ring is disconnected. So we only need to consider case (3b). However, in case (3b), all the critical points of are contained in the Fatou set. This means that has no Herman rings. ∎
7. The case for
In this section, we make some brief comments on the family with , i.e.
We can obtain the same results on the symmetric properties of the dynamical behaviors as proved in §2. However, Lemmas 3.1 and 3.2 become invalid when . Define
If , then is compactly contained in by Lemmas 3.1 and 3.2. However, if , we have . See Figure 4.
Actually, we can prove
Lemma 7.1.
If , then .
Proof.
Indeed, if and , solving the equation
we obtain two fixed points and , and both of them have multiplicity two. This means that both and are parabolic fixed points of with multiplier . Since each fixed parabolic basin attracts at least one critical value, it follows that and are not attracted by . Therefore, . It is sufficient to prove that . Similar to the case of quadratic polynomials , one can check that has no solutions in if . This means that if and hence . We omit the details here. ∎
We conjecture that and . See Figure 5 for the Julia set of with and .
A possible method to prove Theorem 1.1 with is to find a “nice” curve separating and (the point is on this curve) such that if the parameter is chosen in the unbound component of , then is attracted by while is not if the parameter is chosen in the bound component of . This strategy is a bit similar to the ideas in Lemmas 3.1 and 3.2.
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