Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets ^{1}^{1}1Published in Discrete and Continuous Dynamical Sistems Series A, Vol. 29, No. 3, 2011, 1205–1244. 2000 Mathematics Subject Classification. 37F10, 30D05. Keywords: Complex dynamical systems, rational semigroup, polynomial semigroup, random iteration, random complex dynamical systems, Julia set, fractal geometry, iterated function systems, surrounding order.
Abstract
We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if and are two connected components of the Julia set, then one of and surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the cardinality of the set of all connected components of the Julia set of such a semigroup is given. By using this, we give a criterion for the Julia set to be connected. Moreover, we show that for any there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials are found and systematically investigated.
1 Introduction
The theory of complex dynamical systems, which has its origin in the important work of Fatou and Julia in the 1910s, has been investigated by many people and discussed in depth. In particular, since D. Sullivan showed the famous “no wandering domain theorem” using Teichmüller theory in the 1980s, this subject has attracted many researchers from a wide area. For a general reference on complex dynamical systems, see Milnor’s textbook [16] or Beardon’s textbook [3].
There are several areas in which we deal with generalized notions of classical iteration theory of rational functions. One of them is the theory of dynamics of rational semigroups (semigroups generated by a family of holomorphic maps on the Riemann sphere ), and another one is the theory of random dynamics of holomorphic maps on the Riemann sphere.
In this paper, we will discuss the dynamics of rational semigroups.
A rational semigroup is a semigroup generated by a family of nonconstant rational maps on , where denotes the Riemann sphere, with the semigroup operation being functional composition ([13]). A polynomial semigroup is a semigroup generated by a family of nonconstant polynomial maps. Research on the dynamics of rational semigroups was initiated by A. Hinkkanen and G. J. Martin ([13, 14]), who were interested in the role of the dynamics of polynomial semigroups while studying various onecomplexdimensional moduli spaces for discrete groups, and by F. Ren’s group([48, 12]), who studied such semigroups from the perspective of random dynamical systems. Moreover, the research on rational semigroups is related to that on “iterated function systems” in fractal geometry. In fact, the Julia set of a rational semigroup generated by a compact family has “ backward selfsimilarity” (cf. Lemma 3.12). For other research on rational semigroups, see [19, 20, 21, 47, 22, 24, 44, 43, 45, 46], and [27]–[40].
The research on the dynamics of rational semigroups is also directly related to that on the random dynamics of holomorphic maps. The first study in this direction was by Fornaess and Sibony ([10]), and much research has followed. (See [4, 6, 7, 5, 11, 33, 34, 37, 38, 39, 40].)
We remark that the complex dynamical systems can be used to describe some mathematical models. For example, the behavior of the population of a certain species can be described as the dynamical system of a polynomial such that preserves the unit interval and the postcritical set in the plane is bounded (cf. [8]). It should also be remarked that according to the change of the natural environment, some species have several strategies to survive in the nature. From this point of view, it is very important to consider the random dynamics of such polynomials (see also Example 1.4). For the random dynamics of polynomials on the unit interval, see [26].
We shall give some definitions for the dynamics of rational semigroups:
Definition 1.1 ([13, 12]).
Let be a rational semigroup. We set
is called the Fatou set of and is called the Julia set of . We let denote the rational semigroup generated by the family The Julia set of the semigroup generated by a single map is denoted by
Definition 1.2.

For each rational map , we set Moreover, for each polynomial map , we set

Let be a rational semigroup. We set This is called the postcritical set of Furthermore, for a polynomial semigroup , we set This is called the planar postcritical set (or finite postcritical set) of We say that a polynomial semigroup is postcritically bounded if is bounded in
Remark 1.3.
Let be a rational semigroup generated by a family of rational maps. Then, we have that , where Id denotes the identity map on , and that for each From this formula, one can figure out how the set (resp. ) spreads in (resp. ). In fact, in Section 2.6, using the above formula, we present a way to construct examples of postcritically bounded polynomial semigroups (with some additional properties). Moreover, from the above formula, one may, in the finitely generated case, use a computer to see if a polynomial semigroup is postcritically bounded much in the same way as one verifies the boundedness of the critical orbit for the maps
Example 1.4.
Let and let be the polynomial semigroup generated by Since for each , and , it follows that each subsemigroup of is postcritically bounded.
Remark 1.5.
It is wellknown that for a polynomial with , is bounded in if and only if is connected ([16, Theorem 9.5]).
As mentioned in Remark 1.5, the planar postcritical set is one piece of important information regarding the dynamics of polynomials. Concerning the theory of iteration of quadratic polynomials, we have been investigating the famous “Mandelbrot set”.
When investigating the dynamics of polynomial semigroups, it is natural for us to discuss the relationship between the planar postcritical set and the figure of the Julia set. The first question in this regard is:
Question 1.6.
Let be a polynomial semigroup such that each element is of degree at least two. Is necessarily connected when is bounded in ?
The answer is NO.
Example 1.7 ([47]).
Let Then (which is bounded in ) and is disconnected ( is a Cantor set of round circles). Furthermore, according to [31, Theorem 2.4.1], it can be shown that a small perturbation of still satisfies that is bounded in and that is disconnected. ( is a Cantor set of quasicircles with uniform dilatation.)
Question 1.8.
What happens if is bounded in and is disconnected?
Problem 1.9.
Classify postcritically bounded polynomial semigroups.
In this paper, we show that if is a postcritically bounded polynomial semigroup with disconnected Julia set, then (cf. Theorem 2.201), and for any two connected components of , one of them surrounds the other. This implies that there exists an intrinsic total order (called the “surrounding order”) in the space of connected components of , and that every connected component of is either simply or doubly connected (cf. Theorem 2.7). Moreover, for such a semigroup , we show that the interior of “the smallest filledin Julia set” is not empty, and that there exists a maximal element and a minimal element in the space endowed with the order (cf. Theorem 2.20). From these results, we obtain the result that for a postcritically bounded polynomial semigroup , the Julia set is uniformly perfect, even if is not generated by a compact family of polynomials (cf. Theorem 2.22).
Moreover, we utilize Green’s functions with pole at infinity to show that for a postcritically bounded polynomial semigroup , the cardinality of the set of all connected components of is less than or equal to that of , where is the “real affine semigroup” associated with (cf. Theorem 2.12). From this result, we obtain a sufficient condition for the Julia set of a postcritically bounded polynomial semigroup to be connected (cf. Theorem 2.14). In particular, we show that if a postcritically bounded polynomial semigroup is generated by a family of quadratic polynomials, then is connected (cf. Theorem 2.15). The proofs of the results in this and the previous paragraphs are not straightforward. In fact, we first prove (1) that for any two connected components of that are included in , one of them surrounds the other; next, using (1) and the theory of Green’s functions, we prove (2) that the cardinality of the set of all connected components of is less than or equal to that of , where is the associated real affine semigroup; and finally, using (2) and (1), we prove (3) that , int, and other results in the previous paragraph.
Moreover, we show that for any , there exists a finitely generated, postcritically bounded, polynomial semigroup such that the cardinality of the set of all connected components of is equal to (cf. Proposition 2.26, Proposition 2.28 and Proposition 2.29). A sufficient condition for the cardinality of the set of all connected components of a Julia set to be equal to is also given (cf. Theorem 2.27). To obtain these results, we use the fact that the map induced by any element of a semigroup on the space of connected components of the Julia set preserves the order (cf. Theorem 2.7). Note that this is in contrast to the dynamics of a single rational map or a nonelementary Kleinian group, where it is known that either the Julia set is connected, or the Julia set has uncountably many connected components. Furthermore, in Section 2.6 and Section 2.4, we provide a way of constructing examples of postcritically bounded polynomial semigroups with some additional properties (disconnectedness of Julia set, semihyperbolicity, hyperbolicity, etc.) (cf. Proposition 2.40, Theorem 2.43, Theorem 2.45). For example, by Proposition 2.40, there exists a generator postcritically bounded polynomial semigroup with disconnected Julia set such that has a Siegel disk.
As we see in Example 1.4 and Section 2.6, it is not difficult to construct many examples, it is not difficult to verify the hypothesis “postcritically bounded”, and the class of postcritically bounded polynomial semigroups is very wide.
Throughout the paper, we will see many new phenomena in polynomial semigroups that do not occur in the usual dynamics of polynomials. Moreover, these new phenomena are systematically investigated.
In Section 2, we present the main results of this paper. We give some tools in Section 3. The proofs of the main results are given in Section 4.
There are many applications of the results of postcritically bounded polynomial semigroups in many directions. In the sequel [36], by using the results in this paper, we investigate the fiberwise (sequencewise) and random dynamics of polynomials and the Julia sets. We present a sufficient condition for a fiberwise Julia set to be of measure zero, a sufficient condition for a fiberwise Julia set to be a Jordan curve, a sufficient condition for a fiberwise Julia set to be a quasicircle, and a sufficient condition for a fiberwise Julia set to be a Jordan curve which is not a quasicircle. Moreover, using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that for a , there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation which are parameterized by the Cantor set, densely inside In the sequel [37], we classify hyperbolic or semihyperbolic postcritically bounded compactly generated polynomial semigroups, in terms of the random complex dynamics. It is shown that in one of the classes, for almost every sequence , the Julia set of is a Jordan curve but not a quasicircle, the unbounded component of is a John domain, and the bounded component of is not a John domain. Moreover, in [37, 36], we find many examples with this phenomenon. Note that this phenomenon does not hold in the usual iteration dynamics of a single polynomial map with In the sequel [38, 42], we investigate the Markov process on associated with the random dynamics of polynomials and we consider the probability of tending to starting with the initial value Applying many results of this paper, it will be shown in [42] that if the associated polynomial semigroup is postcritically bounded and the Julia set is disconnected, then the function defined on has many interesting properties which are similar to those of the Cantor function. In fact, under certain conditions, is continuous on and varies precisely on the Julia set, of which Hausdorff dimension is strictly less than two. (For example, if we consider the random dynamics generated by two polynomials where , then is continuous on and varies precisely on the Julia set (Figure 1) of the semigroup generated by . See [38, 33].) Such a kind of “singular functions on the complex plane” appear very naturally in random dynamics of polynomials, and the results of this paper (for example, the results on the space of all connected components of a Julia set) are the keys to investigating that. (The above results have been announced in [33, 34, 39].)
Moreover, as illustrated before, it is very important for us to recall that the complex dynamics can be applied to describe some mathematical models. For example, the behavior of the population of a certain species can be described as the dynamical systems of a polynomial such that preserves the unit interval and the postcritical set in the plane is bounded. When one considers such a model, it is very natural to consider the random dynamics of polynomial with bounded postcritical set in the plane (see Example 1.4).
In the sequel [24], we give some further results on postcritically bounded polynomial semigroups, by using many results in this paper and [36, 37]. Moreover, in the sequel [35], we define a new kind of cohomology theory, in order to investigate the action of finitely generated semigroups (iterated function systems), and we apply it to the study of the dynamics of postcritically bounded finitely generated polynomial semigroups . In particular, by using this new cohomology theory, we can describe the space of connected components of Julia sets of , we can give some estimates on the cardinality of , and we can give a sufficient condition for the cardinality of the space of connected components of the Fatou set of to be infinity. In [38, 40, 41], we investigate the random complex dynamics and the dynamics of transition operator, by developing the theory of random complex dynamics and that of dynamics of rational semigroups, simultaneously. It is shown that regarding the random dynamics of complex polynomials, generically the chaos of the averaged system disappears due to the cooperation of the generators, even though each map itself in the system has a chaotic part. We call this phenomenon “cooperation principle”. Moreover, we see that under certain conditions, in the limit state, complex analogues of singular functions (continuous functions on which vary only on the Julia set of associated rational semigroup ) naturally appear. The above function is a typical example of this complex analogue of singular function.
Acknowledgement: The author thanks R. Stankewitz for many valuable comments.
2 Main results
In this section we present the statements of the main results. Throughout this paper, we deal with semigroups that might not be generated by a compact family of polynomials. The proofs are given in Section 4.
2.1 Space of connected components of a Julia set, surrounding order
We present some results concerning the connected components of the Julia set of a postcritically bounded polynomial semigroup. The proofs are given in Section 4.1.
The following theorem generalizes [47, Theorem 1].
Theorem 2.1.
Let be a rational semigroup generated by a family Suppose that there exists a connected component of such that and Moreover, suppose that for any such that is a Möbius transformation of finite order, we have Then, is connected.
Definition 2.2.
We set Rat : endowed with the topology induced by uniform convergence on with respect to the spherical distance. We set Poly : endowed with the relative topology from Rat. Moreover, we set Poly endowed with the relative topology from Rat.
Remark 2.3.
Let , a sequence of polynomials of degree , and a polynomial. Then, in Poly if and only if the coefficients converge appropriately and is of degree
Definition 2.4.
Let be the set of all polynomial semigroups with the following properties:

each element of is of degree at least two, and

is bounded in , i.e., is postcritically bounded.
Furthermore, we set and
Notation: For a polynomial semigroup , we denote by the set of all connected components of such that Moreover, we denote by the set of all connected components of
Remark 2.5.
If a polynomial semigroup is generated by a compact set in Poly, then and thus
Definition 2.6.
For any connected sets and in “” indicates that , or is included in a bounded component of Furthermore, “” indicates and Note that “” is a partial order in the space of all nonempty compact connected sets in This “” is called the surrounding order.
Theorem 2.7.
Let (possibly generated by a noncompact family). Then we have all of the following.

is totally ordered.

Each connected component of is either simply or doubly connected.

For any and any connected component of , we have that is connected. Let be the connected component of containing If , then If and then and
2.2 Upper estimates of
Next, we present some results on the space and some results on upper estimates of The proofs are given in Section 4.2 and Section 4.3.
Definition 2.8.

For a polynomial , we denote by the coefficient of the highest degree term of

We set RA endowed with the topology such that, if and only if and The space RA is a semigroup with the semigroup operation being functional composition. Any subsemigroup of RA will be called a real affine semigroup. We define a map Poly RA as follows: For a polynomial Poly, we set
Moreover, for a polynomial semigroup , we set ( RA).

We set endowed with the topology such that makes a fundamental neighborhood system of , and such that makes a fundamental neighborhood system of For a real affine semigroup , we set
where the closure is taken in the space Moreover, we denote by the set of all connected components of

We denote by RA Poly the natural embedding defined by , where and

We define a map Poly Poly as follows. For a polynomial , we set Moreover, for a polynomial semigroup , we set
Remark 2.9.

The map Poly RA is a semigroup homomorphism. That is, we have Hence, for a polynomial semigroup , the image is a real affine semigroup. Similarly, the map Poly Poly is a semigroup homomorphism. Hence, for a polynomial semigroup , the image is a polynomial semigroup.

The maps Poly RA, RA Poly, and Poly Poly are continuous.
Definition 2.10.
For any connected sets and in , “” indicates that , or each satisfies Furthermore, “” indicates and
Remark 2.11.
The above “” is a partial order in the space of nonempty connected subsets of Moreover, for each real affine semigroup , is totally ordered.
The following theorem gives us some upper estimates of
Theorem 2.12.

Let be a polynomial semigroup in Then, we have More precisely, there exists an injective map such that if and , then

If , then we have that and

Let be a polynomial semigroup in Then,
Corollary 2.13.
Let be a polynomial semigroup in Then, we have More precisely, there exists an injective map such that if and , then , , and
The following three theorems give us sufficient conditions for the Julia set of a to be connected.
Theorem 2.14.
Let be a finitely generated polynomial semigroup in For each , let be the coefficient of the highest degree term of polynomial Let and We set If , then is connected.
Theorem 2.15.
Let be a polynomial semigroup in generated by a (possibly noncompact) family of polynomials of degree two. Then, is connected.
Theorem 2.16.
Let be a polynomial semigroup in generated by a (possibly noncompact) family of polynomials. Let be the coefficient of the highest degree term of the polynomial Suppose that for any , we have Then, is connected.
Remark 2.17.
In [35], a new cohomology theory for (backward) selfsimilar systems (iterated function systems) was introduced by the author of this paper. By using this new cohomology theory, for a postcritically bounded finitely generated polynomial semigroup , we can describe the space of connected components of and we can give some estimates on and
2.3 Properties of
In this section, we present some results on The proofs are given in Section 4.3.
Definition 2.18.
For a polynomial semigroup , we set
and call the smallest filledin Julia set of For a polynomial , we set
Notation: For a set , we denote by int the set of all interior points of
Proposition 2.19.
Let If is a connected component of such that , then int and is simply connected. Furthermore, we have int
Notation: For a polynomial semigroup with , we denote by the connected component of containing Moreover, for a polynomial with , we set
The following theorem is the key to obtaining further results of postcritically bounded polynomial semigroups in this paper, and those of related random dynamics of polynomials in the sequel [36, 42]. We remark that Theorem 2.205 generalizes [47, Theorem 2].
Theorem 2.20.
Let (possibly generated by a noncompact family). Then, under the above notation, we have the following.

We have that (thus ) and the connected component of containing is simply connected. Furthermore, the element containing is the unique element of satisfying that for each

There exists a unique element such that for each element Furthermore, let be the unbounded component of Then, and

If is generated by a family then there exist two elements and of satisfying:

there exist two elements and of with such that for each ;

;

for each , we have and ; and

has an attracting fixed point in , int consists of only one immediate attracting basin for , and int Furthermore, int


For each with , we have that has an attracting fixed point in , int consists of only one immediate attracting basin for , and int Note that it is not necessarily true that when are such that and (see Proposition 2.26).

We have that Moreover,

is disconnected, for each , and

for each with , we have that , , int, and the unique attracting fixed point of in belongs to


Let be the set of all doubly connected components of Then, and is totally ordered.
We present a result on uniform perfectness of the Julia sets of semigroups in
Definition 2.21.
A compact set in is said to be uniformly perfect if and there exists a constant such that each annulus that separates satisfies that mod , where mod denotes the modulus of (See the definition in [15]).
Theorem 2.22.

Let be a polynomial semigroup in Then, is uniformly perfect. Moreover, if is a superattracting fixed point of an element of , then int

If and , then and int

Suppose that Let be a superattracting fixed point of Then int and
We remark that in [14], it was shown that there exists a rational semigroup such that is not uniformly perfect.
We now present results on the Julia sets of subsemigroups of an element of
Proposition 2.23.
Let and let with Let be the unbounded component of for each Then, we have the following.

Let and let be the subsemigroup of generated by Then

Let and let be the subsemigroup of generated by Then

Let and let be the subsemigroup of generated by Then
Proposition 2.24.
Let be a polynomial semigroup generated by a compact subset of Poly Suppose that Then, there exists an element with and there exists an element with
2.4 Finitely generated polynomial semigroups such that
In this section, we present some results on various finitely generated polynomial semigroups such that The proofs are given in Section 4.4.
It is wellknown that for a rational map with , if is disconnected, then has uncountably many connected components (See [16]). Moreover, if is a nonelementary Kleinian group with disconnected Julia set (limit set), then has uncountably many connected components. However, for general rational semigroups, we have the following examples.
Theorem 2.25.
Let be a polynomial semigroup in generated by a (possibly noncompact) family in Poly Suppose that there exist mutually distinct elements such that for each and each , there exists an element with Then, we have
Proposition 2.26.
For any with , there exists a finitely generated polynomial semigroup in satisfying In fact, let and we set for each and Then, for any sufficiently large , there exists an open neighborhood of in (Poly) such that for any , the semigroup satisfies that and
Theorem 2.27.
Let be a polynomial semigroup with Suppose that there exists an element such that , and such that for each we have Then, we have all of the following.


, or

If , then , , and for any with , there exists no sequence of mutually distinct elements of such that as

If , then , , and for any with , there exists no sequence of mutually distinct elements of such that as
Proposition 2.28.
There exists an open set in (Poly such that for any , satisfies that , , , for each , and
Proposition 2.29.
There exists a generator polynomial semigroup in such that , , , there exists a superattracting fixed point of some element of with , and int
As mentioned before, these results illustrate new phenomena which can hold in the rational semigroups, but cannot hold in the dynamics of a single rational map or Kleinian groups.
For the figure of the Julia set of a generator polynomial semigroup such that , see figure 2.
Remark 2.30.
In [35], a new cohomology theory for (backward) selfsimilar systems (iterated function systems) was introduced by the author of this paper. By using it, for a finitely generated , we can describe the space of connected components of , and we can give some estimates on . Moreover, by using this new cohomology, a sufficient condition for the cardinality of the set of all connected components of the Fatou set of a postcritically bounded finitely generated polynomial semigroup to be infinity was given.
2.5 Hyperbolicity and semihyperbolicity
In this section, we present some results on hyperbolicity and semihyperbolicity.
Definition 2.31.
Remark 2.32.
Let be a polynomial semigroup generated by a compact subset of Poly Suppose Then, by Proposition 2.24, we have and Moreover, is a compact subset of For, if and in , then for a repelling periodic point of , we have that as , which implies that and thus
The following Proposition 2.33 means that for a polynomial semigroup generated by a compact subset of Poly, we rarely have the situation that “ is not compact.”
Proposition 2.33.
Let be a polynomial semigroup generated by a nonempty compact subset of Poly Suppose that and that is not compact. Then, both of the following statements 1 and 2 hold.

Let Then, , and int is a nonempty connected set.

Either

for each , is hyperbolic and is a quasicircle; or

for each , int is an immediate parabolic basin of a parabolic fixed point of

Definition 2.34.
Let be a rational semigroup.

We say that is hyperbolic if

We say that is semihyperbolic if there exists a number and a number such that for each and each , we have for each connected component of , where denotes the ball of radius with center with respect to the spherical distance, and denotes the degree of a finite branched covering.
Remark 2.35.
There are many nice properties of hyperbolic or semihyperbolic rational semigroups. For example, for a finitely generated semihyperbolic rational semigroup , there exists an attractor in the Fatou set ([27, 30]), and the Hausdorff dimension of the Julia set is less than or equal to the critical exponent of the Poincaré series of ([30]). If we assume further the “open set condition”, then ([32, 45]). Moreover, if is generated by a compact set and if is semihyperbolic, then for each sequence , the basin of infinity for is a John domain and the Julia set of is locally connected ([30]). In [37], by using the above result, we classify hyperbolic or semihyperbolic postcritically bounded compactly generated polynomial semigroups, in terms of the random complex dynamics. It is shown that in one of the classes, for almost every sequence , the Julia set of is a Jordan curve but not a quasicircle, the unbounded component of is a John domain, and the bounded component of is not a John domain. Moreover, in [37, 36], we find many examples with this phenomenon. Note that this phenomenon does not hold in the usual iteration dynamics of a single polynomial map with
We now present some results on semihyperbolic or hyperbolic polynomial semigroups in These results are used to construct examples of semihyperbolic or hyperbolic polynomial semigroups (see the proof of Proposition 2.40). Therefore these are important in terms of the sequel [36, 37].
Theorem 2.36.
Let be a polynomial semigroup generated by a nonempty compact subset of Poly Suppose that If is semihyperbolic, then is semihyperbolic.
Theorem 2.37.
Let be a polynomial semigroup generated by a nonempty compact subset of Poly Suppose that If is hyperbolic and , then is hyperbolic.
Remark 2.38.
Proposition 2.39.
Let be a polynomial semigroup generated by a nonempty compact subset of Poly Suppose that and that is not compact. Suppose that statement 2a in Theorem 2.33 holds. Then, both of the following statements hold.

We have that is hyperbolic and is semihyperbolic.

Suppose further that . Then is hyperbolic.
2.6 Construction of examples
In this section, we present a way to construct examples of semigroups in (with some additional properties). These examples are important in terms of the sequel [36, 37].
Proposition 2.40.
Let be a polynomial semigroup generated by a compact subset of Poly Suppose that and int Let int Moreover, let be any positive integer such that , and such that for each Then, there exists a number such that for each with , there exists a compact neighborhood of in Poly satisfying that for any nonempty subset of