Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets ^{1}^{1}1Date: November 29, 2013. Published in J. London Math. Soc. (2) 88 (2013) 294–318. 2010 Mathematics Subject Classification. 37F10, 37C60. This work was partially supported by JSPS GrantinAid for Scientific Research (C) 21540216. Keywords: Polynomial semigroups, random complex dynamics, random iteration, skew product, Julia sets, fiberwise Julia sets.
Abstract
We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with bounded planar postcritical set. Using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that if the Julia set of such a semigroup is disconnected, then there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation which are parameterized by the Cantor set, densely inside the Julia set of the semigroup. Moreover, we give a sufficient condition for a fiberwise Julia set to satisfy that 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. We show that under certain conditions, a random Julia set is almost surely a Jordan curve, but not a quasicircle. Many new phenomena of polynomial semigroups and random dynamics of polynomials 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 [14].
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 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 these subjects. 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 ([11]). 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 ([11]), 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 and Z. Gong ([10]) and others, 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. [22, 23]). [17] is a very nice (and short) article for an introduction to the dynamics of rational semigroups. For other research on rational semigroups, see [37, 18, 19, 35, 36], and [21]–[33].
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 ([8]), and much research has followed. (See [2, 4, 5, 3, 9, 27, 30, 31, 32, 33, 34].)
We remark that 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. [7]). From this point of view, it is very important to consider the random dynamics of such polynomials (see also Example 1.4). The results of this paper might have applications to mathematical models. For the random dynamics of polynomials on the unit interval, see [20].
We shall give some definitions for the dynamics of rational semigroups:
Definition 1.1 ([11, 10]).
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 For 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 Using this formula, one can understand how the set (resp. ) spreads in (resp. ). In fact, in Section 3.4, 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 ([14, Theorem 9.5]).
As mentioned in Remark 1.5, the planar postcritical set is one piece of important information regarding the dynamics of polynomials.
When investigating the dynamics of polynomial semigroups, it is natural for us to discuss the relationship between the planar postcritical set and the Julia set. The first question in this regard is: “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. In fact, in [37, 29, 30, 19, 31, 32], we find many examples of postcritically bounded polynomial semigroups with disconnected Julia set such that for each , Thus, it is natural to ask the following questions.
Problem 1.6.
(1) What properties does have if is bounded in and is disconnected? (2) Can we classify postcritically bounded polynomial semigroups?
Applying the results in [29, 30], we investigate the dynamics of every sequence, or fiberwise dynamics of the skew product associated with the generator system (cf. Section 3.1). Moreover, we investigate the random dynamics of polynomials acting on the Riemann sphere. Let us consider a polynomial semigroup generated by a compact family of polynomials. For each sequence , we examine the dynamics along the sequence , that is, the dynamics of the family of maps . We note that this corresponds to the fiberwise dynamics of the skew product (see Section 3.1) associated with the generator system We show that if is postcritically bounded, is disconnected, and is generated by a compact family of polynomials, then, for almost every sequence , there exists exactly one bounded component of the Fatou set of , the Julia set of has Lebesgue measure zero, there exists no nonconstant limit function in for the sequence , and for any point the orbit of along tends to the interior of the smallest filledin Julia set (see Definition 2.7) of (cf. Theorem 3.11, Corollary 3.21). Moreover, using uniform fiberwise quasiconformal surgery ([30]), we find subskew products such that is hyperbolic (see Definition 3.10) and such that every fiberwise Julia set of is a quasicircle, where is a constant not depending on the fibers (cf. Theorem 3.11, statement 3). Reusing the uniform fiberwise quasiconformal surgery, we show that if is a postcritically bounded polynomial semigroup with disconnected Julia set, then for any nonempty open subset of , there exists a generator subsemigroup of such that is the disjoint union of a “Cantor family of quasicircles” (a family of quasicircles parameterized by a Cantor set) with uniform distortion, and such that (cf. Theorem 3.14). Note that the uniform fiberwise quasiconformal surgery is based on solving uncountably many Beltrami equations.
We also investigate (semi)hyperbolic (see Definition 3.12), postcritically bounded, polynomial semigroups generated by a compact family of polynomials. Let be such a semigroup with disconnected Julia set, and suppose that there exists an element such that is not a Jordan curve. Then, we give a (concrete) sufficient condition for a sequence to give rise to the following situation (): the Julia set of is a Jordan curve but not a quasicircle, the basin of infinity is a John domain, and the bounded component of the Fatou set is not a John domain (cf. Theorem 3.18, Corollary 3.22). From this result, we show that for almost every sequence , situation holds. In fact, in this paper, under the above assumption, we find a set of with which is much larger than a set of with given in [30]. Moreover, we classify hyperbolic twogenerator postcritically bounded polynomial semigroups with disconnected Julia set and we also completely classify the fiberwise Julia sets in terms of the information of (Theorem 3.19). Note that situation cannot hold in the usual iteration dynamics of a single polynomial map with (Remark 3.23).
The key to investigating the dynamics of postcritically bounded polynomial semigroups is the density of repelling fixed points in the Julia set ([11, 10]), which can be shown by an application of the Ahlfors five island theorem, and the lower semicontinuity of ([12]), which is a consequence of potential theory. Moreover, one of the keys to investigating the fiberwise dynamics of skew products is, the observation of nonconstant limit functions (cf. Lemma 5.4 and [23]). The key to investigating the dynamics of semihyperbolic polynomial semigroups is, the continuity of the map (this is highly nontrivial; see [23]) and the Johnness of the basin of infinity (cf. [25]). Note that the continuity of the map does not hold in general, if we do not assume semihyperbolicity. Moreover, one of the original aspects of this paper is the idea of “combining both the theory of rational semigroups and that of random complex dynamics”. It is quite natural to investigate both fields simultaneously. However, no study (except the works of the author of this paper) thus far has done so.
Furthermore, in Section 3.4 and [29, 30], we provide a way of constructing examples of postcritically bounded polynomial semigroups with some additional properties (disconnectedness of the Julia set, semihyperbolicity, hyperbolicity, etc.) (cf. Proposition 3.24, [29, 30]). For example, by Proposition 3.24, 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, Section 3.4, and [29, 30], it is not difficult to construct many examples for which we can verify the hypothesis “postcritically bounded”, so the class of postcritically bounded polynomial semigroups is very wide.
Throughout the paper, we will see many new phenomena in polynomial semigroups or random dynamics of polynomials that do not occur in the usual dynamics of polynomials. Moreover, these new phenomena are systematically investigated.
In Section 3, we present the main results of this paper. We give some tools in Section 4. The proofs of the main results are given in Section 5.
There are many applications of the results of postcritically bounded polynomial semigroups in many directions. In the sequel [31, 27, 33, 34], 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 [29], it will be shown in [34] that if the associated polynomial semigroup is postcritically bounded and the Julia set is disconnected, then the chaos of the averaged system disappears due to the cooperation of generators (cooperation principle), and the function defined on has many interesting properties which are similar to those of the devil’s staircase (the Cantor function). Such “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 them. (The above results have been announced in [31, 27, 26, 32].)
In [29], we find many fundamental and useful results on the connected components of Julia sets of postcritically bounded polynomial semigroups. In [30], we classify (semi)hyperbolic, postcritically bounded, compactly generated polynomial semigroups. In the sequel [19], we give some further results on postcritically bounded polynomial semigroups, by using many results in [29, 30], and this paper. Moreover, in the sequel [28], 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 polynomial semigroups.
2 Preliminaries
In this section we give some basic notations and definitions, and we present some results in [29, 30], which we need to state the main results of this paper.
Definition 2.1.
We set Rat : endowed with distance defined as , where denotes the spherical distance on We set Poly : endowed with the relative topology from Rat. Moreover, we set Poly endowed with the relative topology from Rat.
Remark 2.2.
Let , be a sequence of polynomials of degree , and be a polynomial. Then in Poly if and only if is of degree and the coefficients of converge appropriately.
Definition 2.3.
Let be the set of all postcritically bounded polynomial semigroups such that each element of is of degree at least two. Furthermore, we set and
Definition 2.4.
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 ([29]).
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.
Definition 2.7 ([29]).
For a polynomial semigroup , we set
and call the smallest filledin Julia set of For a polynomial , we set For a set , we denote by int the set of all interior points of For a polynomial semigroup with , we denote by the connected component of containing Moreover, for a polynomial with , we set
The following three results in [29] are needed to state the main result in this paper.
Theorem 2.8 ([29]).
Let (possibly generated by a noncompact family). Then we have all of the following.

We have 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
Theorem 2.9 ([29]).
Let (possibly generated by a noncompact family). Then we have all of the following.

We have . Thus

The 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

We have that
For the figures of the Julia sets of semigroups , see Figure 1.
Proposition 2.10 ([29]).
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
3 Main results
In this section, we present the main results of this paper. The proofs of the results are given in Section 5.
3.1 Fiberwise dynamics and Julia sets
We present some results on the fiberwise dynamics of the skew product related to a postcritically bounded polynomial semigroup with disconnected Julia set. In particular, using the uniform fiberwise quasiconformal surgery on a fiber bundle, we show the existence of families of quasicircles with uniform distortion parameterized by the Cantor set densely inside the Julia set of such a semigroup. The proofs are given in Section 5.1.
Definition 3.1 ([23, 25]).

Let be a compact metric space, a continuous map, and a continuous map. We say that is a rational skew product (or fibered rational map on the trivial bundle ) over , if where denotes the canonical projection, and if, for each , the restriction of is a nonconstant rational map, under the canonical identification for each Let , for each Let be the rational map defined by: , for each and , where is the projection map.
Moreover, if is a polynomial for each , then we say that is a polynomial skew product over

Let be a compact subset of Rat. We set endowed with the product topology. This is a compact metric space. Let be the shift map, which is defined by Moreover, we define a map by: where This is called the skew product associated with the family of rational maps. Note that
Remark 3.2.
Regarding item 1 of Definition 3.1, the map is equal to the rational map under the canonical identification for each Thus, if we consider the dynamics of , then we can investigate the dynamics of all sequences generated by the family and the map
Remark 3.3.
Let be a rational skew product over Then the function is continuous in For, since is continuous, the map is continuous. Moreover, the function is continuous ([1, Theorem 2.8.2]). Thus, is continuous.
Definition 3.4 ([23, 25]).
Let be a rational skew product over Then, for each and , we set For each , we denote by the set of points which has a neighborhood in such that is normal. Moreover, we set We set Moreover, we set These sets and are called the fiberwise Julia sets. Moreover, we set , where the closure is taken in the product space For each , we set Moreover, we set We set
Remark 3.5 ([23, 25]).

We have , , , and However, for the last one, strict containment can occur. For example, let be a polynomial having a Siegel disk with center Let be a polynomial such that is a repelling fixed point of Let Let be the skew product associated with the family Let Then, and

Let be a compact subset of Rat and let be the skew product associated with Let be the rational semigroup generated by (thus ). If , then ([30, Lemma 3.5]). From this result, we can apply the results of the dynamics of to the dynamics of
Definition 3.6 ([30]).
Let be a polynomial skew product over Then for each , we set in , and Moreover, we set and
Definition 3.7 ([29]).
Remark 3.8.
Let be a polynomial semigroup generated by a compact subset of Poly Suppose Then, by Proposition 2.10, we have and Moreover, is a compact subset of For, if and in , then for each repelling periodic point of , we have that as , which implies that and thus
Notation: Let be a sequence of meromorphic functions in a domain We say that a meromorphic function is a limit function of if there exists a strictly increasing sequence of positive integers such that locally uniformly on , as
Definition 3.9.
Let and be nonempty subsets of Poly with We set
Definition 3.10.
Let be a rational skew product over We set
Moreover, we set where the closure is taken in the product space This is called the fiberpostcritical set of
We say that is hyperbolic (along fibers) if
We present a result which describes the details of the fiberwise dynamics along in We recall that a Jordan curve in is said to be a quasicircle, if is the image of under a quasiconformal homeomorphism (For the definition of a quasicircle and a quasiconformal homeomorphism, see [13].)
Theorem 3.11.
Let be a polynomial semigroup generated by a compact subset of Poly Suppose Let be the skew product associated with the family of polynomials. Then, all of the following statements 1,2, and 3 hold.

Let Then, each limit function of in each connected component of is constant.

Let be a nonempty compact subset of Then, for each , we have the following.

There exists exactly one bounded component of Furthermore,

For each , there exists a number such that int

Moreover, the map defined on is continuous at , with respect to the Hausdorff metric in the space of nonempty compact subsets of

The 2dimensional Lebesgue measure of is equal to zero.


Let be a nonempty compact subset of For each we denote by the set of elements such that for each , at least one of belongs to Let Then, is a hyperbolic skew product over the shift map , and there exists a constant such that for each is a quasicircle.
Definition 3.12.
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 finite branched covering. (For background on semihyperbolicity, see [23] and [25].)
Theorem 3.13.
Let be a polynomial semigroup generated by a compact subset of Poly Let be the skew product associated with the family Suppose that and that is semihyperbolic. Let be any element. Then, and is a Jordan curve. Moreover, for each point int, there exists an such that int
We next present a result which states that there exist families of uncountably many mutually disjoint quasicircles with uniform distortion, densely inside the Julia set of a semigroup in
Theorem 3.14.
(Existence of a Cantor family of quasicircles.) Let (possibly generated by a noncompact family) and let be an open subset of with Then, there exist elements and in such that all of the following hold.

satisfies that

There exists a nonempty open set in such that , and such that

is a hyperbolic polynomial semigroup.

Let be the skew product associated with the family of polynomials. Then, we have the following.

(disjoint union). Each is connected and is totally ordered.

For each connected component of , there exists an element such that

There exists a constant independent of such that each connected component of is a quasicircle.

The map , defined for all , is continuous with respect to the Hausdorff metric in the space of nonempty compact subsets of , and injective.

For each element

Let be an element such that and such that Then, does not meet the boundary of any connected component of

Remark 3.15.
This “Cantor family of quasicircles” in the research of rational semigroups was introduced by the author of this paper. By using this idea, in [19] (which was written after this paper), it is shown that for a polynomial semigroup which is generated by a (possibly noncompact) family of Poly, if and are two different doubly connected components of , then there exists a Cantor family of quasicircles in such that each element of separates and . In Theorem 3.14 of this paper, we show that there exist Cantor families of quasicircles densely inside the Julia set of a semigroup , which is of independent value.
3.2 Fiberwise Julia sets that are Jordan curves but not quasicircles
We present a result on a sufficient condition for a fiberwise Julia set to satisfy that 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. Note that we have many examples of this phenomenon (see Proposition 3.24,Remark 3.25,Example 3.27), and note also that this phenomenon cannot hold in the usual iteration dynamics of a single polynomial map with (see Remark 3.23). The proofs are given in Section 5.2.
Definition 3.16.
Let be a subdomain of such that We say that is a John domain if there exists a constant and a point ( when ) satisfying the following: for all there exists an arc connecting to such that for any , we have
Remark 3.17.
Theorem 3.18.
Let be a polynomial semigroup generated by a compact subset of Poly Suppose that Let be the skew product associated with the family of polynomials. Let and suppose that there exists an element such that is not a quasicircle. Let be the element such that for each with , Then, the following statements 1 and 2 hold.

Suppose that is hyperbolic. Let be an element such that there exists a sequence of positive integers satisfying that as Then, is a Jordan curve but not a quasicircle. Moreover, the unbounded component of is a John domain, but the unique bounded component of is not a John domain.

Suppose that is semihyperbolic. Let be any element and let Let be an element such that there exists a sequence of positive integers satisfying that as Then, is a Jordan curve but not a quasicircle. Moreover, the unbounded component of is a John domain, but the unique bounded component of is not a John domain.
We now classify hyperbolic twogenerator polynomial semigroups in Moreover, we completely classify the fiberwise Julia sets in terms of the information on
Theorem 3.19.
Let Let Suppose and that is hyperbolic. Let be the skew product associated with Then, for each connected component of , there exists a unique such that Moreover, exactly one of the following statements 1, 2 holds.

There exists a constant such that for each , is a quasicircle.

There exists a unique such that is not a Jordan curve. In this case, for each , exactly one of the following statements (a),(b), (c) holds.

There exists a such that for each , at least one of is not equal to Moreover, is a quasicircle.

and there exists a strictly increasing sequence in such that as Moreover, 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.

There exists an such that . Moreover, is not a Jordan curve.

3.3 Random dynamics of polynomials
In this section, we present some results on the random dynamics of polynomials. The proofs are given in Section 5.3.
Let be a Borel probability measure on Poly
We consider the i.i.d. random dynamics on such that,
at every step, we choose a polynomial map
according to the distribution
(Hence, this defines a kind of Markov process on such that,
at every step, the transition probability
from a point to a Borel subset of is equal to .)
Notation:
For a Borel probability measure on Poly,
we denote by the topological support of on
Poly (Hence, is a closed set in
Poly)
Moreover, we denote by the infinite product measure
This is a Borel probability measure on
Definition 3.20.
Let be a complete metric space. A subset of is said to be residual if is a countable union of nowhere dense subsets of Note that by Baire Category Theorem, a residual set is dense in
Corollary 3.21.
(Corollary of Theorem 3.112) Let be a nonempty compact subset of Poly Let be the skew product associated with the family of polynomials. Let be the polynomial semigroup generated by Suppose Then, there exists a residual subset of such that for each Borel probability measure on Poly with , we have , and such that each satisfies all of the following.

There exists exactly one bounded component of Furthermore,

Each limit function of in is constant. Moreover, for each , there exists a number such that int

We have Moreover, the map