Dynamics of postcritically bounded polynomial semigroups III: classification of semihyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles ^{1}^{1}1Published in Ergodic Theory Dynam. Systems. (2010), 30, No. 6, 1869–1902. 2000 Mathematics Subject Classification. 37F10, 30D05. Keywords: Complex dynamics, polynomial semigroup, rational semigroup, Random complex dynamics, Julia set.
Abstract
We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere ) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups such that is generated by a compact family , the planar postcritical set of is bounded, and is (semi) hyperbolic. In one of the classes, we have that 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. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups such that the planar postcritical set of is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.
1 Introduction
This is the third paper in which the dynamics of semigroups of polynomial maps with bounded planar postcritical set in are investigated. This paper is selfcontained and the proofs of the results of this paper are independent from the results in [37, 38].
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].
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 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 ([12]). 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 ([12, 13]), 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([45, 11]), 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 [20, 21, 22, 44, 23, 24, 42, 41, 43], and [27]–[39].
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 ([9]), and much research has followed. (See [1, 3, 4, 2, 10, 34, 39].)
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 ([12, 11]).
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 More generally, for a family of nonconstant rational maps, we denote by the rational semigroup generated by The Julia set of the semigroup generated by a single map is denoted by Similarly, we set
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 Thus for each From this formula, one can figure out how the set (resp. ) spreads in (resp. ). In fact, in Section 5, 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 two or more. Is necessarily connected when is bounded in ?
The answer is NO.
Example 1.7 ([44]).
Let Then (which is bounded in ) and is disconnected ( is a Cantor set of round circles). Furthermore, according to [32, 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.
Definition 1.10.
Let be the set of all polynomial semigroups with the following properties:

each element of is of degree two or more, and

is bounded in , i.e., is postcritically bounded.
Furthermore, we set and
We also investigate the dynamics of hyperbolic or semihyperbolic polynomial semigroups.
Definition 1.11.
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 the background of semihyperbolicity, see [27] and [30].)
Remark 1.12.
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 ([27]). If we assume further the “open set condition”, then ([33, 43]). 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 connected and locally connected ([30]). This fact will be used in the proofs of the main results of this paper.
In this paper, we classify the semihyperbolic, postcritically bounded, polynomial semigroups generated by a compact family of polynomials. We show that such a semigroup satisfies either (I) every fiberwise Julia set is a quasicircle with uniform distortion, or (II) for almost every sequence , the Julia set 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, or (III) for every , the intersection of the Julia sets and is not empty, and is arcwise connected (cf. Theorem 2.19). Furthermore, we also classify the hyperbolic, postcritically bounded, polynomial semigroups generated by a compact family of polynomials. We show that such a semigroup satisfies either (I) above, or (II) above, or (III)’: for every , the intersection of the Julia sets and is not empty, is arcwise connected, and for every sequence , there exist infinitely many bounded components of (cf. Theorem 2.21). We give some examples of situation (II) above (cf. Example 2.22, figure 1, Example 2.23, and Section 5). Note that situation (II) above is a special phenomenon of random dynamics of polynomials that does not occur in the usual dynamics of polynomials.
The key to investigating the dynamics of postcritically bounded polynomial semigroups is the density of repelling fixed points in the Julia set (cf. Theorem 3.2), which can be shown by an application of the Ahlfors five island theorem, and the lower semicontinuity of (Lemma 3.42), which is a consequence of potential theory. The key to investigating the dynamics of semihyperbolic polynomial semigroups is, the continuity of the map (this is highly nontrivial; see [27]) and the Johnness of the basin of infinity (cf. [30]). 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 thus far has done so.
Furthermore, in Section 5, we provide a way of constructing examples of postcritically bounded polynomial semigroups with some additional properties (disconnectedness of Julia set, semihyperbolicity, hyperbolicity, etc.) (cf. Lemma 5.1, 5.2, 5.4, 5.5, 5.6). By using this, we will see how easily situation (II) above occurs, and we obtain many examples of situation (II) above.
As wee see in Example 1.4 and Section 5, 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 some phenomena in polynomial semigroups or random dynamics of polynomials that do not occur in the usual dynamics of polynomials. Moreover, those phenomena and their mechanisms 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. In Section 5, we present many examples.
There are many applications of the results of postcritically bounded polynomial semigroups in many directions. In subsequent papers [39, 40], we will investigate Markov process on associated with the random dynamics of polynomials and we will consider the probability of tending to starting with the initial value It will be shown in [39, 40] 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. For example, under certain conditions, is continuous on , varies precisely on which is a thin fractal set, and has a kind of monotonicity. Such a kind of “singular functions on the complex plane” appear very naturally in random dynamics of polynomials and the study of the dynamics of postcritically polynomial semigroups are the keys to investigating that. (The above results have been announced in [34, 35].)
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 polynomials with bounded postcritical set in the plane (see Example 1.4).
In [37], we investigate the dynamics of postcritically bounded polynomial semigroups which is possibly generated by a noncompact family. The structure of the Julia set is deeply studied, and for such a with disconnected Julia set, it is shown that , and that if and are two connected components of , then one of them surrounds the other. Therefore the space of all connected components of has an intrinsic total order. Moreover, we show that for each , there exists a finitely generated postcritically bounded polynomial semigroup such that the cardinality of the space of all connected components of is equal to In [38], by using the results in [37], we investigate the fiberwise (random) dynamics of polynomials which are associated with a postcritically bounded polynomial semigroup We will present some sufficient conditions for a fiberwise Julia set to be a Jordan curve but not a quasicircle. Moreover, we will investigate the limit functions of the fiberwise dynamics. In the subsequent paper [24], we will give some further results on postcritically bounded polynomial semigroups, based on [37] and this paper. Moreover, in the subsequent paper [36], we will define a new kind of cohomology theory, in order to investigate the action of finitely generated semigroups, and we will apply it to the study of the dynamics of postcritically bounded polynomial semigroups.
Acknowledgement: The author thanks R. Stankewitz for many valuable comments.
2 Main results
In this section we present the statements of the main results. The proofs are given in Section 4. In order to present the main results, we need some notations and definitions.
Definition 2.1.
We set Rat : endowed with the distance which is defined by , 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 , 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 2.3.
For a polynomial semigroup , we set
and call the smallest filledin Julia set of For a polynomial , we set
Definition 2.4.
For a set , we denote by int the set of all interior points of
Definition 2.5 ([27, 30]).

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 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 2.6.
Let be a rational skew product over Then, the function is continuous on
Definition 2.7 ([27, 30]).
Let be a rational skew product over Then, we use the following notation.

For each and , we set

For each , we denote by the set of points which have a neighborhood in such that is normal. Moreover, we set

For each , we set Moreover, we set These sets and are called the fiberwise Julia sets.

We set , where the closure is taken in the product space

For each , we set Moreover, we set

We set
Remark 2.8.
We have and However, 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
Definition 2.9.
Let be a polynomial skew product over Then for each , we set in , and Moreover, we set and
Definition 2.10.
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
Definition 2.11 ([27]).
Let be a rational skew product over Let We say that a point belongs to if there exists a neighborhood of in and a positive number such that for any , any , any , and any connected component of , Moreover, we set We say that is semihyperbolic (along fibers) if
Remark 2.12.
Definition 2.13.
Let 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 [15].)
Definition 2.14.
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 (Note: in this paper, if we consider a John domain , we require that However, in the original notion of John domain, more general concept of John domains was given, without assuming ([17]).)
Remark 2.15.
Definition 2.16.
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
Definition 2.17.
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.
Let be a Borel probability measure on Poly We consider the independent and identically distributed (abbreviated by i.i.d.) random dynamics on such that at every step we choose a polynomial map according to the distribution (Hence, this is a kind of Markov process on )
Definition 2.18.
For a Borel probability measure on Poly, we denote by the topological support of in Poly (Hence, is a closed set in Poly) Moreover, we denote by the infinite product measure This is a Borel probability measure on Furthermore, we denote by the polynomial semigroup generated by
We present a result on compactly generated, semihyperbolic, polynomial semigroups in
Theorem 2.19.
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 that and that is semihyperbolic. Then, exactly one of the following three statements 1, 2, and 3 holds.

is hyperbolic. Moreover, there exists a constant such that for each , is a quasicircle.

There exists a residual Borel subset of such that, for each Borel probability measure on Poly with , we have , and such that, for each , is a Jordan curve but not a quasicircle, is a John domain, and the bounded component of is not a John domain. Moreover, there exists a dense subset of such that, for each , is not a Jordan curve. Furthermore, there exist two elements such that (Remark: by Lemma 3.6, for each , is connected.)

There exists a dense subset of such that for each , is not a Jordan curve. Moreover, for each , Furthermore, is arcwise connected.
Corollary 2.20.
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 that and that is semihyperbolic. Then, either statement 1 or statement 2 in Theorem 2.19 holds. In particular, for any Borel Probability measure on Poly with , for almost every with respect to , is a Jordan curve.
We now classify compactly generated, hyperbolic, polynomial semigroups in
Theorem 2.21.
Let be a nonempty compact subset of Poly Let be the skew product associated with the family Let be the polynomial semigroup generated by Suppose that and that is hyperbolic. Then, exactly one of the following three statements 1, 2, and 3 holds.

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

There exists a residual Borel subset of such that, for each Borel probability measure on Poly with , we have , and such that, for each , is a Jordan curve but not a quasicircle, is a John domain, and the bounded component of is not a John domain. Moreover, there exists a dense subset of such that, for each , is a quasicircle. Furthermore, there exists a dense subset of such that, for each , there are infinitely many bounded connected components of

For each , there are infinitely many bounded connected components of . Moreover, for each , Furthermore, is arcwise connected.
Example 2.22.
Let and Let Let be the skew product associated with Moreover, let be the polynomial semigroup generated by Let Then, it is easy to see Hence, Let be a small disk around Then and Therefore Moreover, by Remark 1.3, we have that Hence, and is hyperbolic. Furthermore, let Then, it is easy to see that and Combining this with Lemma 3.16 and Lemma 3.12, we obtain that is disconnected. Therefore, Let for each Let with Let Since is not a Jordan curve, from Theorem 2.21, it follows that for almost every with respect to , is a Jordan curve but not a quasicircle, and is a John domain but the bounded component of is not a John domain. (See figure 1: the Julia set of .) In this example, for each connected component of , there exists a unique such that
Example 2.23.
Let and , where with Let Moreover, let Let Then, it is easy to see that and Hence, It follows that int Therefore, and is hyperbolic. Since is not a Jordan curve and is a Jordan curve, Theorem 2.21 implies that there exists a residual subset of such that, for each Borel probability measure on Poly with , we have , and such that, for each , is a Jordan curve but not a quasicircle. Moreover, for each , is a John domain, but the bounded component of is not a John domain.
Remark 2.24.
Let Poly be a polynomial. Suppose that is a Jordan curve but not a quasicircle. Then, it is easy to see that there exists a parabolic fixed point of in and the bounded connected component of is the immediate parabolic basin. Hence, is not semihyperbolic. Moreover, by [5], is not a John domain.
Thus what we see in statement 2 in Theorem 2.19 and statement 2 in Theorem 2.21, as illustrated in Example 2.22 and Example 2.23 (see also Section 5), is a phenomenon which can hold in the random dynamics of a family of polynomials, but cannot hold in the usual iteration dynamics of a single polynomial. Namely, it can hold that for almost every , is a Jordan curve and fails to be a quasicircle all while the basin of infinity is still a John domain. Whereas, if , for some polynomial , is a Jordan curve which fails to be a quasicircle, then the basin of infinity is necessarily not a John domain.
In Section 5, we will see how easily situation 2 in Theorem 2.19 and situation 2 in Theorem 2.21 occur.
Pilgrim and Tan Lei ([18]) showed that there exists a hyperbolic rational map with disconnected Julia set such that “almost every” connected component of is a Jordan curve but not a quasicircle.
Proposition 2.25.
Let be a nonempty compact subset of Poly Let be the skew product associated with the family Let be the polynomial semigroup generated by Suppose that is included in a connected component of int Then, there exists a constant such that for each , is a quasicircle.
Example 2.26.
We have also many examples of such that statement 3 in Theorem 2.19 or statement 3 in Theorem 2.21 holds.
Example 2.27.
Let Suppose that and is hyperbolic. Suppose also that has at least two attracting periodic points in Let be a small compact neighborhood of in Poly Then and is hyperbolic (see Lemma 5.4). Moreover, by the argument in the proof of Lemma 5.6, we see that for each , has at least two bounded connected components, where is the skew product associated with . Thus statement 3 in Theorem 2.21 holds. We remark that by using Lemma 5.5, 5.6 and their proofs, we easily obtain many examples of such that statement 3 in Theorem 2.19 or statement 3 in Theorem 2.21 holds.
3 Tools
To show the main results, we need some tools in this section.
3.1 Fundamental properties of rational semigroups
Notation:
For a rational semigroup , we set
This is called the exceptional set of
Notation: Let
For a subset of , we set
, where is the spherical distance.
For a subset of , we set
, where is the Euclidean distance.
Lemma 3.1 ([12, 11, 29, 27]).
Let be a rational semigroup.

For each we have and Note that we do not have that the equality holds in general.

If , then More generally, if is generated by a compact subset of Rat, then (We call this property of the Julia set of a compactly generated rational semigroup “backward selfsimilarity.” )

If , then is a perfect set.

If , then

If a point is not in then In particular if a point belongs to then

If , then is the smallest closed backward invariant set containing at least three points. Here we say that a set is backward invariant under if for each
Theorem 3.2 ([12, 11, 29]).
Let be a rational semigroup.
If , then
, where
the closure is taken in
In particular,
Remark 3.3.
If a rational semigroup contains an element with , then , which implies that
3.2 Fundamental properties of fibered rational maps
Lemma 3.4.
Let be a rational skew product over Then, we have the following.

([27, Lemma 2.4]) For each , Furthermore, we have Note that equality does not hold in general.
If is a surjective and open map, then , and for each ,

If for each , then diam, where diam denotes the diameter with respect to the spherical distance.

If is a polynomial skew product and for each , then there exists a ball around such that for each , , and for each , Moreover, for each , is connected.

If is a polynomial skew product and