Transversality family of Expanding Rational Semigroups
Abstract.
We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than , then typically the dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give nontrivial examples of families of semigroups of nonlinear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than . We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each with , the family of small perturbations of the semigroup generated by satisfies that for a typical parameter value, the dimensional Lebesgue measure of the Julia set is positive.
Key words and phrases:
Complex dynamical systems, rational semigroups, expanding semigroups, Julia set, transversality condition, Hausdorff dimension, Bowen parameter, random complex dynamics, random iteration, iterated function systems with overlaps, selfsimilar setsHiroki Sumi
Department of Mathematics, Graduate School of Science, Osaka University, 11 Machikaneyama, Toyonaka, Osaka, 5600043, Japan
Email: sumi@math.sci.osakau.ac.jp
Web: http://www.math.sci.osakau.ac.jp/sumi/
Mariusz Urbański
Department of Mathematics, University of North Texas, Denton, TX 762031430, USA
Email: urbanski@unt.edu
Web: http://www.math.unt.edu/urbanski/
Mathematics Subject Classification (2001). Primary 37F35;
Secondary 37F15.
Date: January 8, 2013. Published in Adv. Math. 234 (2013) 697–734.
1. Introduction
A rational semigroup is a semigroup generated by a family of nonconstant rational maps , where denotes the Riemann sphere, with the semigroup operation being functional composition. A polynomial semigroup is a semigroup generated by a family of nonconstant polynomial maps on The work on the dynamics of rational semigroups was initiated by A. Hinkkanen and G. J. Martin ([8]), who were interested in the role of the dynamics of polynomial semigroups while studying various onecomplexdimensional moduli spaces for discrete groups of Möbius transformations, and by F. Ren’s group ([44]), who studied such semigroups from the perspective of random dynamical systems.
The theory of the dynamics of rational semigroups on has developed in many directions since the 1990s ([8, 44, 22, 24, 25, 26, 27, 28, 29, 30, 39, 31, 32, 23, 33, 34, 35, 36, 37]). We recommend [22] as an introductory article. For a rational semigroup , we denote by the maximal open subset of where is normal. The set is called the Fatou set of . The complement is called the Julia set of Since the Julia set of a rational semigroup generated by finitely many elements has backward selfsimilarity i.e.
(1.1) 
(see [24, 26]), rational semigroups can be viewed as a significant generalization and extension of both the theory of iteration of rational maps (see [14, 2]) and conformal iterated function systems (see [11]). Indeed, because of (1.1), the analysis of the Julia sets of rational semigroups somewhat resembles “backward iterated functions systems”, however since each map is not in general injective (critical points), some qualitatively different extra effort in the case of semigroups is needed. The theory of the dynamics of rational semigroups borrows and develops tools from both of these theories. It has also developed its own unique methods, notably the skew product approach (see [26, 27, 28, 29, 31, 38, 32, 34, 35, 36, 37, 40, 39, 41]).
The theory of the dynamics of rational semigroups is intimately related to that of the random dynamics of rational maps. The first study of random complex dynamics was given in [6]. In [3, 7], random dynamics of quadratic polynomials were investigated. The paper [12] develops the thermodynamic formalism of random distance expanding maps and, in particular, applies it to random polynomials. The deep relation between these fields (rational semigroups, random complex dynamics, and (backward) IFS) is explained in detail in the subsequent papers ([30, 31, 38, 32, 33, 34, 35, 36, 37]) of the first author. For a random dynamical system generated by a family of polynomial maps on , let be the function of probability of tending to In [34, 36, 37] it was shown that under certain conditions, is continuous on and varies only on the Julia set of the associated rational semigroup (further results were announced in [35]). For example, for a random dynamical system in Remark 1.5, is continuous on and the set of varying points of is equal to the Julia set of Figure 1, which is a thin fractal set with Hausdorff dimension strictly less than . From this point of view also, it is very interesting and important to investigate the figure and the dimension of the Julia sets of rational semigroups.
In this paper, for an expanding finitely generated rational semigroup , we deal at length with the relation between the Bowen parameter (the unique zero of the pressure function, see Definition 2.13) of the multimap and the Hausdorff dimension of the Julia set of . In the usual iteration of a single expanding rational map, it is well known that the Hausdorff dimension of the Julia set is equal to the Bowen parameter and they are strictly less than two. For a general expanding finitely generated rational semigroup , it was shown that the Bowen parameter is larger than or equal to the Hausdorff dimension of the Julia set ([25, 28]). If we assume further that the semigroup satisfies the “open set condition” (see Definition 3.1), then it was shown that they are equal ([28]). However, if we do not assume the open set condition, then there are a lot of examples for which the Bowen parameter is strictly larger than the Hausdorff dimension of the Julia set. In fact, the Bowen parameter can be strictly larger than two ([28, 41]). Thus, it is very natural to ask when we have this situation and what happens if we have such a case. Let Rat be the set of nonconstant rational maps on endowed with distance defined by , where denotes the spherical distance on For each , we set
Note that is an open subset of (see Lemma 2.9). Let be a nonempty bounded open subset of . For each , let be an element in . We set
We assume that the map is continuous for each For every , let be the zero of the pressure function for the system generated by Note that the function is continuous (see Theorem 2.16). For a family in , we define the transversality condition (see Definition 3.7). The transversality condition was introduced and investigated for a family of contracting IFSs in [16] (one of first studies of transversality type conditions and applications to Bernoulli convolutions), [17] (case of IFSs in ), [19] (case of finite IFSs of similitudes in general Euclidean spaces ), [20] (case of infinite hyperbolic or parabolic IFSs in ), [21] (case of finite parabolic IFSs in ), and [13] (case of skew products and application to Bowen formulas, examples, partial derivative conditions, etc.). Among these papers there are several types of definitions of the transversality condition. Our definition of the transversality condition is similar to that given in [20], though in the present paper we work on a family of semigroups of rational maps which are not contracting and are not injective. Note that there are many works of contracting IFSs with overlaps. See the above papers and [15, 4], etc. Some results of this paper are applicable to the study of contracting IFSs with overlaps and infinitely many new examples of contracting families of IFSs that satisfy the transversality condition are found (see Theorem 1.7, Examples 1.8, 4.13, 4.14, 4.15, Remarks 4.9,4.16).
For any , we denote by Leb the dimensional Lebesgue measure on a dimensional manifold. In this paper, we prove the following.
Theorem 1.1 (Theorem 3.12).
Let be a family in as above. Suppose that satisfies the transversality condition. Then we have all of the following.

for Leba.e. , where HD denotes the Hausdorff dimension.

For Leba.e. we have that Leb
It is very interesting to investigate the Hausdorff dimension of the exceptional set of parameters in the above theorem. In order to do that, we define the strong transversality condition (see Definition 3.15), and we prove the following.
Theorem 1.2 (Theorem 3.19).
Let be a family in as above. Suppose that satisfies the strong transversality condition. Let be a subset of . Let . Suppose Then we have
Since for each , if we further assume in the above theorem, then
It is very important to study sufficient conditions for a family of expanding semigroups to satisfy the strong transversality condition. Let be a bounded open subset of . We say that a family in as above is a holomorphic family in if is holomorphic for each For a holomorphic family in , we define the analytic transversality condition (see Definition 3.21). We prove the following.
Proposition 1.3 (Proposition 3.22).
Let be a holomorphic family in . Suppose that satisfies the analytic transversality condition. Then for each nonempty, relatively compact, open subset of , the family satisfies the strong transversality condition and, hence, the transversality condition.
By using Proposition 1.3, some calculations involving partial derivatives of conjugacy maps with respect to the parameters (Lemma 3.24–Corollary 3.27), and some observation about the combinatorics of the Julia set (Lemma 3.28), we can produce an abundance of examples of holomorphic families satisfying the analytic transversality condition, and hence the strong transversality condition and ultimately the transversality condition. Combining the above and some further observations, we prove Theorem 1.4 which is formulated below. We consider the space
endowed with the relative topology from Rat. We are interested in families of small perturbations of elements in the boundary of the parameter space in , where
Theorem 1.4 (Theorem 4.1).
Let be such that and Let , where and Let be a number such that there exists a number with Let For each , let Then there exists a point and an open neighborhood of in such that the family with satisfies all of the following conditions (i)–(iv).

is a holomorphic family in satisfying the analytic transversality condition, the strong transversality condition and the transversality condition.

For each , .

There exists a subset of with such that for each ,

is connected and Moreover, satisfies the open set condition. Furthermore, for each , the semigroup satisfies the open set condition, , the Julia set is disconnected, and
where denotes the Bowen parameter of
Moreover, there exists an open neighborhood of in such that the family satisfies all of the following conditions (v)–(viii).

is a holomorphic family in satisfying the analytic transversality condition, the strong transversality condition and the transversality condition.

For each , , where is the Bowen parameter of

There exists a subset of with such that for each ,

For each neighborhood of in there exists a nonempty open set in such that for each , we have that and that is connected.
Remark 1.5.
For each and with , we consider the random dynamical system such that for each step, we choose with probability For each , let be the probability of tending to starting with the initial value Then the function is locally constant on Moreover, this function provides a lot of information about the random dynamics generated by (See [34, 37].) Let be as in Theorem 1.4. Let Let Then we can show that is continuous on and the set of varying points of is equal to (For the figure of , see Figure 1.) Moreover, there exists a neighborhood of in such that for each , is continuous on and locally constant on It is a complex analogue of the devil’s staircase and is called a “devil’s coliseum.” (These results are announced in the first author’s papers [35, 34].) From this point of view also, it is very natural and important to investigate the Hausdorff dimension of the Julia set of a rational semigroup.
In Theorem 1.4 we deal with generator polynomial semigroups with , for which the planar postcritical set is bounded. In fact, it is very important to investigate the dynamics of polynomial semigroups with bounded planar postcritical set (see [31, 38, 32, 23]). There appear many new phenomena (for example, the Julia sets of such semigroups can be disconnected) in the dynamics of such semigroups which cannot hold in the usual iteration dynamics of a single polynomial. In the proof of Theorem 1.4, we use some idea from the study of dynamics of such semigroups. In the family of Theorem 1.4, for a typical parameter value the Hausdorff dimension of the Julia set is strictly less than and is equal to the Bowen parameter. Thus it is very natural to ask what happens for polynomial semigroups with for which the planar postcritical set is bounded. In this case, by [31, Theorem 2.15], is connected and Combining Proposition 1.3 and the lower estimate of the Bowen parameter from [41], which was obtained by using thermodynamic formalisms, potential theory, and some results from [43], we prove the following.
Theorem 1.6 (Corollary 4.5).
For each with , there exists an open neighborhood of in such that is a holomorphic family in satisfying the analytic transversality condition, the strong transversality condition and the transversality condition, and for a.e. with respect to the Lebesgue measure on , we have that
Note that in the usual iteration dynamics of a single expanding rational map , the Hausdorff dimension of the Julia set is strictly less than two. In particular, Leb
For any with , is equal to the closed annulus between and , thus However, regarding Theorem 1.6, it is an open problem to determine, for any other parameter value with , whether or not. We have some partial answers though. At least we can show that for each with and for each neighborhood of in there exists a nonempty open subset of such that for each , the Fatou set has at least three connected components, and thus the Julia set is not a closed annulus. If with , then we can show that for each neighborhood of in and for each with , there exists a nonempty open subset of such that for each , has at least connected components and is not a closed annulus (see Remark 4.6).
We now consider the expanding semigroups generated by affine maps. Let For each , let , where Let Since , Hence, by (1.1), is a compact subset of which satisfies Since is a contracting similitude on , it follows that is equal to the selfsimilar set constructed by the family of contracting similitudes. For the definition of selfsimilar sets, see [4, 5, 9]. Note that the Bowen parameter of is equal to the unique solution of the equation . Thus is the similarity dimension of Conversely, any selfsimilar set constructed by a finite family of contracting similitudes on is equal to the Julia set of the rational semigroup By using Proposition 1.3 and some calculations of the partial derivatives of the conjugacy maps with respect to the parameters, we prove the following.
Theorem 1.7 (Theorem 4.8).
Let with For each , let where , . Let We suppose all of the following conditions hold.

For each with and , there exists a number such that

If are mutually distinct elements in , then

For each with , we have
Then, there exists an open neighborhood of , where Aut, such that is a holomorphic family in satisfying the analytic transversality condition, the strong transversality condition and the transversality condition.
Note that in the above theorem, for each ,
Note also that even if we replace “” by , similar results hold (see Remark 4.9).
By using Theorem 1.7, we can obtain many examples of families of systems of affine maps satisfying the analytic transversality condition. In fact, we have the following.
Example 1.8 (Example 4.11).
Let be such that makes an equilateral triangle. For each , let Let Then is equal to the Sierpinski gasket. It is easy to see that satisfies the assumptions of Theorem 1.7. Moreover, By Theorems 1.7, 1.2 and 2.15, there exists an open neighborhood of in and a Borel subset of with such that (1) is a holomorphic family in satisfying the analytic transversality condition, the strong transversality condition and the transversality condition, and (2) for each ,
For some other examples including the families related to the Snowflake, Pentakun, Hexakun, Heptakun, Octakun and so on, see Examples 4.10, 4.13, 4.14, 4.15 and Remark 4.16. (For the definition of Snowflake, Pentakun, etc., see [9].) We remark that, up to our best knowledge, these examples (Examples 1.8, etc.) have not been explicitly dealt with in any literature of contracting IFSs with overlaps.
In section 2, we introduce and collect some fundamental concepts, notation, and definitions. In section 3, we prove the main results of this paper. In section 4, we describe some applications and examples. In section 5, we make a remark on similar results for families of conformal contracting iterated function systems in arbitrary dimensions.
2. Preliminaries
In this section we introduce notation and basic definitions. Throughout the paper, we frequently follow the notation from [26] and [28].
Definition 2.1 ([8, 44]).
A “rational semigroup” is a semigroup generated by a family of nonconstant rational maps , where denotes the Riemann sphere, with the semigroup operation being functional composition. A “polynomial semigroup” is a semigroup generated by a family of nonconstant polynomial maps of For a rational semigroup , we set
and we call the Fatou set of . Its complement,
is called the Julia set of If is generated by a family (i.e., ), then we write For each , we set and
Note that for each , For the fundamental properties of and , see [8, 22, 26]. For the papers dealing with dynamics of rational semigroups, see for example [8, 44, 22, 24, 25, 26, 27, 28, 29, 30, 40, 39, 41, 31, 38, 32, 23, 33, 34, 35, 36, 37], etc.
We denote by Rat the set of all nonconstant rational maps on endowed with distance defined by , where denotes the spherical distance on For each , we set Rat Note that each is a connected component of Rat. Hence Rat has countably many connected components. In addition, each connected component of Rat is an open subset of Rat and has a structure of a finite dimensional complex manifold. Similarly, we denote by the set of all polynomial maps with endowed with the relative topology inherited from Rat. We set endowed with the relative topology inherited from Rat. For each with , we set Note that each is a connected component of Hence has countably many connected components. In addition, each connected component of is an open subset of and has a structure of a finite dimensional complex manifold. Moreover, Aut is a connected, complextwodimensional complex manifold. We remark that as in if and only if there exists a number such that

for each , and

the coefficients of converge to the coefficients of appropriately as
Thus
For more information on the topology and complex structure of Rat and , the reader may consult [2].
For each , we denote by the complex tangent space of at Let be a holomorphic map defined on an open set of and let We denote by the derivative of at Moreover, we denote by the norm of the derivative at with respect to the spherical metric on
Definition 2.2.
For each , let be the space of onesided sequences of symbols endowed with the product topology. This is a compact metrizable space. For each , we define a map
by the formula
where and denotes the shift map. The transformation is called the skew product map associated with the multimap We denote by the projection onto and by the projection onto . That is, and For each and , we put
We define
for each and we set
where the closure is taken with respect to the product topology on the space is called the Julia set of the skew product map In addition, we set and We also set (disjoint union). For each let be the length of For each we write For each and each , we put
For every let . If , we put
If , is the longest initial subword common for both and . Let be a fixed number with We endow the shift space with the distance defined as with the standard convention that . The distance induces the product topology on . Denote the spherical distance on by and equip the product space with the distance defined as follows.
Of course induces the product topology on . If and , we set For a , we set
Remark 2.3.
By definition, the set is compact. Furthermore, if we set , then, by [26, Proposition 3.2], the following hold:

is completely invariant under ;

is an open map on ;

if and is contained in , then the dynamical system is topologically exact;

is equal to the closure of the set of repelling periodic points of if , where we say that a periodic point of with period is repelling if .

Definition 2.4 ([28]).
A finitely generated rational semigroup is said to be expanding provided that and the skew product map associated with is expanding along fibers of the Julia set , meaning that there exist and such that for all ,
(2.1) 
Definition 2.5.
Let be a rational semigroup. We put
and we call the postcritical set of . A rational semigroup is said to be hyperbolic if
We remark that if and is generated by , then
(2.2) 
Therefore for each ,
Definition 2.6.
Let be a polynomial semigroup. We set This set is called the planar postcritical set of We say that is postcritically bounded if is bounded in
Remark 2.7.
Let be a rational semigroup such that there exists an element with and such that each Möbius transformation in is loxodromic. Then, it was proved in [25] that is expanding if and only if is hyperbolic.
Definition 2.8.
For each , we define
Then we have the following.
Lemma 2.10 (Theorem 2.14 in [27]).
For each , and
Definition 2.11.
We set