Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups
We estimate the Bowen parameters and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map and the Lyapunov exponent of with respect to the maximal entropy measure for . Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form by a Möbius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than .
Key words and phrases:Complex dynamical systems, rational semigroups, expanding semigroups, Julia set, Hausdorff dimension, Bowen parameter, random complex dynamics
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan
Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA
Mathematics Subject Classification (2001). Primary 37F35; Secondary 37F15.
A rational semigroup is a semigroup generated by a family of non-constant 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 non-constant polynomial maps on The work on the dynamics of rational semigroups was initiated by A. Hinkkanen and G. J. Martin (), who were interested in the role of the dynamics of polynomial semigroups while studying various one-complex-dimensional moduli spaces for discrete groups of Möbius transformations, and by F. Ren’s group (), 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 ([7, 35, 15, 17, 18, 19, 20, 21, 22, 23, 30, 32, 25, 27, 16, 28, 29]). Since the Julia set of a rational semigroup generated by finitely many elements has backward self-similarity i.e.
(see [17, 19]), it can be viewed as a significant generalization and extension of both the theory of iteration of rational maps (see ) and conformal iterated function systems (see ). 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 cases 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 [19, 20, 21, 22, 25, 26, 27, 29, 30, 31, 32]).
The theory of the dynamics of rational semigroups is intimately related to that of the random dynamics of rational maps. For the study of random complex dynamics, the reader may consult [5, 3, 4, 2, 1, 6, 29]. The deep relation between these fields (rational semigroups, random complex dynamics, and (backward) IFS) is explained in detail in the subsequent papers ([23, 25, 26, 27, 24, 28, 29]) of the first author.
In this paper, we deal at length with Bowen’s parameter (the unique zero of the pressure function) of expanding finitely generated rational semigroups (see Definition 2.12). In the usual iteration dynamics of a single expanding rational map, it is well known that the Hausdorff dimension of the Julia set is equal to the Bowen’s parameter. For a general expanding finitely generated rational semigroup , it was shown that the Bowen’s parameter is larger than or equal to the Hausdorff dimension of the Julia set ([18, 21]). If we assume further that the semigroup satisfies the “open set condition” (see Definition 3.2), then it was shown that they are equal (). However, if we do not assume the open set condition, then there are a lot of examples such that the Bowen’s parameter is strictly larger than the Hausdorff dimension of the Julia set. In fact, the Bowen’s parameter can be strictly larger than two. Thus, it is very natural to ask when we have this situation and what happens if we have such a case. We will show the following.
Theorem 1.1 (Theorem 3.1).
For an expanding rational semigroup , the Bowen’s parameter satisfies
where denotes the skew product map associated with the multi-map (see section 2), and denotes the unique maximal entropy measure for (see [12, 19]). Moreover, the equality in the (1.2) holds if and only if we have a very special condition, i.e., there exists a Möbius transformation and a positive integer such that for each , is of the form .
Note that is equal to the entropy of The above result (Theorem 3.1) generalizes a weak form of A. Zdunik’s theorem (), which is a result for the usual iteration of a single rational map. In fact, in the proof of the main result of our paper, Zdunik’s theorem is one of the key ingredients. We emphasize that in the main result of our paper, we can take the Möbius map which does not depend on
If each is a polynomial with , then by using potential theory, we can calculate in (1.2) in terms of and an integral related to fiberwise Green’s functions (see Lemmas 3.13, 3.14). From this calculation, we can prove the following.
Theorem 1.2 (Theorem 3.17).
Let and for each , let be a polynomial with If is an expanding polynomial semigroup, the postcritical set of in is bounded, where and , and , then there exists a Möbius transformation such that for each , is of the form
Thus, if the postcritical set of in is bounded and , then typically we have that Note that in the usual iteration dynamics of a single rational map, we always have
Therefore, we can say that there are plenty of expanding finitely generated polynomial semigroups for which the Bowen’s parameter is strictly larger than 2.
Note that combining these estimates of Bowen’s parameter and the “transversal family” type arguments, we will show that we have a large amount of expanding -generator polynomial semigroups such that the Julia set of has positive 2-dimensional Lebesgue measure ().
A “rational semigroup” is a semigroup generated by a family of non-constant 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 non-constant polynomial maps on 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 , then we write
We denote by Rat the set of all non-constant rational maps on endowed with the topology induced by uniform convergence on Note that 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 from Rat. Note that 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.
Let be an open subset of and let For a holomorphic map , we denote by the norm of the derivative of at with respect to the spherical metric on
For each , let be the space of one-sided 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 multi-map We denote by the projection onto and by the projection onto . That is, and For each and , we put
Moreover, we denote by the norm of the derivative of at with respect to the spherical metric on 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
By definition, the set is compact. Furthermore, if we set , then, by [19, 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 ().
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 ,
Let be a rational semigroup. We put
and we call the postcritical set of . A rational semigroup is said to be hyperbolic if
Let be a polynomial semigroup. We set We say that is postcritically bounded if is bounded in
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  that is expanding if and only if is hyperbolic.
We also set (disjoint union). For every let be the length of For each and each , we put
Then we have the following.
Let and let be the skew product map associated with For each , let be the topological pressure of the potential with respect to the map (For the definition of the topological pressure, see .) We denote by the unique zero of (Note that the existence and the uniqueness of the zero of was shown in .) The number is called the Bowen parameter of the semigroup
We have the following fact, which is one of the main results of .
Theorem 2.13 ().
The function is real-analytic and plurisubharmonic.
For a subset of , we denote by the Hausdorff dimension of with respect to the spherical metric. For a Riemann surface , we denote by the set of all holomorphic isomorphisms of For a compact metric space , we denote by the space of all continuous complex-valued functions on , endowed with the supremum norm.
Let Let and let for each Let be the maximal entropy measure for . Then the following statements (1) and (2) hold.
then, the following items (a),(b),(c) hold.
There exist an automorphism and complex numbers with such that for each ,
From the convexity of , we obtain that
We now assume that and
Because of the convexity of again, we infer that
for all Let be the unique -conformal measure on for (see ). Let
be the operator, called the transfer operator, defined by the following formula
In virtue of , the limit exists, where denotes the constant function taking its only value Let Then
(see ), it follows that By the uniqueness of maximal entropy measure of , we obtain that
Let be the operator defined as follows
Since , (3.1) implies that Thus, for any open subset of such that is injective, if is a Borel subset of , then Moreover, we have
Thus for -a.e. Since supp (see ), it follows that
Therefore, for each there exists a continuous function such that
Thus, for each -invariant Borel probability measure on , we have
Let be the topological pressure of with respect to the potential function It follows that for each with ,
In particular, is linear. Hence,
where denotes the maximal entropy measure for Therefore, by Zdunik’s theorem (), it follows that for each with , there exists an and an element such that
In particular, there exists an element such that for each Suppose that there exists a such that If , then since each point of is a critical point of and , it contradicts (3.6). If , then since each point of is a critical point of and , it contradicts (3.6) again. Therefore, for each ,
Therefore, Thus, we have completed the proof. ∎
Regarding Theorem 3.1, we give several remarks. In order to relate the Bowen parameter to the geometry of the Julia set we need the concept of the open set condition. We define it now.
Let and let . Let also be a non-empty open set in We say that (or ) satisfies the open set condition (with ) if
for each with There is also a stronger condition. Namely, we say that (or ) satisfies the separating open set condition (with ) if
for each with
We remark that the above concept of “open set condition” (for “backward IFS’s”) is an analogue of the usual open set condition in the theory of IFS’s.
We introduce two other analytic invariants.
Definition 3.3 ().
Let be a countable rational semigroup. For any and , we set
counting multiplicities. We also set
(if no exists with , then we set ). Furthermore, we put
The number is called the critical exponent of the Poincaré series of
Definition 3.4 ().
Let , , and We put
counting multiplicities. Moreover, we set
(if no exists with , then we set ). Furthermore, we set
The number is called the critical exponent of the Poincaré series of
Let , , and let Then, and Note that for almost every with respect to the Lebesgue measure, is a free semigroup and so we have and
Lemma 3.6 ().
Let be a rational semigroup. Then, we define
Let us record the following fact proved in  .
Theorem 3.8 ().
In order to prove our second main theorem (see Theorem 3.17), we need some notation and lemmas from . We shall provide the full proofs of these lemmas for the sake of completeness of our exposition and convenience of the readers.
For each , we set
Let Let be the skew product map associated with For any , we set
For any , let
where for each By the arguments in , for each , the limit exists, the function is subharmonic on , and is equal to the Green’s function on with pole at . Moreover, is continuous on Let , where Note that by the argument in [8, 9], is a Borel probability measure on such that Furthermore, for each , let , where runs over all critical points of in , counting multiplicities.
Remark 3.11 ().
Let Let be the skew product map associated with Also, let and let be the Bernoulli measure on with respect to the weight Suppose that for each Then, there exists a unique -invariant Borel probability ergodic measure on such that and
where denotes the relative metric entropy of with respect to , and denotes the space of ergodic measures for (see ). The measure is called the maximal relative entropy measure for with respect to
Lemma 3.12 ().
Let and let Let Let be the skew product associated with Let be the Bernoulli measure on with respect to the weight Let be a Borel probability measure on defined by
for any continuous function on , where is the measure coming from Definition 3.10. Then, is an -invariant ergodic measure, , and is the maximal relative entropy measure for with respect to (see Remark 3.11).
Lemma 3.13 ().
Let . Let