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Dynamics of postcritically bounded polynomial semigroups

II: fiberwise dynamics and the Julia sets ∗

Hiroki Sumi Department of Mathematics, Graduate School of Science

Osaka University 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

E-mail: sumi@math.sci.osaka-u.ac.jp http://www.math.sci.osaka-u.ac.jp/∼sumi/welcomeou-e.html

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 Jγ to satisfy that Jγ is a Jordan curve but not a quasicircle, the unbounded component of Ĉ \ Jγ is a John domain and the bounded component of C \ Jγ 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 Ĉ), 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 non-constant rational maps on Ĉ, where Ĉ denotes the Riemann sphere, with the semigroup operation being functional composition ([11]). A polynomial semigroup is a semigroup generated by a family of non-constant polynomial maps. Research on the dynamics of

∗Date: 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 Grant-in-Aid for Scientific Research (C) 21540216. Keywords: Polynomial semigroups, random complex dynamics, random iteration, skew product, Julia sets, fiberwise Julia sets.

1

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 one-complex-dimensional 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 self-similarity” (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 f(z) = az(1−z) such that f 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 G be a rational semigroup. We set

F (G) = {z ∈ Ĉ | G is normal in a neighborhood of z}, J(G) = Ĉ \ F (G).

F (G) is called the Fatou set of G and J(G) is called the Julia set of G. We let 〈h1, h2, . . .〉 denote the rational semigroup generated by the family {hi}. The Julia set of the semigroup generated by a single map g is denoted by J(g).

Definition 1.2. For each rational map g : Ĉ → Ĉ, we set CV (g) := {all critical values of g : Ĉ → Ĉ}. Moreover, for each polynomial map g : Ĉ → Ĉ, we set CV ∗(g) := CV (g) \ {∞}. For a rational semigroup G, we set

P (G) := ∪ g∈G

CV (g) (⊂ Ĉ).

This is called the postcritical set of G. Furthermore, for a polynomial semigroup G, we set P ∗(G) := P (G) \ {∞}. This is called the planar postcritical set (or finite postcritical set) of G. We say that a polynomial semigroup G is postcritically bounded if P ∗(G) is bounded in C.

Remark 1.3. Let G be a rational semigroup generated by a family Λ of rational maps. Then, we have that P (G) = ∪g∈G∪{Id} g(∪h∈ΛCV (h)), where Id denotes the identity map on Ĉ, and that g(P (G)) ⊂ P (G) for each g ∈ G. Using this formula, one can understand how the set P (G) (resp. P ∗(G)) spreads in Ĉ (resp. C). 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 G is postcritically bounded much in the same way as one verifies the boundedness of the critical orbit for the maps fc(z) = z2 + c.

Example 1.4. Let Λ := {h(z) = cza(1−z)b | a, b ∈ N, c > 0, c( aa+b ) a( ba+b )

b ≤ 1} and let G be the polynomial semigroup generated by Λ. Since for each h ∈ Λ, h([0, 1]) ⊂ [0, 1] and CV ∗(h) ⊂ [0, 1], it follows that each subsemigroup H of G is postcritically bounded.

Remark 1.5. It is well-known that for a polynomial g with deg(g) ≥ 2, P ∗(〈g〉) is bounded in C if and only if J(g) is connected ([14, Theorem 9.5]).

2

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 G be a polynomial semigroup such that each element g ∈ G is of degree at least two. Is J(G) necessarily connected when P ∗(G) is bounded in C?” The answer is NO. In fact, in [37, 29, 30, 19, 31, 32], we find many examples of postcritically bounded polynomial semigroups G with disconnected Julia set such that for each g ∈ G, deg(g) ≥ 2. Thus, it is natural to ask the following questions.

Problem 1.6. (1) What properties does J(G) have if P ∗(G) is bounded in C and J(G) is discon- nected? (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 G generated by a compact family Γ of polynomials. For each sequence γ = (γ1, γ2, γ3, . . .) ∈ ΓN, we examine the dynamics along the sequence γ, that is, the dynamics of the family of maps {γn ◦ · · · ◦ γ1}∞n=1. We note that this corresponds to the fiberwise dynamics of the skew product (see Section 3.1) associated with the generator system Γ. We show that if G is postcritically bounded, J(G) is disconnected, and G is generated by a compact family Γ of polynomials, then, for almost every sequence γ ∈ ΓN, there exists exactly one bounded component Uγ of the Fatou set of γ, the Julia set of γ has Lebesgue measure zero, there exists no non-constant limit function in Uγ for the sequence γ, and for any point z ∈ Uγ the orbit of z along γ tends to the interior of the smallest filled-in Julia set K̂(G) (see Definition 2.7) of G (cf. Theorem 3.11, Corollary 3.21). Moreover, using uniform fiberwise quasiconformal surgery ([30]), we find sub-skew products f such that f is hyperbolic (see Definition 3.10) and such that every fiberwise Julia set of f is a K-quasicircle, where K is a constant not depending on the fibers (cf. Theorem 3.11, statement 3). Reusing the uniform fiberwise quasiconformal surgery, we show that if G is a postcritically bounded polynomial semigroup with disconnected Julia set, then for any non-empty open subset V of J(G), there exists a 2-generator subsemigroup H of G such that J(H) is the disjoint union of a “Ca