COMPLEX DYNAMICS OF MÖBIUS SEMIGROUPS

DAVID FRIED, SEBASTIAN M. MAROTTA, AND RICH STANKEWITZ

Abstract. We study the dynamics of semigroups of Möbius transformations on the Riemann sphere, especially their Julia sets and attractors. This theory relates to the dynamics of rational functions, rational semigroups, and Möbius groups and we compare and contrast these theories. We particularly examine Caruso’s family of Möbius semigroups, based on a random dynamics variant of the Fibonacci sequence.

Contents

1. Introduction 1 2. Rational semigroups 3 3. Möbius groups 11 4. Contracting iterated function systems 16 5. Thick attractors 18 6. Some Julia sets of Möbius semigroups 24 7. Caruso semigroups 26 7.1. Preliminaries 26 7.2. Connected J_{β} 30 7.3. When does J_{β} equal Λ_{β}? 32 7.4. The closure of G_{β} 33 8. Some J_{β} and Λ_{β} 34 References 37

1. Introduction

This paper examines a random dynamics variant of the Fibonacci ratios. Specifically, we let z_{n} be a recursively defined sequence such that z_{n+1} = z_{n−1} ± βz_{n}, where the terms and the nonzero constant β are complex numbers and the sign ± is chosen at random. The ratio of two consecutive terms r_{n} = z_{n}/z_{n−1} then satisfies the recursion relation r_{n+1} = ±β +1/r_{n}. When β = 1 and we always choose the sign + (as in the Fibonacci sequence) the limit as n → ∞ of r_{n} is the golden ratio ϕ = (1 + \sqrt{5})/2, as long as r_{0} ≠ (1 − \sqrt{5})/2. By introducing the randomness mentioned above, and allowing for other values of β, we will see that the set of possible long term states may be a complicated and beautiful fractal set. In particular, the set of possible long term states is often the attractor set A_{β} of a Contracting Iterated Figure 1. The sets obtained by randomly iterating an arbitrary point in the complex plane ℂ using the semigroup generated by {β + 1/z, −β + 1/z}. We plot a number of points in the randomly generated orbit, dropping the first few iterates as transient. In some cases, the pictures that we get do not seem to depend on the chosen seed or the chosen composition sequence. The unit circle (dashed) is shown for reference. On the left, the parameter β = e^{iπ/6}, and on the right β = 1.25e^{iπ/3}. In both cases we computed one million points and dropped the first hundred points as transient. See Remark 5.13.

Date: June 7, 2011.

Function System (CIFS) generated by {β +1/z, −β +1/z}.^{1} See Figure 1. Some background on these relationships can be found in [8, 14, 40].

This paper is motivated by Caruso’s problem: given a value of β, in what sense does there exist an attractor set A_{β}, how does such a set depend on β, and what are the properties of A_{β}? In [8], the set of long term states was described by means of numerical experiments. The frequency distributions of the modulus and argument of these states were presented for particular values of the parameter β. In this paper we deal instead with the topological properties of the set of possible long term states.

It turns out that the attractor A_{β} (when it exists) of the CIFS generated by {β+1/z, −β+ 1/z} is the Julia set of the semigroup S′_{β} generated under function composition by the inverse transformations 1/(z − β) and 1/(z + β) of these CIFS generators. Thus, one of our goals in this paper is to characterize the Julia set of S′_{β} for different values of the complex parameter β.

During the course of our investigation it became apparent that the study of S′_{β} is intimately related to the study of two other algebraic structures, namely, the semigroup S_{β} and the group G_{β} generated by β + 1/z and −β + 1/z. Thus, we also focus on the topological characteristics of the Julia set of S_{β} and the limit set of G_{β} for different values of β. The study of these structures, as they arise in Caruso’s problem, led us to find several interesting

±β + \frac{1}{±β + \frac{1}{±β+...}} .

1Notice that the sequences generated with this system are also generalized continued fractions of the form

general results about semigroups of Möbius transformations. We have expanded our focus to present these results.

The article is organized in the following manner. In Sections 2 and 3, we develop, in a general setting, the background material relevant to our problem. We introduce nonelementary semigroups and explain their basic properties and characteristics. For example, Theorem 3.3 shows that when the Julia set J(S) of a non-elementary Möbius semigroup S is connected, the limit set of the group generated by S is also connected. The concepts of attractors and blocks are discussed in Sections 4 and 5. In particular, in Theorem 5.7 we give necessary and sufficient conditions for a finitely generated Möbius semigroup to have what we call a thick attractor, which attracts any random orbit whose initial value is near the attractor. Further properties and comparisons between limit sets and various sorts of Julia sets are discussed via examples presented in Section 6.

In Section 7 we study the Caruso semigroups S_{β}. In Section 7.1 we prove that each S_{β} is non-elementary and we study the symmetries that arise in dynamical and parameter planes. In Section 7.2 we study the connectedness of the Julia sets J(S_{β}). In particular, Corollary 7.12 shows that J(S_{β}) must be bounded and disconnected when a thick attractor exists. We present sufficient conditions for J(S_{β}) to equal the limit set of G_{β} in Section 7.3. In Section 7.4 we calculate this limit set when G_{β} is not discrete. We use reflection groups in Section 8 to treat the cases when β^{4} is real.

In a subsequent paper we will use Riley groups to study the Caruso problem when β^{4} is not real.

We would like to thank Linda Keen for introducing us to reference [19] while visiting Boston University in 2005. We also thank Dan Goodman for sharing his computer programs and for giving detailed descriptions of how they work.

2. Rational semigroups

The dynamics of iteration of a complex analytic map has been studied deeply in various contexts, such as rational, entire, and meromorphic maps. This theory generalizes to the setting where the map may be changed at each point of the orbit, exactly as in a random walk. More precisely, instead of repeatedly applying the same map over and over again, one may start with a family of maps {h_{i} : i ∈ I}, and consider the dynamics of any iteratively defined composition sequence of maps, that is, any sequence h_{i_{n}} ∘ ⋯ ∘ h_{i_{1}} where i_{1}, ..., i_{n} ∈ I. Randomly choosing the map at each stage is the setting for random dynamics (see [11, 4, 6, 7, 5, 37, 34]). Restricting one’s attention to the case where all h_{i} are rational, one is lead to study the dynamics of rational semigroups.

A rational semigroup is a nonempty semigroup of nonconstant rational functions defined on the Riemann sphere^{2} ℂ̂ = ℂ ∪ {∞} with the semigroup operation being functional composition.^{3} When a semigroup S is generated by a family of functions h_{i}, i ∈ I, for some nonempty index set I, we write S = ≺ h_{i} : i ∈ I ≻. Thus S is exactly the set of all finite compositions of the maps h_{i}. Note that many papers use the notation ⟨·⟩ for rational semigroups but, since we discuss both groups and semigroups, we will reserve the notation ⟨·⟩ for groups, so ⟨h_{i} : i ∈ I⟩ denotes the group generated by h_{i}, i ∈ I. We say S is cyclic when S is generated by some one function, so S =≺ h ≻.

2As usual, the letter ℂ denotes the complex plane and ℂ^{∗} = ℂ\{0}.

3Beware that some authors require that a rational semigroup contain at least one rational map of degree two or more, but we do not impose such a restriction here.

A rational mapping m(z) of ℂ̂ is invertible if and only if it is Möbius, meaning that the degree of m(z) is one. Clearly the Möbius transformations form a group M. A rational semigroup S whose elements are invertible lies in M and is called a Möbius semigroup. A Möbius group is a subgroup of M.

We recall the connection between ℂ̂ and hyperbolic 3-space H^{3}. The upper halfspace model of hyperbolic 3-space H^{3} is the subset of R^{3} consisting of triples (x_{1}, x_{2}, x_{3}) with x_{3} > 0. In this model, ℂ̂ is the boundary of H^{3}. M acts by isometries on H^{3} and this action extends by continuity to the the Möbius action on ℂ̂. This Möbius action on H^{3} is called the Poincare extension of the Möbius action on ℂ̂ [2].

Each m ∈ M has the form m(z) = (az + b)/(cz + d), where the vector of coefficients (a, b, c, d) lies in the open subset V ⊂ ℂ^{4} defined by ad − bc ≠ 0. The mapping m(z) only determines (a, b, c, d) up to a nonzero scalar factor. It follows that M is a Lie group of real dimension 6 diffeomorphic to the quotient space V/ℂ^{∗}. This quotient topology on M is also the topology of uniform convergence on ℂ̂ ( [2], p. 78). See Theorem 3.1 for a classification of Möbius maps and their dynamical behavior under iteration.

After its inception in the paper [16] of Hinkkanen and Martin, the study of rational semigroups has grown in the past decade (see, for example, [13, 29, 30, 36, 39] as well as the references therein). These works generally focus, however, on semigroups that contain at least one non-Möbius map. Möbius groups, especially discrete ones, have been studied since the pioneering work of Fricke and Klein (see, for example, [2, 20, 21, 23]. This paper thus fills in a gap in the literature by focusing on Möbius semigroups. When convenient, however, we will treat rational semigroups in general.

A rational semigroup divides the Riemann sphere into two complementary sets, an open Fatou set and a closed Julia set with opposite dynamical properties. In order to define these sets we first recall the notion of a normal family of holomorphic maps. Let U be an open set in ℂ̂. A family of holomorphic maps F on U with values in ℂ̂ is normal when every sequence {h_{n}} ∈ F has a subsequence that converges uniformly on each compact set in U. We will make particular use of three types of normal families of Möbius transformations.

\underline{Type 0)} Suppose F consists of isometries of the spherical metric,^{4} so each h ∈ F is a spherical rotation of the form h(z) = (az − b̄)/(bz + ā) for some a, b ∈ ℂ with |a|^{2} + |b|^{2} = 1. Then F is a normal family on ℂ̂.

\underline{Type 1)} Suppose F stabilizes ∞, so each h ∈ F has the form h(z) = az +b (while such maps are sometimes called complex affine we prefer the term linear). If |a| = 1 then h(z) is an isometry of ℂ with its usual metric. If |a| = 1 for all h ∈ F then F is a normal family on ℂ.

\underline{Type 2)} Suppose that F stabilizes \{0, ∞\}. Each h ∈ F is monomial, that is h(z) = az or h(z) = a/z for some a. Then each h(z) is an isometry of the metric |dz|/|z| on ℂ^{∗} (i.e., for ρ(z) = 1/|z| we have h^{∗}ρ(z) =^{def} ρ(h(z))|h′(z)| = ρ(z)) and so F is a normal family on ℂ^{∗}.

4For z, w ∈ ℂ, 2|z − w|(1 + |z|^{2})^{−1/2}(1 + |w|^{2})^{−1/2} is the chordal distance from z to w (see [2], p.22).

Let S be a rational semigroup. As in [16] we define the Fatou set^{5} F (S) as the union of all open sets U ⊂ ℂ̂ on which S forms a normal family. The Julia set J(S) is defined as the complement of F (S) in ℂ̂. When S =≺ h ≻ is cyclic we let F (h) = F (S) and J(h) = J(S).

Remark 2.1. We note that semigroups which are conjugate by a Möbius map (or the complex conjugate of a Möbius map) transfer normality in the obvious way. Specifically, if h is such a map and S_{1} and S_{2} are rational semigroups such that S_{2} = hS_{1}h^{−1} = {hsh^{−1} : s ∈ S_{1}}, then F (S_{2}) = h(F (S_{1})) and J(S_{2}) = h(J(S_{1})) (see [3, 16]).

The dynamics near the Julia set exhibits sensitive dependence on initial conditions. On the Fatou set, however, nearby points have similar dynamic behavior For instance, when a Möbius semigroup S is conjugate in M to one of the 3 types of families discussed above, Remark 2.1 shows that F (S) contains the region where S acts isometrically. This easily gives the following.

Proposition 2.2. Let S be a Möbius semigroup.

0) If S, or some semigroup conjugate to S in M, preserves spherical distance then F (S) = ℂ̂, so |J(S)|^{6} = 0.

1) If p is a neutral fixed point of each s ∈ S then p is the only possible Julia point, so |J(S)| ≤ 1.

2) If S stabilizes a two-point set {p, q} then p and q are the only possible Julia points, so |J(S)| ≤ 2.

The behavior of a Möbius semigroup near a Fatou point is quite restricted, as the following lemma shows. The proof uses the well-known fact that M is simply triply transitive on ℂ̂, that is any two distinct ordered triples of points of ℂ̂ are related by a unique Möbius transformation ( [2], Theorem 4.1.1).

Lemma 2.3. If D is an open set in ℂ̂ and s_{1}, s_{2}, ... ∈ M converge uniformly to h on compact subsets of D then either h is locally constant, with |h(D)| ≤ 2, or s_{n} → s in M and h = s|D.

Proof. If |h(D)| ≥ 3 then we can choose p_{i} ∈ D, i = 0, 1, 2, with h(p_{i}) distinct. Let s, m_{1}, m_{2}, ... be the unique elements of M with s(p_{i}) = h(p_{i}) and m_{n}(s_{n}(p_{i})) = h(p_{i}) for all i. Thus m_{n} → Id, since s_{n}(p_{i}) → s(p_{i}) for all i, and s = m_{n}s_{n} in M for all n, since they agree at 3 distinct points. Thus s_{n} → s in M so s_{n}|D converges uniformly to s|D. □

We will see that the Julia set of a rational semigroup can be described in terms of fixed points. Recall that at a fixed point p = h(p) ∈ ℂ̂ of a rational mapping h(z) we call λ = h′(p) the multiplier of h at p (with the usual convention when p = ∞). The fixed point p is a sink (or attracting) if |λ| < 1 and a source (or repelling) if |λ| > 1. When |λ| = 1 we say p is neutral (or indifferent).

Suppose a rational semigroup S is normal on open set U and some s ∈ S has a fixed point p ∈ U. If p were repelling, then |(s^{n})′(p)| = |s′(p)|^{n} → +∞ and so no subsequence of the iterates s^{n} could converge uniformly on any compact neighborhood of p. Hence p must either be attracting or neutral. Thus any source of any s ∈ S must be a Julia point.

5Some authors use N(S) for the Fatou set since it is the set of normality.

6|X| denotes the cardinality of a set X.

We let Source S (respectively Sink S) be the smallest closed set containing all the sources (respectively sinks) of elements of S. Then we see that Source S ⊂ J(S). Except for one special circumstance, mentioned in Proposition 2.2, equality must hold.

Theorem 2.4. Let S be a rational semigroup. Either Source S = J(S) or S is a Möbius semigroup such that some p ∈ ℂ̂ is a neutral fixed point of each s ∈ S and J(S) = {p}. Proof. [16] proves that sources are dense in J(S) whenever |J(S)| ≥ 3 and S is not a Möbius semigroup. Their proof remains valid, however, when S is a Möbius semigroup and |J(S)| ≥ 3. [31], p. 5, treats both cases.

Clearly if J(S) is empty then Source S is empty as well. The remaining cases, when J(S) consists of 1 or 2 points, are easily dealt with using Theorem 2.11 below. □

This powerful theorem reexpresses the Julia set, which was defined via complex analysis, in purely topological terms. For whether a fixed point p of s ∈ S is attracting, repelling, or neutral depends only on the local topological dynamics of s near p.

Remark 2.5. Theorem 2.11 for a Möbius semigroup S follows from the fact that any Julia point p that is not fixed by S must lie in Source S. We sketch an elementary proof of this fact (c.f. [32]).

We may suppose p = 0 ≠ s(0), for some s ∈ S. As 0 ∈ J(S), the Arzela-Ascoli Theorem implies that there is no neighborhood of 0 on which S is equicontinuous as a family of maps to ℂ̂. So we may choose f_{n} ∈ S and z_{n} ∈ ℂ, z_{n} → 0, such that f_{n} expands the spherical metric at z_{n} by at least n. Passing to a subsequence, and replacing f_{n} by sf_{n} if need be, we may suppose that |f_{n}(∞)| > c for all n, where c > 0. We now show that f_{n} has a source near 0 for n large, so 0 ∈ Source S.

Let D be the closed disc centered at 0 of radius c/2. Let q_{n} be a spherical rotation with q_{n}(∞) = f_{n}(∞). As |q_{n}(∞)| > c, the disc D_{n} = q_{n}^{−1}(D) lies in a bounded region in ℂ.

We factor f_{n} = q_{n}r_{n} in M, so r_{n} fixes ∞ and r_{n} expands the spherical metric at z_{n} by at least n. As z_{n} → 0, r_{n}^{−1}(z) = a_{n}z + b_{n} where (a_{n}, b_{n}) → (0, 0). So for n large, r_{n}^{−1}(D_{n}) is a small disc near 0. Thus f_{n}^{−1}(D) = r_{n}^{−1}(D_{n}) ⊂ Int D, so f_{n}^{−1} has a sink near 0.

A Möbius semigroup S determines an inverse semigroup S^{−1} = {h^{−1} : h ∈ S}. We denote its Julia and Fatou sets by

J′(S) = J(S^{−1}) and F′(S) = F (S^{−1}),

respectively.

We can easily describe the Julia set of a Möbius semigroup S in terms of the action of S^{−1} on hyperbolic space.

Proposition 2.6. Let S be a Möbius semigroup. Then for each x ∈ H^{3}, J(S) = \overline{S^{−1}x} ∩ ℂ̂.

Proof. Let K = \overline{S^{−1}x} ∩ ℂ̂. We first show that K contains J(S).

Consider first the special case when S has a unique Julia point p ∈ ℂ̂ that is a neutral fixed point of each s ∈ S. We may assume p = ∞. S^{−1}x lies on the horosphere Σ through x with center p and the only accumulation point of Σ in ℂ̂ is p. Also S is unbounded in M (see Proposition 2.12) and so S^{−1}x is unbounded in Σ. Thus p ∈ K so J(S) ⊂ K.

For any S one easily sees that Source S ⊂ K. Combining Theorem 2.4 with the preceding paragraph shows that K contains J(S) for all S.

If K does not equal J(S) then some k ∈ K is a Fatou point. Suppose D is an open disc containing k such that \overline{D} lies in F (S) and that s_{1}, s_{2}, ... ∈ S with s_{n}^{−1}(x) → k. Passing to a subsequence, we may assume s_{n}|\overline{D} converges uniformly to h : \overline{D} → ℂ̂. As s_{n}^{−1}(x) → k, s_{n} does not converge in M. By Lemma 2.3, h is constant. Say h(\overline{D}) = \{q\}.

Let H be the open halfspace of H^{3} bounded by D. Then s_{n}(H) is bounded by s_{n}(D) and so it is located very near to q for large n. But for n large enough, H contains s_{n}^{−1}(x) and so x ∈ s_{n}(H). As x ∈ H^{3} and q ∈ ℂ̂, we have x ≠ q so this is a contradiction. Thus K = J(S). □

We now turn to the invariance properties of Julia and Fatou sets. We distinguish, as usual, three kinds of invariance. If h is a map of a set X into itself, a subset Y of X is:

i) forward invariant under h if h(Y) ⊂ Y;
ii) backward invariant under h if h^{−1}(Y) ⊂ Y;
iii) completely invariant under h if h(Y) ⊂ Y and h^{−1}(Y) ⊂ Y.

Let F be a family of maps from X to X. Y is forward, backward, or completely invariant under F if it is forward, backward, or completely invariant under each h ∈ F, respectively.

We will mainly be concerned with a closed nonempty set X ⊂ ℂ̂ and a rational semigroup S. When X is forward invariant under S and X is compact and nonempty we say that X is S-invariant.

The natural invariance properties of Julia and Fatou sets are as follows.

Proposition 2.7 ([16], p. 360). Let S be a rational semigroup. F (S) is forward invariant under S and consequently J(S) is backward invariant under S.

Proposition 2.7 and Theorem 2.4 give the following.

Corollary 2.8. Source S is backward invariant under S for any rational semigroup S.

This is surprising since the set of sources of elements of S is not backward invariant, in general, but its closure Source S must be so.

Example 2.9. Let s_{1}(z) = 2z, s_{2}(z) = 2z − 1, and S =≺ s_{1}, s_{2} ≻ . Note that the interval (0, 1) is backward invariant under S. Although q = 1 is a source for s_{2}, 1/2 = s_{1}^{−1}(1) is not fixed by any s ∈≺ s_{1}, s_{2} ≻ . For if s(1/2) = 1/2, we would have the contradiction 1/2 ∈ s^{−1}((0, 1)) ⊂ s_{1}^{−1}((0, 1)) ∪ s_{2}^{−1}((0, 1)) = (0, 1/2) ∪ (1/2, 1).

For most rational semigroups S, neither F (S) nor J(S) is completely invariant. But there are exceptions, such as cyclic semigroups ([22], Invariance Lemma 4.1) and certain Möbius s emigroups (Section 7.3).

The following straightforward proposition regarding finitely generated semigroups has been noted by many people (c.f. [35], p.719).

Proposition 2.10. For a rational semigroup S =≺ h_{1}, ..., h_{n} ≻, we have J(S) = ∪_{i=1}^{n}h_{i}^{−1}(J(S)) and F (S) = ∩_{i=1}^{n}h_{i}^{−1}(F (S)).

We refer to the above condition on the set J(S) as backward self-similarity. We also note that it implies a partial forward invariance in the sense that for any z ∈ J(S), there exists some generator h_{i} such that h_{i}(z) ∈ J(S).

A rational semigroup S is elementary when |J(S)| ≤ 2 and otherwise non-elementary. In fact, as we now show, S is elementary whenever J(S) is finite, in which case it must be a Möbius semigroup conjugate to one of three special types, as follows.

Theorem 2.11. Let S be a rational semigroup such that J(S) is finite. Then S ⊂ M and S is elementary. Further, S is conjugate in M to a semigroup S^{∗} whose elements s^{∗} are as follows, depending on |J(S)|, |J(S)| = 2 : monomials s^{∗}(z) = az^{n}, with n = ±1, |J(S)| = 1 : linear maps s^{∗}(z) = az + b, with |a| ≤ 1, or |J(S)| = 0 : spherical rotations s^{∗}(z) = (az − b̄)/(bz + ā), with a, b ∈ ℂ and |a|^{2} + |b|^{2} = 1.

Proof. For each s ∈ S, J(s) ⊂ J(S) is finite. So s must be Möbius ( [22], Lemma 4.5 and Corollary 4.11). Thus S ⊂ M. As s is bijective and J(S) is finite and backward invariant by s, s permutes J(S).

First suppose |J(S)| = n ≥ 3. Since s^{−1}(J(S)) ⊂ J(S) and a Möbius transformation is determined by its values at any 3 points, |S| ≤ n(n − 1)(n − 2). As S is finite, S is a normal family on ℂ̂ and so J(S) is empty. This contradiction proves S is elementary.

Next suppose |J(S)| = 2. Let S^{∗} be the semigroup obtained by conjugating S by a Möbius map which takes J(S) to {0, ∞}. Then each s ∈ S^{∗} stabilizes {0, ∞} so s is of the form z ↦ az^{n}.

Now suppose |J(S)| = 1. Let S^{∗} be the semigroup obtained by conjugating S by a Möbius map which takes J(S) to {∞}. Each s ∈ S^{∗} fixes ∞ and so s is of the form z ↦ az + b. Further, |a| ≤ 1 else s would have a repelling fixed point in ℂ, thus contradicting the fact that J(S^{∗}) = {∞}.

The cases when J(S) have 1 or 2 elements conclude our proof of Theorem 2.4.

Our final assertion, when J(S) is empty, is just (1) ⇒ (3) in the following proposition. □

Proposition 2.12. Let S be a Möbius semigroup and let G be the group generated by S. The following are equivalent:

(1) J(S) = ∅

(2) S is bounded, i.e., its closure \overline{S} ⊂ M is compact

(3) S fixes a point in H^{3} and thus S is conjugate to a semigroup of spherical rotations (4) J(G) = ∅

(5) G is bounded, i.e., its closure \overline{G} ⊂ M is compact

(6) G fixes a point in H^{3} and thus G is conjugate to a group of spherical rotations

(7) Every nonidentity element of G is elliptic.

Proof. (1) ⇒ (2). Suppose J(S) is empty, so S is a normal family on ℂ̂. Then every sequence in S has a subsequence that converges in M so S is bounded in M.

(2) ⇒ (3). Consider a bounded Möbius semigroup S. Choose a point p ∈ H^{3} and consider S(p) = \{s(p)|s ∈ S\}, the S-orbit of p. S(p) is bounded in H^{3} since \overline{S} is compact. Let C(p) be the closed convex hull of S(p) in H^{3}, that is C(p) is the intersection of all hyperbolic halfspaces containing S(p).

Since S(p) is bounded in H^{3}, C(p) is compact as well as convex. Moreover C(p) has the same diameter as S(p). For each s ∈ S we have s(S(p)) ⊂ S(p) and so s(C(p)) ⊂ C(p). As s ∈ S acts on the compact metric space C(p) by an isometry, we have s(C(p)) = C(p) ([24], Exer. 7 p. 182).

Clearly C(p′) ⊂ C(p) for all p′ ∈ C(p). Moreover, if p′ ∈ C(p) is not an extreme point of C(p) then no point of S(p′) is an extreme point of C(p). It follows that the diameter of C(p′) is strictly less than the diameter of C(p).

We may choose b ∈ C(p) so that the diameter of C(b) is minimal. Then every point of C(b) is an extreme point, so C(b) consists of just one point. Thus S fixes b. By moving b to (0, 0, 1) by a Möbius transformation we may conjugate S to a semigroup S^{∗} of the required form (see [2], p. 63).

(3) ⇒ (1). Clearly J(S^{∗}) is empty, since S^{∗} consists only of isometries of the spherical metric. It follows that J(S) is empty.

Thus (1)-(3) are equivalent. Since G is also a semigroup in its own right, the equivalence of (4)-(6) then follows. Clearly (3) is equivalent to (6) and (6) ⇒ (7), so it suffices to show the following.

(7) ⇒ (4). Suppose G fixes a point p in ℂ̂ that is a neutral fixed point of each g ∈ G. We may conjugate G so that p = ∞ and G is a group of transformations az + b with |a| = 1. As G contains no translations, the commutator of any two elements of G is trivial. Thus G is abelian and any two nonidentity elements of G must fix the same point in ℂ. Thus G is bounded and so Λ(G) is empty. Now (4) follows by Theorem 2.4, which was proven above. □

Remark 2.13. (a) The equivalence of (6) and (7) in Proposition 2.12 is Theorem 4.3.7 of [2]. We have given a self-contained proof for the sake of completeness.

(b) When S is bounded and p ∈ H^{3} we set d(p) = dim C(p) and give C(p) the measure defined by the hyperbolic d(p)-volume. The mean squared distance to points of C(p) is a strictly convex function on H^{3} whose unique minimum b(p) is the barycenter, or center of mass, of C(p) ([26], Theorem 1). As S preserves this measure on C(p) and S preserves hyperbolic distance, S fixes b(p). This gives another proof that (2) ⇒ (3).

Example 2.14. There is no condition for semigroups in Proposition 2.12 analogous to (7) because an unbounded Möbius semigroup S may consist entirely of elliptic elements. For example, let S =≺ az, az + 1 ≻ where |a| = 1 and a is not a root of unity. Each s ∈ S has the form s(z) = a^{n}z +b for some n > 0, b ∈ ℂ, so s is elliptic. But the axes of az and az +1 do not meet in H^{3}. Thus S fixes no point in H^{3} and so, since (2) ⇒ (3) in Proposition 2.12, S is unbounded.

Remarkably, all non-elementary rational semigroups have many features in common. This follows from Montel’s Theorem, which is the major tool in the study of normal families.

Theorem 2.15 ([3], p. 57). (Montel) Let F be a family of meromorphic functions on an open set U ⊂ ℂ̂ with values in ℂ̂ \ {a, b, c} where a, b, c ∈ ℂ̂ are three distinct points. Then F is normal.

We first use Montel’s Theorem to quantify the sensitive dependence on initial conditions near J(S).

Proposition 2.16 (Transitivity for semigroups). Let S be a rational semigroup. Suppose z ∈ J(S) and let N be a neighborhood of z. Then the union U of the images s(N) for s ∈ S can omit at most two points of ℂ̂. If S is Möbius then U can omit at most one point of ℂ̂.

It is easy to see that these results are optimal. For the cyclic semigroup S generated by z^{2} and for any small neighborhood N of the Julia point z = 1 we have U = ℂ^{∗}. For a semigroup of linear maps containing h(z) = 2z and any small neighborhood of the Julia point z = 0 we have U = ℂ.

Proof. Replacing N by its interior, we may suppose that N is open. Note that s(N) ⊂ U for all s ∈ S but S is not normal on N since z ∉ F (S). By Montel’s theorem, U omits at most two points of ℂ̂.

Suppose S is Möbius and U omits exactly two points p and q. Then the set {p, q} must be stabilized by S, and so by Proposition 2.2, we have either z = p or z = q. But then clearly for any s ∈ S we would have z = s^{2}(z) ∈ s^{2}(N) ⊂ U which contradicts the fact that U omits p and q. □

When S is cyclic, the set U in Proposition 2.16 contains J(S), as in [22], Transitivity Theorem 4.7. This is not so, however, for S =≺ 2z, (z + 1)/2 ≻ with z = 0 ∈ J(S). For as already noted, U = ℂ for a small neighborhood N of 0 and so U omits the Julia point ∞.

The classification of elementary semigroups in Theorem 2.11 suggests that any backward invariant set X ⊂ ℂ̂ for a rational semigroup S should be quite special. We give three results concerning such an X, two of which use Montel’s Theorem.

Proposition 2.17. Let X ⊂ ℂ̂ be backward invariant for a rational semigroup S. If X is finite then X is completely invariant under S.

Proof. Let s ∈ S. Since s maps ℂ̂ onto itself we must have s(s^{−1}(X)) = X and so |s^{−1}(X)| ≥ |X|. Since s^{−1}(X) ⊂ X, by hypothesis, we see that s^{−1}(X) = X. Thus s(X) = s(s^{−1}(X)) = X. □

Proposition 2.18. Let X ⊂ ℂ̂ be backward invariant for a rational semigroup S. Then J(S) ⊂ \overline{X} or |X| ≤ 2.

Proof. Suppose n = |X| ≥ 3. As X is backward invariant, so is its closure \overline{X}. Montel’s Theorem applies to the open forward invariant set U = ℂ̂\\overline{X} and gives U ⊂ F (S), so J(S) ⊂ \overline{X}. □

Proposition 2.19. Let X ⊂ ℂ̂ be backward invariant for a rational semigroup S. If X is finite and contains 3 or more points then S is Möbius and finite.

Proof. Suppose n = |X| ≥ 3. By Proposition 2.18, J(S) ⊂ X so J(S) is finite. By Proposition 2.11, each s ∈ S is Möbius. The values of s^{−1} at three points of X determine s, so |S| ≤ n(n − 1)(n − 2). □

In addition to the methods used in the given references, the following remark may be justified by the previous propositions. In particular, one can take X to be the derived set of J(S) in Proposition 2.18 in order to show that J(S) is perfect.

Remark 2.20. (see [16] p.363, [31] p.765). Let S be a non-elementary rational semigroup. Then J(S) is the smallest closed backward invariant set containing at least three points. Moreover J(S) is perfect and thus uncountable.

Suppose the S in this remark is Möbius. Propositions 2.2 and 2.17 show that any closed nonempty backward invariant set that does not contain J(S) must contain exactly one point. In the following example ∞ is such a point, which happens to be a Fatou point. We also observe the existence of a closed set X ≠ ℂ̂ strictly containing J(S) that is backward selfsimilar.

Remark 2.21. Let h_{1}(z) = 3z, h_{2}(z) = 3z − 2, and S =≺ h_{1}, h_{2} ≻ . Example 4.4 shows that J(S) is the standard middle third Cantor set, yet X = ℝ̂ satisfies X = ∪_{i=1}^{2}h_{i}^{−1}(X). Also {∞} is backward invariant but does not contain J(S).

With the aid of Theorem 2.11 we now see that each elementary S has a finite orbit in the 3-ball H^{3} ∪ ℂ̂.

Corollary 2.22. An elementary rational semigroup S either fixes a point in H^{3} or has an orbit in ℂ̂ with one or two points.

Proof. Any such S must be a Möbius semigroup. Since J(S) is backward invariant with |J(S)| ≤ 2, it is completely invariant by Proposition 2.17. Thus when J(S) is nonempty, any Julia point must have an orbit contained in J(S). When J(S) is empty, Theorem 2.11 shows that S fixes a point in H^{3}. □

3. Möbius groups

We recall the following definitions and results which can be found in [2], our main reference for Möbius groups. For every complex matrix

M = ( \begin{matrix}a b \\ c d \end{matrix} )

with ad − bc ≠ 0 there is a corresponding m(z) = (az + b)/(cz + d) in M. We write tr(M) = a + d and det(M) = ad − bc for the trace and determinant of M and note that trace is homogeneous of degree 1 while determinant is homogeneous of degree 2. Since m(z) determines M up to a scalar factor, we may unambiguously define tr^{2}[m] = tr^{2}(M)/det(M). This invariant of m(z) determines its dynamical properties, as follows.

Theorem 3.1 ([2], p. 67). Let Id denote the identity map on ℂ̂ and let m ∈ M, m ≠ Id. Then

(1) tr^{2}[m] = 4 if and only if m is \underline{parabolic}, i.e., m has only one neutral fixed point and m is conjugate to the translation z ↦ z + 1.

(2) tr^{2}[m] ∈ [0, 4) if and only if m is \underline{elliptic}, i.e., m has two neutral fixed points and m is conjugate to a rotation z ↦ kz with |k| = 1.

(3) tr^{2}[m] ∈ (4, +∞) if and only if m is \underline{hyperbolic}, i.e., m has an attracting and a repelling fixed point and m is conjugate to z ↦ kz with k ∈ ℝ and |k| > 1.

(4) tr^{2}[m] ∉ [0, +∞) if and only if m is \underline{strictly loxodromic}, i.e., m has an attracting and a repelling fixed point and m is conjugate to z ↦ kz with |k| > 1 and k ∉ ℝ.

Finally, a loxodromic map is any m ∈ M which is either hyperbolic or strictly loxodromic. Beware that this terminology is not universal! Some authors use loxodromic to mean strictly loxodromic and use hyperbolic to encompass all loxodromic maps. Others use hyperbolic to encompass both loxodromic maps and elliptic maps.

Let G be a Möbius group. We say G is an elementary Möbius group if it has a finite orbit in either ℂ̂ or H^{3} and otherwise we say that G is a non-elementary Möbius group ([2], p.83).

Remark 3.2. Corollary 2.22 (as well as Theorem 5.1.3 in [2]) shows that if the Möbius semigroup G is elementary then G is also an elementary Möbius group. Alas G may have a finite orbit in ℂ̂ and an infinite Julia set, as in Example 3.10, in which case the elementary Möbius group G is not elementary! As our main interest is in semigroups, we will avoid ambiguity by treating the phrases “elementary Möbius group” and “non-elementary Möbius group” as indivisible, just as if they were single words. With this convention “the Möbius group G is non-elementary must mean that |J(G)| > 2 (although we will avoid such confusing statements altogether).

The class of elementary Möbius groups contains all abelian subgroups of M, all finite subgroups of M and the stabilizer^{7} of each point in ℝ̂^{3}. For example, the groups generated by the non-elementary semigroups studied in Examples 3.10, 6.1, 6.3 and 6.4 are subgroups of the stabilizer of ∞ in M. For a more complete discussion of elementary Möbius groups see [2, 27].

When G is a non-elementary Möbius group, the limit set (or unstable set) Λ(G) is defined in [2], p. 97, as the closure of the set of all sinks and sources of elements of G. It follows from Theorem 2.4 that Λ(G) = J(G). However [12] defines the limit set for any Möbius group G to be ℂ̂ ∩ \overline{Gx}, where x is any point in H^{3}. By Proposition 2.6, ℂ̂ ∩ \overline{Gx} = J(G), so this coincides with the previous definition when G is a non-elementary Möbius group. We will adopt the latter definition of the limit set, so that Λ(G) = J(G) for any Möbius group G. The ordinary set (or stable set) Ω(G) is defined as the open set in ℂ̂ complementary to Λ(G) ([2], p. 97). Thus Ω(G) = F (G).

Recall that a Möbius group G is discrete if it is discrete as a subspace of the topological space M or, equivalently, if Id is an isolated point of G.

Two Möbius groups Γ and Γ′ are commensurable if they share a finite index subgroup, that is if the intersection Γ ∩ Γ′ has finite index both in Γ and in Γ′ ([20], p. 170, or [12]). Commensurability is an equivalence relation on Möbius groups. If Γ′ is commensurable with a non-elementary Möbius group Γ then Γ′ is also a non-elementary Möbius group and Λ(Γ′) = Λ(Γ). If Γ and Γ′ are commensurable then so are the closed groups \overline{Γ} and \overline{Γ′}. Any group Γ′ commensurable with a discrete group Γ is also discrete.

Each Möbius semigroup S generates a Möbius group G. If h_{i}, i ∈ I, are semigroup generators of S then h_{i}, i ∈ I, are also group generators of G. We are interested quite generally in the relation between the dynamics of S and the dynamics of G.

For example, we can compare and contrast the connectivity of Julia sets and limit sets.

Theorem 3.3. Let G be a Möbius group generated by a non-elementary semigroup S. If J(S) is connected then Λ(G) is also connected.

This theorem is based on the following lemma.

Lemma 3.4. Assume that the Möbius semigroup S =≺ h_{i} : i ∈ I ≻ is non-elementary and that some set K ⊂ J(S) meets h_{i}^{−1}(K) for all i. If K is connected and |K| > 1 then J(S) is connected.

See Theorem 1 of [39] and Theorem 2.1 of [38] for similar results regarding semigroups which allow non-Möbius rational maps in the semigroup.

Proof of Lemma 3.4. As K is connected, K is contained in some connected component C of J(S). C is closed in J(S) and hence compact. As K is connected and |K| > 1, both K and C are infinite.

7Let G be a subgroup of M and let x be a point in ℝ̂^{3}. The stabilizer of x in G is the subgroup G_{x} = {g ∈ G : g(x) = x}.

Fix i ∈ I and let C_{i} = h_{i}^{−1}(C). Clearly C_{i} is connected. As J(S) is backward invariant under S, C_{i} ⊂ J(S). By our assumption that K meets h_{i}^{−1}(K), C_{i} must meet C so C ∪ C_{i} is connected. As C is a component of J(S), it follows that C_{i} ⊂ C.

As the h_{i} generate S, we see that C is backward invariant under S. As C ⊂ J(S) is infinite and compact, Remark 2.20 shows that C = J(S). Thus J(S) is connected. □

Proof of Theorem 3.3. We take generators h_{i}, i ∈ I, for S and regard G as a semigroup with generators h_{i}, h_{i}^{−1}, i ∈ I. We will show that we may apply Lemma 3.4 to the semigroup G with K = J(S). Note that J(S) ⊂ J(G), since S ⊂ G. Also |J(S)| ≥ 3, since S is non-elementary, and J(S) is connected, by hypothesis.

Now fix i ∈ I. h_{i}^{−1}(J(S)) ⊂ J(S), since J(S) is backward invariant under S, so applying h_{i} to both sides we also get J(S) ⊂ h_{i}(J(S)). Hence both generators h_{i} and h_{i}^{−1} meet the conditions of Lemma 3.4. We deduce that Λ(G) = J(G) is connected. □

Consider S =≺ 2z ≻, so G = ⟨2z⟩. Then J(G) = {0, ∞} is disconnected even though J(S) = {∞} is connected. This shows that one must rule out elementary Möbius groups in Theorem 3.3.

Note that the converse of Theorem 3.3 is false. For in Example 3.10 J(S) is a Cantor set but Λ(G) is connected.

Theorem 3.3 shows that when Λ(G) is disconnected then J(S) and J′(S) are also disconnected. Since J(S_{1}) ⊂ J(S_{2}) whenever S_{1} ⊂ S_{2}, we obtain a similar, but simpler, result.

Proposition 3.5. Let S be a Möbius semigroup and G the group generated by S. If Λ(G) is totally disconnected, then J(S) and J′(S) are also totally disconnected.

We now consider sensitive dependence on initial conditions near a limit point. Since a limit set is a Julia set, Proposition 2.16 implies

Corollary 3.6 (Transitivity for groups). Let G be a Möbius group. Suppose N is a neighborhood of z ∈ Λ(G). Then the union U of the images of g(N) for g ∈ G contains all but at most one point of ℂ̂.

If the Julia set J(h) of a rational function h has an interior point then J(h) = ℂ̂, as in [22], Corollary 4.8. We will now show that limit sets behave similarly.

Corollary 3.7. If the limit set Λ(G) of a Möbius group G contains an interior point then Λ(G) = ℂ̂.

Proof. Let N ⊂ Λ(G) be nonempty and open. The union U of the forward images g(N) is a subset of Λ(G), by forward invariance. U is dense in ℂ̂ (by Corollary 3.6). Since Λ(G) is a closed set, it follows that Λ(G) = ℂ̂. □

On the other hand, Example 6.1 below shows that J(S) ≠ ℂ̂ may have nonempty interior. This shows that the forward invariance of the limit set is essential in the last corollary.

Non-elementary Möbius groups are necessarily complicated.

Theorem 3.8 ([2], p. 90). Every non-elementary Möbius group contains infinitely many loxodromic elements, no two of which have a common fixed point in ℂ̂.

The limit set of a non-elementary Möbius group must also be complicated, as shown by the following consequence of Remark 2.20.

Theorem 3.9. Let G be a Möbius group such that |Λ(G)| ≥ 3. Then Λ(G) is a perfect set and hence uncountable. Moreover each G-invariant set in ℂ̂ with three or more points contains Λ(G). ^{8}

In particular, Λ(G) is the unique minimal G-invariant set, as shown in [2], p. 97, in the case that G is a non-elementary Möbius group.

Theorem 2.4 implies that every non-elementary Möbius semigroup S contains infinitely many loxodromic elements. When these all share a fixed point p, however, S must fix p. This implies that S generates an elementary Möbius group, as in the following example.

Example 3.10. Let S =≺ 3z, 3z − 2 ≻ so S^{−1} =≺ z/3, (z + 2)/3 ≻ and the group generated by S is G = ⟨3z, 3z − 2⟩. J(S) is the Cantor middle-thirds set (see Example 4.4) which implies that S and G are non-elementary semigroups. But S^{−1} is a normal family in ℂ so J′(S) = {∞}. Thus S^{−1} is an elementary semigroup. Since G fixes ∞, G is an elementary Möbius group.

We show Λ(G) = ℝ̂ as follows. First, we see that Λ(G) ⊂ ℝ̂ by Theorem 3.9 since ℝ̂ is completely invariant under G. If Λ(G) ≠ ℝ̂, then there exist a, b ∈ ℝ such that a < b, the interval (a, b) ⊂ ℝ \ Λ(G) and a ∈ Λ(G). Letting p(z) = 3z and q(z) = 3z − 2, we see a_{n} = p^{−n}(q^{−1}(p^{n+1}(a))) = a + 2/3^{n+1}. By complete invariance each a_{n} ∈ Λ(G) and a_{n} ↘ a. However, this contradicts the fact that (a, b) does not meet Λ(G), and so the result follows.

It is also possible to characterize the stable set of a discrete Möbius group using the concept of discontinuous group action.

Definition 3.11. A subgroup G of M acts discontinuously on a completely invariant open set X ⊂ ℂ̂ if for every compact subset K of X there are only finitely many g ∈ G such that g(K) meets K.

When G acts discontinuously on X, this action does not have any sensitive dependence on initial condition. When G is a discrete non-elementary Möbius group, G acts discontinuously on Ω(G) and Ω(G) contains any completely invariant open set X on which G acts discontinuously (see [2] p. 99 and 104). This clearly contrasts with the sensitive dependence on initial conditions that holds near a limit set.

The limit set of a Möbius group G with |Λ(G)| ≥ 3 is constrained by a dichotomy.

Corollary 3.12. Let G be a Möbius group with |Λ(G)| ≥ 3. Then Λ(G) is either connected or else has uncountably many connected components.

Note that the example G = ⟨2z⟩ with Λ(G) = {0, ∞} shows that the condition |Λ(G)| ≥ 3 above is critical.

Proof. We imitate the proof of the corresponding result for the Julia set of a rational function of degree two or more ([22], p.47). Suppose Λ(G) can be expressed as the union A_{0} ∪ A_{1} of two disjoint, non-empty compact subsets. Since Λ(G) is perfect by Theorem 3.9, we have |A_{0}| = +∞ and |A_{1}| = +∞.

We show that neither A_{0} nor A_{1} can be connected. Choose an open set U which meets A_{0} but not A_{1}. Let k ∈ G be a map with a repelling fixed point in U ∩ A_{0} (see Theorem 2.4).

8Recall, that an S-invariant set is by definition a non-empty, compact set which is forward invariant under S.

Since k is loxodromic and |A_{1}| > 1, we may choose n to be large enough so that k^{n}(U) meets both A_{0} and A_{1}. Set h = k^{n}. Since Λ(G) is backward invariant, we must have that h(A_{0}) meets both A_{0} and A_{1}. We can therefore use the forward invariance of Λ(G) to express A_{0} as the disjoint union of non-empty compact subsets A_{00} = A_{0}∩h^{−1}(A_{0}) and A_{01} = A_{0}∩h^{−1}(A_{1}).

We have just shown the following.

Lemma 3.13. Let G be a Möbius group with |Λ(G)| ≥ 3. When Λ(G) is disconnected, each nonempty closed and open subspace of Λ(G) is also disconnected.

Using this lemma, we can similarly express A_{1} as a disjoint union of non-empty compact subsets A_{10} and A_{11}. Again applying this technique we can express any of the sets A_{00}, A_{01}, A_{10} and A_{11} as the disjoint union of non-empty compact subsets. In particular, for any sequence α_{1} … α_{k} where each α_{j} is 0 or 1 we can construct a non-empty compact subset A_{α_{1}...α_{k}} such that for any of the uncountable infinite sequences α_{1} … α_{k} … the infinite decreasing intersection

∩_{k}A_{α_{1}...α_{k}}

is non-empty and contains at least one connected component of Λ(G). Since each such intersection is disjoint from any other such intersection the result follows. □

The preceding proof shows that the Cantor set {0, 1}^{ℕ} is a quotient space of Λ(G) when this limit set is disconnected and |Λ(G)| ≥ 3. More is true, however.

Proposition 3.14. When Λ(G) is disconnected and |Λ(G)| ≥ 3 there is a natural map π : Λ(G) → Q, where Q is a Cantor set, π is onto, and each fiber π^{−1}(q), q ∈ Q, is connected.

Proof. Let X be any topological space and let I be the set of all open and closed subspaces A ⊂ X. For each such A let χ_{A} be the characteristic function of A. In combination, these characteristic functions define a mapping χ from X to the product space {0, 1}^{I}. The image of this map is Q(X), the space of quasicomponents of X, and we have a natural projection π : X → Q(X) that maps each x ∈ X to its quasicomponent.

When X is a compact metric space, I is countable and so Q(X) is compact, metric, and totally disconnected. For such X, every quasicomponent is a component ([24] Section 5-1, Exer. 4) so the fibers of π are just the components of X.

When X = Λ(G) is disconnected, |Q(X)| > 1. By Lemma 3.13, Q(X) has no isolated point. Thus Q(X) is a Cantor set. □

Corollary 3.12 does not necessarily hold in the more general setting of rational semigroups. We record this and other such differences in the following table which compares the dynamics of different classes of rational semigroups S. The first column shows three potential properties of the Julia and Fatou sets. The second column refers to both the dynamics of a rational function of degree two or more (i.e., S =≺ h ≻ with deg(h) ≥ 2) and the dynamics of a Möbius group S with |J(S)| ≥ 3 (i.e., S =≺ h_{i}, h_{i}^{−1} : i ∈ I ≻ with each h_{i} Möbius). The third column refers to the dynamics of a rational semigroup S with no restrictions on the degree of its generators.

The results in the table for rational functions can be found in [22], and the results for Möbius groups and rational semigroups can be found in this paper as indicated (see also [16]).

rational function;  rational semigroup

Möbius group

J(S) and F (S) are  F (S) is forward invariant;
completely invariant  J(S) is backward invariant^{10}

J(S) is connected or

has uncountably many  yes^{11}
connected components  no^{12}

J(S) has empty interior or J(S) = ℂ̂

We now study when the inclusion J(S) ⊂ Λ(G) is an equality.

Proposition 3.15. Let S be a non-elementary Möbius semigroup such that J(S) is completely invariant. Then J(S) = Λ(G) where G is the group generated by S.

For Theorem 3.9 gives the opposite inclusion Λ(G) ⊂ J(S).

Beware that J(S) may be completely invariant with J(S) ≠ J′(S), as illustrated in the following example.

Example 3.16. Let q_{n} be a dense sequence of real numbers and for each n ∈ ℕ let s_{n} be a linear map with real coefficients with source q_{n}. Let S =≺ s_{n} : n ∈ ℕ ≻. Theorem 3.9 gives Λ(G) ⊂ ℝ̂. Since q_{n} ∈ J(S) for all n, ℝ̂ ⊂ J(S). Thus J(S) = Λ(G) = ℝ̂ and so J(S) is completely invariant. But S^{−1} is normal on ℂ and so J′(S) = {∞} ≠ J(S).

4. Contracting iterated function systems

Let S be a rational semigroup and K ⊂ ℂ̂ an S-invariant set equipped with a metric d, consistent with the topology on K. We say that S is an IFS (iterated function system) on (K, d). S uniformly contracts (K, d) if there exists a constant c, 0 < c < 1, such that for each s ∈ S we have s^{∗}d ≤ cd (that is, d(s(z), s(w)) ≤ cd(z, w) for all z, w ∈ K). When S uniformly contracts (K, d) we say that S is a CIFS (contracting iterated function system) on (K, d) or, less precisely, a CIFS on K.

We define the attractor A = A(K, S) as the closure of the set of all fixed points in K of the maps s ∈ S. We suppress the dependence on S and K when there is no chance for confusion.

For example consider the quadratic family S_{c} =≺ z^{2} + c ≻ . The set {∞} is an attractor for all c ∈ ℂ. For small c, S_{c} uniformly contracts the disc K = Δ(0, 1/2) in the Euclidean metric, so S_{c} has multiple attractors.

The situation simplifies for Möbius semigroups, however.

Proposition 4.1. Suppose S is a Möbius semigroup and K is an S-invariant set that is uniformly contracted by S. Either K is a fixed point of S or A(K, S) = J′(S).

Proof. We assume |K| > 1 and show A(K, S) = J′(S).

Suppose s ∈ S is loxodromic with sink p and source q. Then for z ∈ K, z ≠ q, the sequence s^{n}(z), n ≥ 0, lies in K and converges to p. Thus p ∈ K, as K is compact. But q ∉ K as s contracts K and both p and q are fixed by s, Suppose s ∈ S is parabolic with fixed point p. Then s^{n}(z) → p, for any z ∈ K, so p ∈ K. Suppose s ∈ S is elliptic. For some sequence n_{k} → ∞, s^{n_{k}} converges uniformly to Id on ℂ̂.

10See Proposition 2.7 and Example 3.10.

11See Proposition 2.7 and Corollaries 3.12 and 3.7.

12See Examples 6.1, 6.3, and 6.4.

For distinct points z, w ∈ K this implies that the distance from s^{n_{k}}z to s^{n_{k}}w does not tend to 0, contradicting the assumption that S uniformly contracts K. Thus no s ∈ S is elliptic.

All told, we see that each fixed point in K is either a sink or a parabolic fixed point, hence a point in J′(S). Thus A(K, S) ⊂ J′(S). Since each sink lies in K we see that Sink S ⊂ A(K, S). By applying Theorem 2.4 to S^{−1}, we see that either A(K, S) = Sink S = Source S^{−1} = J(S^{−1}) = J′(S) or some p ∈ ℂ̂ is a neutral fixed point of each s ∈ S.

Suppose such a p exists. Then as S contains no elliptic elements, each s ∈ S is parabolic with fixed point p. In this case J′(S) = {p}, by Proposition 2.2. But the parabolic fixed point p lies in K and hence lies in A(K, S). So one still has A(K, S) = J′(S) in this case. □

As mentioned in the introduction of this paper, we are interested in the properties of A in relation to the random dynamics (or random walk) when S =≺ h_{1}, …, h_{N} ≻ is a finitely generated CIFS. Specifically, it is known that A is the ω-limit set^{12} of the random walk given by the following description: starting at a point x_{0} in K, randomly select (we may for simplicity assume a uniform probability) a number ι_{1} from {1, …, n} and define x_{1} = h_{ι_{1}}(x_{0}). Then randomly pick ι_{2} from {1, …, n} and define x_{2} = h_{ι_{2}}x_{1} = h_{ι_{2}}h_{ι_{1}}(x_{0}). Continue in this way to generate an orbit x_{j} for j = 0, 1, . . .. With probability one, the ω-limit set of the orbit x_{j} is exactly A. We should mention that there exists a unique stationary Borel measure μ for this process whose support is exactly A and that an ergodic theorem holds for μ-almost all orbits. See [17, 1, 9] or [28] for details and precise statements of all the above results.

We define the compact image S(N) of a compact set N ⊂ ℂ̂ as the smallest compact set that contains each image s(N), s ∈ S. Compact image commutes with finite unions in the sense that if N_{1}, ..., N_{k} are compact with union N then S(N) is the union of S(N_{1}), ..., S(N_{k}). When S is finitely generated and N is S-invariant then S(N) is the union of the images h(N) where h runs over a finite set of generators of S.

Let S =≺ h_{1}, …, h_{n} ≻ be a CIFS on K, as above. One can construct the attractor A iteratively and characterize it as a fixed point, as follows.

Theorem 4.2 ([17], p. 724). Consider the space of all non-empty compact subsets D ⊂ K, endowed with the Hausdorff metric. Then each sequence D, S(D), S^{2}(D) = S(S(D)), ... converges to A. In particular, for any open set U ⊂ ℂ̂ containing A there exists N > 0 such that S^{k}(D) ⊂ U for all k ≥ N. Moreover A is the unique nonempty compact set in K such that

A = S(A) = ∪_{i=1}^{n}h_{i}(A). (4.1)

We refer to (4.1) as the self-similarity property of A since it expresses A is the union of small (contracted) copies of itself h_{i}(A). Beware that the assumption of uniform contraction on K is critical here, as illustrated by Remark 2.21 (using the contracting inverse semigroup S^{−1} instead of the semigroup S given there).

Example 4.3 (Linear contractions). Consider S =≺ h_{1}, …, h_{n} ≻ where each h_{i} is a linear contraction, i.e., for each i = 1, …, n we have h_{i}(z) = a_{i}z + b_{i} where |a_{i}| < 1. In this case, it is not hard to see that for R sufficiently large, the set K = \overline{Δ(0, R)} is forward invariant under each h_{i}, and therefore under S. Since these maps are uniformly contracting with respect to the Euclidean metric (with contraction coefficient c = max_{i=1,...,n} |a_{i}|), we have that S is a CIFS with an attractor A ⊂ K. We also note that for any compact set D ⊂ ℂ, we have S^{k}(D) → A (since one can choose R large enough that D ⊂ K). It follows that A is independent of R, for R sufficiently large.

12The ω-limit set of a sequence x_{j} is the set of all y such that there exists a subsequence x_{j_{k}} → y.

Under the conditions of Proposition 4.1, with S finitely generated, Theorem 4.1 implies that any self-similar S-invariant set B ⊂ K must equal J′(S). This allows us to calculate many Julia sets.

Example 4.4. Consider S =≺ h_{1}, h_{2} ≻ for h_{1}(z) = 3z and h_{2}(z) = 3z − 2. The inverse semigroup S^{−1} is a CIFS on, say, K = \overline{Δ(0, 10)}. Since the middle third Cantor set C satisfies C = ∪_{i=1}^{2}h_{i}^{−1}(C), we conclude J(S) = A(S^{−1}) = C.

5. Thick attractors

Let S be a finitely generated Möbius semigroup that is a CIFS on some compact set K^{∗}. When the interior of K^{∗} contains A = A(K^{∗}, S) we say this attractor A is thick.^{13} As a thick attractor for S attracts the orbits of all points in the large set K^{∗}, we see that there are many seed points whose random orbit will generate a good picture of A.

Thick attractors arise whenever certain simple dynamical conditions are met.

Theorem 5.1. Let K and K′ be nonempty disjoint compact sets in ℂ̂, not both singleton sets, and let h_{1}, ..., h_{n}, n ≥ 1, be loxodromic Möbius transformations such that for each j, h_{j}(K) ⊂ K and K′ ⊂ h_{j}(K′). Let S be the semigroup generated by the h_{j}, so the Sinvariant set K is disjoint from the S^{−1}-invariant set K′. Then S uniformly contracts K and A(K, S) = J′(S) = Sink S is a thick attractor.

The following example illustrates the importance of the conditions put on |K| and |K′| above.

Example 5.2. Let S =≺ 2z ≻, K = {0}, and K′ = {∞}. Here J′(S) = {∞} = Sink S, but since it is not a subset of K, it cannot be A(K, S).

Although Example 5.2 shows how the conclusion of Theorem 5.1 can fail when |K| = |K′| = 1, this exceptional case can be fully analyzed. Indeed, suppose without loss of generality that K = {0} and K′ = {∞}. Then both 0 and ∞ are fixed by S. Thus each generator h_{j} ∈ S is of the form z ↦ a_{j}z. Since, as noted above, each h_{j} is then an isometry on ℂ^{∗} endowed with the metric |dz|/|z| we have that ℂ^{∗} ⊂ F (S). If max |a_{j}| < 1, then J(S) = \{∞\} and K = \{0\} = J′(S) is a thick attractor (using K^{∗} = \overline{Δ(0, r)} for any r > 0). If min |a_{j}| > 1, then J(S) = {0} and K′ = {∞} = J′(S) is a thick attractor (using K^{∗} = ℂ̂ \ Δ(0, r) for any r > 0). If min |a_{j}| < 1 and max |a_{j}| > 1, then S does not have a thick attractor and {0, ∞} = J′(S) = J(S).

Proof of Theorem 5.1. Since h_{1} has no neutral fixed point, Theorem 2.4 shows that Sink S = Source S^{−1} = J(S^{−1}) = J′(S).

We first prove the theorem in the cases where K or K′ has at most two points.

13Beware that S may also be a CIFS on another set K with the same attractor A for which Int K does not contain A.

Consider the case |K′| = 2. We may suppose, by conjugating S in M, that K′ = {0, ∞}. As K′ is backward invariant by h_{1} and as h_{1} is loxodromic, h_{1}(z) = az for some a with |a| ≠ 1. Now K is nonempty so we may choose z ∈ K ⊂ ℂ^{∗}. The sequence z, h_{1}(z), h_{1}^{2}(z), ... lies in K and converges to either the origin (if |a| < 1) or to ∞ (if |a| > 1), contradicting the fact that K is closed and disjoint from K′. Thus |K′| ≠ 2. Likewise one can show |K| ≠ 2.

Now suppose that |K′| = 1 and |K| > 1. We may suppose, by conjugating S in M, that K′ = {∞} so h_{j}(z) = a_{j}z + b_{j} for some a_{j}, b_{j}. As h_{j} is loxodromic, h_{j}(K) ⊂ K, K ⊂ ℂ is compact, and |K| > 1 we must have |a_{j}| < 1 (else there would exist a point in K which is not the repelling fixed point of h_{j} and such a point would iterate under h_{j} to ∞). Thus S uniformly contracts ℂ, with its usual distance function and with contraction constant equal to the maximum of |a_{1}|, ..., |a_{n}|, and so S uniformly contracts K. When s ∈ S, s has a source at ∞ so the only fixed point of s in K is its sink. Thus Sink S = A(K, S). Consider the compact set K^{∗} of all complex numbers whose distance from K is at most 1. Then S is also a CIFS on K^{∗} with A(K^{∗}, S) = A(K, S) ⊂ K ⊂ K^{∗}, so this attractor is thick. Thus the theorem holds when |K′| = 1 and |K| > 1.

When |K| = 1 and |K′| > 1 we may, in a similar fashion, suppose K = {∞} so that h_{j}(z) = a_{j}z + b_{j} for some a_{j}, b_{j} with |a_{j}| > 1. Then each s ∈ S is loxodromic with sink at ∞. It is trivial that S uniformly contracts K and the rest of the theorem is clear in this case.

For the rest of the proof we shall assume K and K′ have at least 3 points each. We need three lemmas in order to show that S uniformly contracts K.

Lemma 5.3. Each s ∈ S is loxodromic with sink in K and source in K′. Moreover s(K) is strictly contained in K and s(K′) strictly contains K′.

Proof. Say s ∈ S is loxodromic with sink p and source q. As |K| > 1 we can choose z ∈ K, z ≠ q. As s(K) ⊂ K, the sequence z, s(z), s^{2}(z), ... approaches p. As K is compact, p ∈ K. Likewise q ∈ K′. As K and K′ are disjoint, we see that q ∉ K. Iterating s^{−1}, we see that s^{−1}(K) is not a subset of K, so s(K) is strictly contained in K. Likewise s(K′) strictly contains K′.

In particular h_{j}(K) is strictly contained in K for all j. This implies s(K) is strictly contained in K for all s ∈ S. Likewise s^{−1}(K′) is strictly contained in K′, so K′ is strictly contained in s(K′).

Suppose s is elliptic. After conjugating s, we may suppose that s preserves spherical distance on ℂ̂. Then s is an isometry of the compact metric space K, so s(K) = K ([24], Exer. 7 p. 182). This contradicts the strict inclusion we just established, so s is not elliptic.

Suppose s is parabolic with fixed point p. Then s(K) ⊂ K and K is compact and nonempty. If z ∈ K then the orbit z, s(z), s^{2}(z), ... approaches p and so p ∈ K. Reasoning likewise with s^{−1}, we see that p ∈ K′, contradicting our assumption that K and K′ are disjoint. Thus s is not parabolic

The only remaining possibility is that s is loxodromic. □

Let U be the complement of K′ in ℂ̂ and let V be a component of U. Then V is open and connected and its complement has at least three points so V has a unique hyperbolic metric. This is a complete Riemannian metric of curvature -1 compatible with the complex structure on V. We let d_{V} denote the distance function on V defined by this hyperbolic metric. We will use the various d_{V} to define via (5.2) a distance function d on K and prove that S uniformly contracts (K, d).

As K ⊂ U is compact and the components of U are disjoint and open, K meets only finitely many components U_{1}, ..., U_{m} of U. We have m ≥ 1 since K is nonempty. We let d_{j} = d_{U_{j}} be the hyperbolic distance function on U_{j}.

Since K′ is backward invariant under S, U is forward invariant under S. Thus S acts on the finite index set F = {1, ..., m} so that s(U_{j}) ⊂ U_{s(j)} for all j ∈ F and s ∈ S. For s(U_{j}) ⊂ U is connected so s(U_{j}) ⊂ V for some component V of U. As K ∩ U_{j} is nonempty and s(K ∩ U_{j}) ⊂ K ∩ V, we must have V = U_{k} for some k ∈ F. As the U_{i}’s are disjoint, k is uniquely determined by j and s and we may define s(j) = k.

The following result is certainly consistent with S being uniformly contracting on K.

Lemma 5.4. There is a λ > 0 such that for every s ∈ S expressed as a word of length at least λ in the generators h_{1}, ..., h_{n} one has s(K) ⊂ U_{k} for some k = k(s).

Proof. We claim that each s ∈ S fixes a unique point of F. Recall from Lemma 5.3 that s is loxodromic with sink p, say, in K. As K is covered by the U_{j}, there is a j ∈ F with p ∈ U_{j} which implies s(j) = j. If k ∈ F and s(k) = k then s(U_{k}) ⊂ U_{k}. Thus p ∈ \overline{U_{k}}. As the U_{j} are disjoint and open, k = j. Thus j is the unique fixed point of s in F.

Consider now the product action of S on F^{2} = F × F. By the uniqueness result just proven, any fixed point in F^{2} of any element of S must lie on the diagonal D ⊂ F^{2}. Let l = m^{2} − m = |F^{2}| − |D| and let s = h_{j_{l}}...h_{j_{1}} ∈ S be a product of l generators h_{j}.

We claim that s(F^{2}) ⊂ D. To see this we define s_{i} = h_{j_{i}}...h_{j_{1}} for 0 ≤ i ≤ l, with the understanding that s_{0} = Id. For (j, k) ∈ F^{2} the pairs s_{i}(j, k) ∈ F^{2}, 0 ≤ i ≤ l, are either distinct or not. If these pairs are distinct then our choice of l implies that s_{i}(j, k) ∈ D for some i. If these pairs s_{i}(j, k) are not distinct then there exist i < i′ with s_{i}(j, k) = s_{i′}(j, k). But s_{i}(j, k) is fixed by h_{j_{i′}}...h_{j_{i+1}} ∈ S and so s_{i}(j, k) ∈ D. Either way, since D is S -invariant and l ≥ i we have s(j, k) = s_{l}(j, k) ∈ D. This proves our claim.

Now let k = s(1) and pick j ∈ F. We have shown that s(j, 1) ∈ D, so we have s(j) = k. Thus s(U_{j}) ⊂ U_{k}. As the U_{j}, j ∈ F, cover K, we find that s(K) ⊂ U_{k} as well. □

The preceding proof shows that we may take λ = m^{2} − m, but in a given example a much smaller λ may suffice.

Now define the open set U_{K} to be the union of all components of U that meet K, so U_{K} = U_{1}∪...∪U_{m}. We define a generalized distance function δ on U_{K} by defining δ(z, w) = ∞ when z, w lie in different components of U_{K} and δ(z, w) = d_{j}(z, w) when z, w ∈ U_{j}. For l ≥ λ, we define a distance function δ_{l} on U_{K} by setting δ_{l}(z, w) equal to the maximum value of δ(s(z), s(w)) over all s ∈ S expressible as a word of length l in the generators h_{1}, ..., h_{n}. Note that our choice of λ assures that δ_{l}(z, w) is finite, and we leave it to the reader to verify that δ_{l} is a distance function. We surely have δ_{l}(h_{j}(z), h_{j}(w)) ≤ δ_{l+1}(z, w) for all z, w ∈ U_{K}, since sh_{j} is expressible as a word of length l + 1 whenever s is expressible as a word of length l. In a more convenient notation, we have

h_{j}^{∗}δ_{l} ≤ δ_{l+1}. (5.1)

Lemma 5.5. There is a μ > 0 such that for every j ∈ F and each s ∈ S expressible as a word of length at least μ in the generators h_{1}, ..., h_{n}, s(U_{j}) is strictly contained in U_{s(j)}.

Proof. Fix s ∈ S and let f ∈ F be the unique fixed point of s and let p and q be the sink and source of s, respectively. We have seen that p ∈ U_{f} ∩ K. As U_{f} is an open neighborhood of p, if s^{−1}(U_{f}) ⊂ (U_{f}) we would find that U_{f} contains ℂ̂\{q}, contradicting the fact that the complement of U contains at least three points. Thus s(U_{f}) is strictly contained in U_{f}.

Suppose s = h_{j_{m}}...h_{j_{1}} ∈ S is a product of m generators h_{j}. Let s_{i} = h_{j_{i}}...h_{j_{1}} for i = 1, ..., m. Given j ∈ F we define j_{0}, ..., j_{m} in F so that j_{0} = j and s_{i}(U_{j}) ⊂ U_{j_{i}}. Since |F | = m we must have j_{a} = j_{b} = f for some a < b and some ℓ ∈ F. Then s^{∗} = s_{b}s_{a}^{−1} ∈ S and s^{∗}(ℓ) = ℓ. Hence from above s^{∗}(U_{ℓ}) is strictly contained in U_{ℓ}. It follows that s(U_{j}) is strictly contained in U_{j_{m}}. □

The preceding proof shows that we may take μ = m, but in a given example a much smaller μ may suffice.

Fix a generator h = h_{i} of S. Suppose j, k ∈ F and h(U_{j}) ⊂ U_{k}. As h is holomorphic, a theorem of Pick shows that h cannot increase hyperbolic distance. That is d_{k}(h(z), h(w)) ≤ d_{j}(z, w) for all z, w ∈ U_{j}, or, more concisely, h^{∗}d_{k} ≤ d_{j}. Moreover Pick shows that if h(U_{j}) is strictly contained in U_{k} then h is uniformly contracting on each compact set X ⊂ U_{j}, so there is a contraction constant c = c(X, i, j) < 1 such that h^{∗}d_{k} ≤ cd_{j} as distance functions on X ([22], Theorem 2.11).

Let c(K) < 1 be the maximum of c(K ∩ U_{j}, i, j) over all i and j such that h_{i}(U_{j}) is strictly contained in some U_{k} (note that this does not necessarily happen for all i, j). If j, k ∈ F with h_{i}(U_{j}) strictly contained in U_{k} then h_{i}^{∗}d_{k} ≤ c(K)d_{j} as distance functions on K ∩ U_{j}. By our choice of μ, we see that for every s = h_{j_{μ}}...h_{j_{1}} ∈ S expressed as a product of μ generators and for every j ∈ F we have s^{∗}d_{k} ≤ c(K)d_{j} as distance functions on K ∩ U_{j}, where k = k(s, j). By our choice of λ, we find δ_{l+μ} ≤ c(K)δ_{l} as distance functions on K for all l ≥ λ.

Let a = a(K) = c(K)^{1/μ} < 1. Define the distance function d on U_{K} by

d = a^{μ−1}δ_{λ} + a^{μ−2}δ_{λ+1} + ... + aδ_{λ+μ−2} + δ_{λ+μ−1}. (5.2)

Using (5.1) together with δ_{λ+μ} ≤ c(K)δ_{λ} = a^{μ}δ_{λ} on K, we see that on K we have

h_{j}^{∗}d = h_{j}^{∗}(a^{μ−1}δ_{λ} + a^{μ−2}δ_{λ+1} + ... + aδ_{λ+μ−2} + δ_{λ+μ−1})
≤ a^{μ−1}δ_{λ+1} + a^{μ−2}δ_{λ+2} + ... + aδ_{λ+μ−1} + δ_{λ+μ}
≤ a^{μ−1}δ_{λ+1} + a^{μ−2}δ_{λ+2} + ... + aδ_{λ+μ−1} + a^{μ}δ_{λ} = ad.

As a < 1, S uniformly contracts (K, d). Now Proposition 4.1 shows that A(K, S) = J′(S).

It only remains to check that the attractor J′(S) is thick. Let K^{∗} ⊂ U_{K} be the compact unit neighborhood of K in the hyperbolic metric, that is, z ∈ K^{∗} when z ∈ U_{K} and d_{K}(z, w) ≤ 1 for some w ∈ K. By the theorem of Pick K^{∗} is forward invariant under S, and so we may repeat the above argument, but with K replaced by K^{∗}, to show that S uniformly contracts (K^{∗}, d). But Sink S ⊂ K lies in the interior of K^{∗} so Sink S is thick. □

Note that the preceding proof simplifies greatly when m = 1, that is when K lies in a single component V ⊂ U or, equivalently, when K′ does not separate K in ℂ̂. In this case S uniformly contracts (K, d_{V}) and so the last two lemmas are not needed at all. In Example 6.2 we show that K′ = J(S) may separate K = J′(S), so J′(S) ⊂ F (S) is a thick attractor that meets two or more components of F (S). Hence the above generality is required for our purposes.

Let S be a finitely generated Möbius semigroup. We define an S-block to be a nontrivial compact set N ⊂ ℂ̂ whose interior contains S(N) (by nontrivial we mean that neither N nor its complement is empty).

Example 5.6. Recall from the introduction that our motivation is to study S_{β} =≺ f_{β}, g_{β} ≻ where f_{β}(z) = β + 1/z and g_{β}(z) = −β + 1/z (see also Section 7.1). We now give a block for S_{β} when |ℜβ−ℑβ| > 2. Let A be the segment from 1 to −i and let^{14} E = A∪1/A∪−A∪−1/A, so E is a convex curve in ℂ. By construction, E = 1/E. Let N be the compact region in ℂ̂ containing ∞ and bounded by E. Then 1/N ± β is the bounded region with boundary E ± β and so N is a block for S_{β}. The set \{\overline{z}|z ∈ N\} is a block for S_{β} when |ℜβ + ℑβ| > 2. Combining these two cases, we see that S_{β} has a block whenever |ℜβ| + |ℑβ| > 2.

The term “block” is suggested by the “isolating blocks” which Conley used to study invariant sets of flows. Conley’s work was based on Smale’s pioneering study of hyperbolic attractors for smooth flows, in which forward invariant compact regions played a prominent role.

Blocks give a practical way to detect thick attractors.

Theorem 5.7. For a finitely generated Möbius semigroup S the following are equivalent:

1) S has a thick attractor J′(S)

2) there is an S-block

3) S^{−1} has a thick attractor J(S)

4) there is an S^{−1}-block

5) J′(S) is disjoint from J(S) and the generators of S are loxodromic.

Proof. We show 1) implies 2). Let S be a CIFS on (K, d) whose attractor A lies in Int K. Let U be the open ϵ-neighborhood of A in (K, d), where ϵ is chosen small enough that U lies in Int K, and let N = \overline{U}. Then since S uniformly contracts (K, d) and N lies in the closed ϵ-neighborhood of the S-invariant set A, S(N) ⊂ U. Thus N is an S-block.

Now suppose 2) holds. Let N be an S-block and U its interior. Since N is compact and ℂ̂\N is nonempty, S is a normal family on U by Montel’s Theorem. Thus U ⊂ F (S). Since N is forward invariant under S, it is backward invariant under S^{−1}, which implies J′(S) = J(S^{−1}) ⊂ N by Lemma 2.18. Since S(N) is also an S-block, J′(S) ⊂ S(N) ⊂ U ⊂ F (S) and so J(S) is disjoint from J′(S). Let s ∈ S. Since s(N) ⊂ U, we see that s is neither elliptic nor the identity. The map s is also not parabolic since its fixed point would have to be in both J(s) ⊂ J(S) and J(s^{−1}) ⊂ J′(S). Thus s is loxodromic and 5) holds.

Now suppose 5) holds. Letting K = J′(S) and K′ = J(S) in Theorem 5.1, we see that if |J′(S) ∪ J(S)| ≥ 3, then 1) must follow. When |J′(S)| = 1 = |J(S)| we quickly see from the discussion immediately following Example 5.2 that using K = J′(S) and K′ = J(S) we must have J′(S) is a thick attractor for S. Thus 1), 2), and 5) are equivalent, which by replacing S by S^{−1} quickly shows that 5) is equivalent to 3) and 4). □

Remark 5.8. We can see directly that 2) implies 4). Let N be an S-block with interior U and let N′ = ℂ̂\U. Then N′ is an S^{−1}-block since for h ∈ S we have N ⊂ h^{−1}(U) so h^{−1}(N′) ⊂ ℂ̂\N ⊂ Int N′.

Remark 5.9. The existence of a block for S restricts the dynamics of nearby semigroups. For suppose N is an S-block and let V be an open set such that S(N) ⊂ V and \overline{V} ⊂ Int N. The Möbius transformations m(z) with m(N) ⊂ V define an open neighborhood Σ of S in M. Σ is a Möbius semigroup and N is a Σ-block. For any semigroup S^{∗} ⊂ Σ it follows that N is also an S^{∗}-block. This implies that in a continuously parametrized family of Möbius semigroups, the condition that N be a block defines an open set of parameters. Letting N vary, we see that the existence of a block also defines an open set of parameters. In this sense, blocks (and hence thick attractors) are a stable phenomenon.

14For a set X ⊂ ℂ̂ we use the notation −X = {−z : z ∈ X}, X^{−1} = 1/X = {1/z : z ∈ X}, X + α = {z + α : z ∈ X} and αX = {αz : z ∈ X}, for any α ∈ ℂ.

For a fixed finitely generated Möbius semigroup S, an S-block N can be used to calculate Julia sets and Fatou sets. We define N^{−n} for n > 0 as the set of all points z ∈ ℂ̂ such that s(z) ∈ N for all s ∈ S that are expressible as a product of n generators. Clearly N^{−n} is compact. N^{−n} is nontrivial as it contains the sink but not the source of any generator of S. We also define N^{0} = N and N^{n} = S^{n}(N) for n > 0. Then for all integers n, S(N^{n}) ⊂ N^{n+1} and N^{n+1} lies in the interior of N^{n}.

We now describe the limiting behavior of these nested compact sets N^{n}.

Proposition 5.10. Let N be an S-block, where S is a finitely generated Möbius semigroup. Then J′(S) = ∩ N^{n} and F (S) = ∪ N^{n}, n ∈ ℤ.

Proof. Let K be the intersection of all the N^{n} and let O be the union of all the N^{n}. Then O is open, since it is the union of all the Int N^{n}. We let K′ = ℂ̂ − O. Since each N^{n} is strictly contained in ℂ̂ and the sequence Int N^{n} is decreasing, K′ is compact and nonempty. Clearly K′ and K are disjoint. Since each N^{n} is S-invariant, K is S-invariant and K′ is S^{−1}-invariant.

Moreover S(K) = K. For if x ∈ K then for each m we have x ∈ s_{1}...s_{m}(N) with each s_{i} a generator of S. By the pigeonhole principle, some generator must occur infinitely often as s_{1}, which implies that x ∈ s_{1}(K).

By Theorem 4.2 we see that K = A(S, N). By Theorem 5.1, with K replaced by N, we see that A(S, N) = J′(S) and so K = J′(S).

Likewise S′(K′) = K′ and K′ = A(S′, N′) = J(S), where N′ is the block of Remark 5.8. □

Remark 5.11. For S as in Proposition 5.10, the dynamics of S on F (S) becomes trivial when we ignore the dynamics of S on J′(S). More precisely, let X = F (S)/J′(S) with the quotient topology and denote the singular point J′(S)/J′(S) ∈ X by p. X is locally compact and metrizable. S acts naturally on X since both F (S) and J′(S) are forward invariant. Fix an integer n. The sets S^{k}(N^{n})/J′(S) ⊂ X, k = 1, 2, 3, ... are nested neighborhoods of p. Since S^{k}(N^{n}) ⊂ S^{k+n}, the intersection of these neighborhoods is p. Thus the family of mappings s : N^{n}/J′(S) → N^{n}/J′(S), s ∈ S, converges uniformly to the constant map p. But each compact K ⊂ F (S) is contained in Int N^{n} for some n, since these open sets form a nested family whose union is F (S). Thus the family of mappings s : X → X, s ∈ S, converges to the constant map p, uniformly on each compact subset of X.

On the other hand, Theorem 2.4 shows that any open set that meets J(S) contains a source of some s ∈ S, whose S-orbit is not attracted to J′(S). In this sense, F (S) is the basin of the attractor J′(S).

In fact J′(S) can be calculated from the orbit of any Fatou point z, as follows by setting Z = {z} in this result.

Corollary 5.12. Let S be a Möbius semigroup generated by a finite number of loxodromic maps such that J′(S) ⊂ F (S). For each compact subset Z ⊂ F (S) the sequence S^{n}(Z) converges to J′(S) in the Hausdorff metric on compact subsets of ℂ̂.

Proof. By Theorem 5.7, there is an S-block N. By Proposition 5.10, Z ⊂ N^{n}, for some n, and so S^{k}(Z) ⊂ N^{n+k} for all k > 0. By Theorem 5.1, there is a compact neighborhood K of J′(S) that is uniformly contracted by S. We can choose k so that Int K contains N^{n+k}. Now apply Theorem 4.2 with D = S^{k}(Z) ⊂ K. We see that S^{n}(Z), n ≥ k, converges to J′(S) in the Hausdorff metric on compact subsets of (K, d). Let d_{S} denote the spherical distance function on ℂ̂. Since d_{S} and d define the same topology on K, we also have convergence in the Hausdorff metric on compact subsets of (K, d_{S}). Thus S^{n}(Z), n ≥ 0, converges to J′(S) in the Hausdorff metric on compact subsets of (ℂ̂, d_{S}). □

Remark 5.13. To conclude, we note that the existence of a block implies that the long term states of a randomly generated orbit must accumulate on J′(S), provided the initial condition is in F (S). Moreover the ω-limit set of such a random orbit is all of J′(S) (with probability one). Hence, plotting the points of such a random orbit (and discarding the first 50 or so as transient) approximates points in J′(S).

6. Some Julia sets of Möbius semigroups

In this section we illustrate various possible properties of the Julia set of a finitely generated non-elementary Möbius semigroup. We particulary present examples that contrast these Julia sets with the Julia sets of rational maps of degree ≥ 2 or with the limit sets of Möbius groups.

We first show that a Julia set can have interior without being all of ℂ̂, contrary to the case for limit sets (Corollary 3.7) and for Julia sets of rational functions of degree greater than or equal to two ([22], p. 58).

Example 6.1 (F neither empty nor dense). Define S to be the semigroup generated by h_{±}(z) = (±b − iz)/a, where 0 < a < 1 and b = 1 − a^{2}. Then S^{−1} =≺ iaz ± ib ≻ is a CIFS on K for any large disc K with the Euclidean metric, as |ia| < 1. Let R be the closed rectangular region with corners ±a ± i. Then h_{+}^{−1}(R) ∪ h_{−}^{−1}(R) = R if a ≥ 1/\sqrt{2}. For such a, Theorem 4.2 implies that A(S^{−1}) = R. By Proposition 4.1 we see that J(S) = R. Thus J(S) ≠ ℂ̂ and Int J(S) is nonempty, as desired.

Furthermore, J′(S) = {∞} since S^{−1} is normal on ℂ, and the group G generated by S has Λ(G) = ℂ̂, by Corollary 3.7, since R = J(S) ⊂ Λ(G).

Corollary 7.22 (with β^{2} ∉ ℝ and G_{β} not discrete) gives many 2-generator Möbius semigroups whose Julia sets equal ℂ̂.

We now show that J(S) may separate J′(S), meaning that J′(S) may lie in the Fatou set F (S) but meet more than one such Fatou component.

Example 6.2 (J separates J′). Choose a closed triangular region T ⊂ ℂ such that the open triangle U whose vertices are the midpoints of T contains 0. Label the vertices of T as c_{i} and consider the doubling dilation h_{i}(z) = 2z − c_{i} centered at c_{i} for i = 1, 2, 3. Let N be a closed r-neighborhood of T \U for r positive and small, so that 0 ∉ N but Int(h_{i}(N)) contains N for all i. Let S_{T} be the expanding linear semigroup generated by h_{i}, i = 1, 2, 3. Then J′(S_{T}) = {∞} and, since T is backward invariant under S_{T}, J(S_{T}) ⊂ T is bounded. Indeed, as in Example 4.4 we see that J(S_{T}) ⊂ N is a Sierpinski triangle and U is a bounded component of F (S_{T}). Thus J(S_{T}) separates 0 from ∞.

Let h(z) = b/(z − a), so h^{−1}(z) = a + b/z. Choose b very small so that h^{−1}(N) is small and then choose a so that h^{−1}(N) is contained in Int N. Let S be the Möbius semigroup generated by h and S_{T} . Then J′(S_{T}) ⊂ J′(S) so ∞ ∈ J′(S) and 0 = h(∞) ∈ J′(S).

By construction Int N contains S^{−1}(N) and so N is an S^{−1}-block. By Theorem 5.7 J′(S) ⊂ F (S). Since J(S) contains J(S_{T}) it follows that J(S) separates the points 0 and ∞ of J′(S).

By adding k appropriate loxodromic generators to S_{T} , with sources in distinct components of F (S_{T}), we can likewise construct a semigroup S as in Theorem 5.1 in which J′(S) meets more than k Fatou components.

Recall from Corollary 3.12 that a disconnected limit set with three or more points has uncountably many components and that the same holds for the Julia set of a rational map of degree ≥ 2. The next two examples show that the situation is not so simple for Möbius semigroups.

Example 6.3 (J with n > 1 components). Let J be the union of n > 1 disjoint closed line segments J_{1}, J_{2}, …, J_{n} in ℝ. Choose m linear maps a_{j} such thavt ∪_{j=1}^{m}a_{j}(J) = [0, 1] (depending on the number and configuration of the J_{k} this may require a fair number of maps a_{j}, but it can always be done). Now, for each k = 1, …, n, let b_{k} be a linear map whose image of [0, 1] is exactly J_{k}. Let S =≺ b_{k} ∘a_{j} : j = 1, …, m; k = 1, …, n ≻ and note that the generators are all linear contractions on ℂ since each b_{k} ∘ a_{j} maps J_{k} strictly inside itself. Since J = ∪_{k=1}^{n}b_{k}([0, 1]) = ∪_{k=1}^{n}b_{k}(∪_{j=1}^{m}a_{j}(J)) = ∪_{k=1}^{n} ∪_{j=1}^{m} b_{k} ∘a_{j}(J), we have by Theorem 4.2 that J is the attractor set of the contracting iterated function system S. By Proposition 4.1, J(S^{−1}) = J has exactly n components.

In this example J(S) = {∞} since S is normal on ℂ. We can also calculate Λ(G) for the group G generated by S. J = J(S^{−1}) ⊂ Λ(G) and Theorem 2.4 implies that some h ∈ S^{−1} has a repelling fixed point in J_{1} that is not an endpoint of J_{1}. Forward invariance shows that ℝ = ∪_{j=1}^{∞}h^{j}(J) ⊂ Λ(G). Since ℝ̂ is completely invariant under G, however, Theorem 3.9 implies that Λ(G) = ℝ̂.

We note that Example 6.3 can be adjusted to generate a Julia set with exactly two components using 4 generators by using J_{1} = [2, 2.49], J_{2} = [2.51, 3], a_{1}(z) = z − 2, and a_{2}(z) = 0.1 + (z − 2)/2. This approach can also be adjusted to produce n components of J(S) with 2n generators (just select intervals J_{k} to be subintervals of [2, 3] with small gaps in between and use a_{1} and a slight change to the constant term in a_{2}). However, also see Example 6.4.

For any n ∈ ℕ, there are examples of polynomial semigroups whose Julia set has exactly n components (such a semigroup may also be chosen so all its elements have degree ≥ 2 and so that its planar postcritical set is bounded). In the first such examples, the semigroup was generated by 2n elements [38]. However, 4 generators are sufficient for any n [33].

The following example shows for any n ∈ ℕ, there is a 5 generator Möbius semigroup whose Julia set has exactly n components. Likewise 5 generators can produce a countably infinite number of Julia components.

Example 6.4 (J with ℵ_{0} components). Let J_{1} = [2, 2.49] and J_{2} = [2.51, 3] and label the gap between them H = (2.49, 2.51). Now consider the linear maps (as in Example 6.3) chosen so that b_{1}([0, 1]) = J_{1}, b_{2}([0, 1]) = J_{2}, a_{1}(z) = z − 2, and a_{2}(z) = 0.1 + (z − 2)/2. Thus for S =≺ b_{k} ∘ a_{j} ≻ we have A(S) = J_{1} ∪ J_{2}.

Set h(z) = 0.02(z − 2.49) + 2.49 and note that h(3) < 2.51. Thus J_{2}, h(J_{2}), h^{2}(J_{2}), … are all disjoint intervals. Let S^{∗} =≺ h, b_{k} ∘ a_{j} ≻. We claim that J^{∗} = J_{1} ∪ J_{2} ∪ ∪_{n=1}^{∞}h^{n}(J_{2}), which has exactly ℵ_{0} components, is the attractor set of S^{∗}.

Since each map a_{j} is linear and maps both J_{1} and J_{2} into [0, 1], each map a_{j} must also map the gap H into [0, 1]. Hence each composite map b_{k} ∘ a_{j} not only maps J = J_{1} ∪ J_{2} into itself, it also maps H into J. Since ∪_{n=1}^{∞}h^{n}(J_{2}) ⊂ H and A(S) = J_{1} ∪ J_{2} is forward invariant under S, we then have that J^{∗} is forward invariant under each b_{k} ∘ a_{j}. Clearly from the definition of h we see that J^{∗} is also forward invariant under h. Thus A(S^{∗}) ⊂ J^{∗}. Since S ⊂ S^{∗} implies J_{1} ∪ J_{2} = A(S) ⊂ A(S^{∗}) and A(S^{∗}) is forward invariant under h, we then see that J^{∗} ⊂ A(S^{∗}). Hence A(S^{∗}) = J^{∗}. Thus, by Proposition 4.1, we see that the inverse semigroup of S^{∗} has Julia set equal to J^{∗}.

We again note that, as in Example 6.3, J(S^{∗}) = {∞} and Λ(G) = ℝ̂, where G is the group generated by S^{∗}.

If the above map h is chosen to instead have its sink in the open interval (2, 2.49), then some h^{k}(J_{2}) would meet J_{1} and thus the attractor set J^{∗} would have a finite number of components. One could then assure that we have exactly n components of J^{∗} by adjusting h carefully. This constructs a Julia set with exactly n components out of only 5 Möbius generators.

7. Caruso semigroups

The previous sections have set up a framework for the study of Möbius semigroups. In this section, we explore the dynamics of a particular family of Möbius semigroups S_{β}, β ∈ ℂ^{∗}. We study the topological properties of the Julia set of S_{β} and the limit set of the group G_{β} it generates. We will especially examine the extreme cases where the Julia sets of S_{β} and S_{β}^{−1} are equal or disjoint.

7.1. Preliminaries. For a fixed nonzero complex number β we let f(z) = β + 1/z and g(z) = −β + 1/z. We define the Möbius group G_{β} = ⟨f, g⟩. the semigroup S_{β} =≺ f, g ≻, and its inverse semigroup S_{β}^{−1} =≺ f^{−1}, g^{−1} ≻ .

Note that f^{−1}(z) = 1/(z − β) and g^{−1}(z) = 1/(z + β). It follows that fg^{−1}(z) = z + 2β. Thus the only point in H^{3} ∪ ℂ̂ with a finite orbit under fg^{−1} is ∞ ∈ ℂ̂. As f(∞) ≠ ∞, G_{β} has no finite orbit in H^{3} ∪ ℂ̂. By definition, this shows that G_{β} is a non-elementary Möbius group.

As S_{β} (respectively S_{β}′) generates G_{β}, it has no finite orbit in H^{3} ∪ ℂ̂. Corollary 2.22 now shows that S_{β} (respectively S_{β}′) is non-elementary.

We denote the Julia set of S_{β} by J_{β} = J(S_{β}), the Julia set of S_{β}^{−1} by J′_{β} = J(S_{β}^{−1}), and the limit set of G_{β} by Λ_{β} = Λ(G_{β}). Similarly, we denote the Fatou set of S_{β} by F_{β} = F (S_{β}), the Fatou set of S_{β}^{−1} by F′_{β} = F (S_{β}^{−1}) and the stable set of G_{β} by Ω_{β} = Ω(G_{β}).

From Remark 2.20 and Theorem 2.4 we obtain the following.

Proposition 7.1. For each β ∈ ℂ^{∗}, S_{β} and S_{β}^{−1} are non-elementary and G_{β} is a nonelementary Möbius group. The sources of S_{β}, S_{β}^{−1} and G_{β} are dense in J_{β}, J′_{β} and Λ_{β}, respectively. Each of these sets is perfect and uncountable.

We now introduce various transformations. We define three involutions^{15} of ℂ̂, m(z) = −z, r(z) = 1/z, and k(z) = \overline{z}, where m, r, and k denote minus, reciprocal, and conjugate, respectively. We let i(z) = iz, so i^{2}(z) = m(z). We define the translation t(z) = t_{β}(z) = z+β and let u = u_{β} = rtr, so t and u are parabolic and u(z) = z/(βz + 1) fixes the origin. Note that f = tr and g = t^{−1}r.

We will need the following Möbius groups later on. Define H_{β} = ⟨t, r⟩ so G_{β} is a subgroup of H_{β} of index at most two. Let P_{β} = ⟨t, u⟩ so P_{β} is a parabolically generated subgroup of H_{β} of index at most two. A superscript + will be used to denote the groups obtained by adjoining m, so G_{β}^{+} = ⟨f, g, m⟩, H_{β}^{+} = ⟨t, r, m⟩ and P_{β}^{+} = ⟨t, u, m⟩. All these groups are commensurable, so each has limit set Λ_{β} and ordinary set Ω_{β} and each is discrete when any is discrete.

Define a Koebe group to be a 2-generator non-elementary Möbius group such that the commutator of the generators is parabolic and one generator has order two (see [41] Fig. 8, or the groups G_{2}[τ] in [25]). G_{β}^{+} is Koebe since it is generated by f and m, m has order 2, and fmf^{−1}m^{−1} = fg^{−1} is parabolic.

Remark 7.2. Conversely, every Koebe group is conjugate to G_{β}^{+} for some nonzero β. For one may assume that the order two generator is m and that the other generator is h(z) = (az +b)/(cz +d) where bc ≠ 0 and where at least one of a, d is nonzero (since m, h generate a non-elementary Möbius group). In order that hmh^{−1}m^{−1} be parabolic one must have ad = 0 and so h is conjugate to an f_{β} by a Möbius transformation kz or k/z that commutes with m.

We represent f and g by the matrices M(β) and M(−β) where

M(β) = ( \begin{matrix}β 1 \\ 1 0 \end{matrix} )

and use them to calculate tr^{2}[f] = tr^{2}[g] = −β^{2}. Thus by Theorem 3.1 f and g have the same type and are classified as a function of the parameter β. Specifically, f and g (as well as f^{−1} and g^{−1}) are

• parabolic if and only if β = ±2i,

• elliptic if and only if β is purely imaginary with |β| < 2,

• hyperbolic if and only if β is purely imaginary with |β| > 2,

• strictly loxodromic for any other value of β. The study of J_{β}, J′_{β} and Λ_{β} is simplified by noticing that the functions f, g and their

inverses are conjugate to each other. This holds as two Möbius transformations u, v, neither the identity, are conjugate if and only if tr^{2}[u] = tr^{2}[v] ([2],p.66). Through simple computations one can make these conjugacies explicit.

Proposition 7.3. We have the following conjugacies:

(1) f ∘ m = m ∘ g,

(2) f ∘ r = r ∘ g^{−1}, and

(3) f ∘ mr = mr ∘ f^{−1}.

We summarize these results (along with some direct consequences) in Figure 2 where we write w ← h → v if w and v are conjugate via an involution h.

15A map h is an involution if h^{2} = Id, i.e., h = h^{−1}.

f g^{−1}  g f^{−1} −z −z −1/z 1/z  1/z

Figure 2. This diagram shows the conjugacies between the elements f, g, f^{−1} and g^{−1} under the involutions 1/z, −z and −1/z.

Proposition 7.3 describes some of the connections between the elements of S_{β}, S_{β}^{−1}, and G_{β}. Together with Remark 2.1, it explains a large amount of symmetry in the structure of J_{β}, J′_{β} and Λ_{β}, which we state as follows. See Figure 3.

Theorem 7.4. J_{β}, J′_{β}, and Λ_{β} satisfy the following properties^{16}:

(1) J_{β} = −J_{β} and J′_{β} = −J′_{β}

(2) J_{β} = 1/J′_{β}

(3) J_{β} = f^{−1}(J_{β}) ∪ g^{−1}(J_{β}) = (J_{β} − β)^{−1} ∪ (J_{β} + β)^{−1}

(4) J′_{β} = f(J′_{β}) ∪ g(J′_{β}) = (J_{β} + β) ∪ (J_{β} − β)

(5) Λ_{β} = −Λ_{β}

(6) Λ_{β} = 1/Λ_{β}

(7) Λ_{β} = Λ_{β} + β

Proof. Proposition 7.3(1) shows that S_{β} = mS_{β}m^{−1} and so by Remark 2.1, we conclude J_{β} = m(J_{β}) = −J_{β}. Similarly, the rest of (1) and (2), (5) and (6) can be shown. Parts (3) and (4) are an application of backward self-similarity (Proposition 2.10) to the semigroups S_{β} and S_{β}^{−1} (for the second equality in (4) we also made use of (2)). Part (7) uses (6) and Proposition 2.7 to compute Λ_{β} = f(Λ_{β}) = 1/Λ_{β} + β = Λ_{β} + β. □

Corollary 7.5. For all values of β ≠ 0, we have 0, ∞ ∈ Λ_{β}. Also, if 0 or ∞ lie in J_{β} or J′_{β} then both 0 and ∞ lie in both J_{β} and J′_{β}.

Note that by Theorems 8.1(3) and 8.2(3), it is possible for J_{β} ∩ J′_{β} ≠ ∅ without containing ∞.

Proof. Parts 6 and 7 of Theorem 7.4 imply 0, ∞ ∈ Λ_{β}. Part 4 of Theorem 7.4 shows that ∞ ∈ J_{β} if and only if ∞ ∈ J′_{β}. Part 2 of Theorem 7.4 shows both 0 ∈ J_{β} if and only if ∞ ∈ J′_{β}, and ∞ ∈ J_{β} if and only if 0 ∈ J′_{β}. The result then follows. □

The relations so far encountered do not exhaust the set of symmetries of S_{β}, S_{β}^{−1} and G_{β}. We focus now on the symmetries that originate in the parameter β-plane with respect to the maps m(z) = −z, k(z) = \overline{z}, and i(z) = iz. For convenience in describing the symmetry

16Recall footnote 14.

J^{β}  J ′β  Λ^{β}

Figure 3. Symmetries and relations in dynamical plane. From left to right we have J_{β}, J′_{β} and Λ_{β} for β = 1.3e^{iπ/6}. It is very easy to spot the symmetries described in Theorem 7.4. The unit circle (dashed) is shown for reference. These pictures were generated by choosing a seed at random, computing one million random iterates in its orbit and dropping the first hundred. See Remark 5.13.

under i(z) we extend S_{β} to a semigroup S_{β}^{+} =≺ m, f_{β}, g_{β} ≻=≺ m, f_{β} ≻, recalling that mf_{β}m = g_{β}.

Using some simple equations in M we can understand how S_{β} and G_{β} (or S_{β}^{+} and G_{β}^{+}) transform under m, k, and i. In the following proposition we use subscripts to make explicit the dependence of the maps f = f_{β}, g = g_{β} on the parameter β.

Proposition 7.6. (1) f_{−β} = g_{β}, g_{−β} = f_{β}, so S_{−β} = S_{β}, G_{−β} = G_{β}.

(2) f\overline{β} = kf_{β}k^{−1}, g\overline{β} = kg_{β}k^{−1}, so S\overline{β} = kS_{β}k^{−1}, G\overline{β} = kG_{β}k^{−1}. (3)g_{iβ} = i(g_{β}m)i^{−1}, f_{iβ} = i(f_{β}m)i^{−1}, m = imi^{−1}, so S_{iβ}^{+} = iS_{β}^{+}i^{−1}, G_{iβ}^{+} = iG_{β}^{+}i^{−1}.

This proposition explains the behavior of the sets J_{β} and Λ_{β} under the symmetries m, k, i of the parameter β-plane.

Theorem 7.7. The Julia set J_{β} and the limit set Λ_{β} have the following relations: (1) J_{β} = J_{−β} and Λ_{β} = Λ_{−β},

(2) k(J_{β}) = Jβ̄ and k(Λ_{β}) = Λ\overline{β},

(3) iJ_{β} = J_{iβ} and iΛ_{β} = Λ_{iβ}.

Proof. By Proposition 7.6(1), we have S_{β} = S_{−β} and G_{β} = G_{−β} and so (1) follows.

Proposition 7.6(2) shows that for k(z) = z̄, we have Sβ̄ = kS_{β}k^{−1} and Gβ̄ = kG_{β}k^{−1} and so (2) follows from Remark 2.1.

Likewise Proposition 7.6(3) implies J(S_{iβ}^{+}) = iJ(S_{β}^{+}) and Λ(G_{iβ}^{+}) = iΛ(G_{β}^{+}). Thus to prove (3) it suffices to show J(S_{β}^{+}) = J_{β} and Λ(G_{β}^{+}) = Λ_{β}.

Since S_{β} = mS_{β}m^{−1} by Proposition 7.3(1), we have m^{−1}(J_{β}) = J_{β} by Remark 2.1. Thus J_{β} is backward invariant by m and by S_{β} and so it is backward invariant by S_{β}^{+}. As S_{β} is non-elementary, its extension S_{β}^{+} is likewise non-elementary. Thus Remark 2.20 applies to S_{β}^{+} and implies that J(S_{β}^{+}) ⊂ J_{β}. But the other inclusion is clear since S_{β} ⊂ S_{β}^{+}. Thus J(S_{β}^{+}) = J_{β}.

In a similar way one shows that Λ(G_{β}^{+}) = Λ_{β}. □

The symmetries of J_{β} and Λ_{β} under the substitution of iβ or \overline{β} for β allow us to focus on values of β in the sector 0 ≤ arg β ≤ π/4. Consider the two rays that bound this sector.

β = 2e^{iπ/4}  β = \sqrt{2}e^{iπ/4}  β = e^{iπ/4}

Figure 4. Top row: the Julia sets J_{β} for different values of β with argument π/4 as in Theorem 8.2. On the left (β = 2e^{iπ/4}), we barely see the Cantor set J_{β} inside the unit circle. In the middle (β = \sqrt{2}e^{iπ/4}), the Cantor set J_{β} intersects the unit circle at the points ±1, ±i. On the right (β = e^{iπ/4}), we have a gasket Julia set that is both forward and backward invariant. Bottom row: the limit sets Λ_{β} for the same values of β. In the middle, we see the Apollonian gasket. These pictures were generated by choosing a seed at random, computing one million random iterates in its orbit and dropping the first hundred. See Remark 5.13.

When arg β = 0, each set J_{β}, J′_{β}, Λ_{β} is contained in the real line. When β = re^{iπ/4} with r ∈ ℝ, however, the diagonal line arg θ = π/4 is a line of symmetry of each of these sets. For instance, since β = i\overline{β}, Theorem 7.7 implies that ik(J_{β}) = iJ\overline{β} = J_{i\overline{β}} = J_{β}. See Figure 4.

7.2. Connected J_{β}. In this section we investigate the consequences of a connected Julia set J_{β}. We already know from Lemma 3.3 that Λ_{β} must be connected when J_{β} is connected. We also have the following result.

Theorem 7.8. Suppose J_{β} is connected. Then J_{β} meets J′_{β}.

The proof is based on 2 lemmas, the first of which relates J′_{β} and F_{β}.

Lemma 7.9. When ∞ ∈ F_{β}, J′_{β} meets the unbounded component V of F_{β}. When 0 ∈ F_{β}, J′_{β} meets \overline{U} where U is the component of 0 in F_{β}

Proof. Suppose ∞ ∈ F_{β}. Then J_{β} is bounded, nonempty, and closed and so there is a point z ∈ J_{β} of maximum modulus. By the triangle inequality, 2|z| ≤ |z + β| + |z − β|. Since the three points z, z + β, z − β are distinct and collinear, they cannot all have the same modulus. Thus for some choice of sign we have |z| < |z ± β| and so z ± β ∈ V. But since z ∈ J_{β}, Theorem 7.4(4) shows that z ± β ∈ J′_{β}. This proves the first statement.

For the second statement, suppose 0 ∈ F_{β} and let U be the component of 0 in F_{β}. Since J′_{β} = 1/J_{β}, we see ∞ ∈ F′_{β}. Let V′ be the unbounded component of F′_{β}, and note that U = 1/V′. We now show J_{β} meets \overline{V′}, thus giving J′_{β} = 1/J_{β} meets \overline{U} = 1/\overline{V′} as desired. Under our assumption, J′_{β} is bounded, closed, and nonempty. We choose a point w^{∗} ∈ J′_{β} that maximizes ℑ(w/β) = dist(w, βℝ)/|β| over all w ∈ J′_{β}. Hence J′_{β} is contained in the closed half plane H containing βℝ having boundary line βℝ + w^{∗}, and so the complement of H lies in V′. Then w^{∗}, w^{∗} + β, w^{∗} − β ∈ βℝ + w^{∗} ⊂ \overline{V′}. But by Theorem 7.4(4), we can choose the sign so that w^{∗} ± β lies in J_{β} as well. □

The second lemma is entirely topological.

Lemma 7.10. Let K be a compact nonempty connected subset of ℂ^{∗}. If −K = K then K separates 0 from ∞ in ℂ̂.

Proof. Let W be the open ϵ-neighborhood of K in ℂ, where ϵ is less than the distance from K to the origin. We first show that W separates 0 from ∞ in ℂ̂.

Clearly W is open, −W = W, and W is connected and nonempty. Thus we may choose a smooth path γ(t) ∈ W, 0 ≤ t ≤ 1, such that γ(1) = −γ(0). The variation in arg(γ(t)) between t = 0 and t = 1 is nπ, for some odd integer n. Concatenating γ(t) with the path^{17} −γ(t) gives a closed loop δ(t) ∈ W, where δ(t) = γ(2t) for 0 ≤ t ≤ 1/2 and δ(t) = −γ(2t−1) for 1/2 ≤ t ≤ 1. The variation in arg(δ(t)) between t = 0 and t = 1 is 2nπ. Since n ≠ 0 (as n is odd) each path c(t) in ℂ̂ from 0 to ∞ must pass through W.

As ϵ → 0 we see, since K is compact, that c(t) must also pass through K. □

From these lemmas we can derive the following.

Proposition 7.11. For β ∈ ℂ^{∗}, J′_{β} is not contained in any simply connected component of F_{β}.

Proof. Suppose to the contrary that U is a simply connected component of F_{β} and J′_{β} ⊂ U. Then J′_{β} and J_{β} are disjoint so, by Corollary 7.5, 0 and ∞ lie in F_{β} ∩ F′_{β}. Using both parts of Lemma 7.9, we see that 0 and ∞ lie in U.

Since J′_{β} = −J′_{β} ⊂ −U, U meets −U. Since −F_{β} = F_{β}, −U is a component of F_{β}. Thus −U = U.

Note that the compact set K = ℂ̂\U is nonempty, since J_{β} ⊂ K. Since −U = U we have −K = K. Moreover K is connected since U is simply connected. This contradicts Lemma 7.10. □

Proof of Theorem 7.8. Suppose to the contrary that J_{β} is connected and disjoint from J′_{β}. Then J′_{β} ⊂ F_{β}. By Theorem 7.4(2), J′_{β} = 1/J_{β}. Thus J′_{β} is connected so J′_{β} lies in some component U of F_{β}. Since J_{β} is connected, however, U must be simply connected. This contradicts Proposition 7.11. □

Theorems 5.1 and 7.8 and Corollary 7.5 give necessary conditions for the existence of a thick attractor.

17Note that in this context −γ(t) doe not denote the reverse path of γ(t), but rather the reflection of γ through the origin.

Corollary 7.12. If S_{β} has a thick attractor then J_{β} and J′_{β} are disjoint and each is disconnected and bounded.

The disconnectedness in this corollary is special to the Caruso family of semigroups S_{β}. For instance, Example 6.1 gives a non-elementary finitely generated Möbius semigroup with a connected thick attractor.

7.3. When does J_{β} equal Λ_{β}? By Proposition 2.7, the Julia set of any rational semigroup S is backward invariant. Examples 3.10 and 6.1 show that the Julia set is not necessarily S-invariant. This lack of invariance occurs for many Caruso semigroups as well, including those considered in Example 5.6.

But when J_{β} is S_{β}-invariant, Propositions 3.15 and 7.1 show that J_{β} = Λ_{β}. In many cases this enables us to calculate J_{β}. Accordingly we will study conditions under which J_{β} is S_{β}-invariant. We begin with a special property of the family S_{β}.

Proposition 7.13. For all values of β, if any two of the three sets J_{β}, J′_{β}, and Λ_{β} are equal, then J_{β} = J′_{β} = Λ_{β}.

Proof. If J_{β} = J′_{β} then J_{β} is S-invariant and so J_{β} = Λ_{β}. If J_{β} = Λ_{β} then J_{β}′ = 1/J_{β} = 1/Λ_{β} = Λ_{β} by Theorem 7.4. Likewise J′_{β} = Λ_{β} implies J_{β} = Λ_{β}. □

By Example 3.16 this proposition does not hold for Möbius semigroups in general. But we can even weaken the hypotheses of Proposition 7.13 with the same conclusion.

Proposition 7.14. If J′_{β} ⊂ J_{β} or J_{β} ⊂ J′_{β}, then J′_{β} = J_{β} = Λ_{β}.

Proof. Suppose J′_{β} ⊂ J_{β} and let z ∈ J_{β}. Then since J_{β} = 1/J′_{β} by Theorem 7.4, we have 1/z ∈ J′_{β} ⊂ J_{β} by hypothesis. But again using J_{β} = 1/J′_{β}, we conclude that z ∈ J′_{β}. Thus J_{β} = J′_{β}, from which Proposition 7.13 gives the desired conclusion.

A similar argument applies when J_{β} ⊂ J′_{β}. □

We now give a different condition under which J_{β} = Λ_{β}.

Proposition 7.15. If Id ∈ \overline{S_{β}} then \overline{S_{β}} = \overline{G_{β}} and J_{β} = Λ_{β}.

Proof. Recall that by Proposition 7.3(1), the involution m(z) = −z satisfies mfm^{−1} = g and mgm^{−1} = f. Thus mS_{β}m^{−1} = S_{β}.

By assumption there is a sequence s_{j} ∈ S_{β} such that s_{j} → Id. We may assume that s_{j} ≠ f and s_{j} ≠ g for all j, since f ≠ Id ≠ g. Each s_{j} can be written as a word in f and g of length at least two. Using the symmetry provided by conjugation by m, we may suppose that each s_{j} begins with the letter f. We factor s_{j} = fs′_{j} with s′_{j} ∈ S. As s′_{j} → f^{−1} we find that f^{−1} ∈ \overline{S_{β}}. Conjugating by m we see that g^{−1} ∈ \overline{S_{β}} as well.

Since \overline{S_{β}} is a semigroup that contains the semigroup generators f, g, f^{−1}, g^{−1} of G_{β}, we find G_{β} ⊂ \overline{S_{β}}. Thus \overline{G_{β}} ⊂ \overline{S_{β}}. As the opposite inclusion is trivial, our first conclusion holds.

It is clear that any family F ⊂ M is normal on an open set U ⊂ ℂ̂ if and only if its closure is normal on U. Thus for any Möbius semigroup S, both S and \overline{S} have the same Fatou set and hence the same Julia set.

This gives J_{β} = J(S_{β}) = J(\overline{S_{β}}) = J(\overline{G_{β}}) = J(G_{β}) = Λ_{β}. □

Remark 7.16. By a similar proof, J(S) = Λ(G) whenever S is a non-elementary Möbius semigroup that generates G and \overline{S} contains a generating set for S^{−1}. The symmetries of S_{β} enabled us to simplify this condition on \overline{S}.

Proposition 7.15 immediately gives the following.

Theorem 7.17. If S_{β} contains an elliptic element then S_{β} is dense in G_{β} and J_{β} = Λ_{β}.

Example 7.18. The word f has tr^{2}[f] = −β^{2}. For β = ix, x ∈ (−2, 2), we see 0 ≤ tr^{2}[f] < 4 and so the preceding theorem gives J_{β} = Λ_{β}. By the symmetry in Theorem 7.7(3), we also have J_{β} = Λ_{β} when β ∈ (−2, 2).

Example 7.19. The word f^{2}g^{2} has tr^{2}[f^{2}g^{2}] = (2 + β^{4})^{2}. If β = (1 ± i)x with −1 < x < 1, then f^{2}g^{2} is elliptic, and again J_{β} = Λ_{β}.

For instance, β = \sqrt{i} fits into this last family of examples with x = 1/\sqrt{2}. The corresponding set J_{β} = Λ_{β} is the gasket presented in Figure 4.

7.4. The closure of G_{β}. Whenever G_{β} is not discrete we will compute \overline{G_{β}} ⊂ M and use this to calculate the limit set Λ_{β}. We recall that M is a Lie group of real dimension 6. It is also connected. \overline{G_{β}} is a closed subgroup of M and hence a Lie subgroup of some dimension d_{β}.

We define M_{ℝ}, the real Möbius group, to be the stabilizer in M of the real circle ℝ̂ ⊂ ℂ̂. This is the 3-dimensional subgroup of M consisting of transformations m(z) = (az+b)/(cz+ d) with real a, b, c, d. When β is real, both G_{β} and its closure lie in M_{ℝ}. M_{ℝ} is a Lie subgroup of M of dimension 3 with two connected components, distinguished by the sign of the nonzero real number ad − bc.

Likewise we define M_{iℝ}, the imaginary Möbius group, to the the stabilizer of the imaginary circle iℝ̂ ⊂ ℂ̂, so M_{iℝ} = iM_{ℝ}i^{−1} where i(z) = iz. This is also a Lie subgroup of M of dimension 3 with two connected components. When β ∈ iℝ, both G_{β} and its closure lie in M_{iℝ}.

Theorem 7.20. (Density) Either G_{β} is discrete or it is dense in

1) M_{ℝ}, with β real,

2) M_{iℝ}, with β imaginary, or

3) M, with β neither real nor imaginary.

Proof. In each case, it is enough to see that G_{β} meets each component of the indicated Lie group and that d_{β} is at least as large as the dimension of that group.

It is easiest to show G_{β} meets the required components. Since M is connected, there is no difficulty when β is neither real nor imaginary. Say β is real. The matrices M(β) and M(−β) that represent the generators of G_{β} have determinant -1, and so G_{β} meets the non-identity component of M_{ℝ}. The argument is similar for imaginary β.

Assume G_{β} is not discrete. We only need to show that d_{β} ≥ 6 whenever β^{2} is not real and that d_{β} ≥ 3 for all β. For these estimates we need to study the Lie algebra L of M.

We can identify L as usual with the space of all 2 × 2 complex matrices of trace 0. The exponential map exp : L → M sends l ∈ L to the Möbius transformation defined by the invertible 2 × 2 matrix e^{l}. For every parabolic Möbius tranformation p there is a unique ν_{p} ∈ L such that ν_{p}^{2} = 0 and p = exp ν_{p}. We call ν_{p} the nilpotent logarithm of p. We call the linear map n_{p} : L → L given by n_{p}(x) = [ν_{p}, x] the nilpotent derivation of p.

Lemma 7.21. Let G be a Möbius group normalized by a parabolic element p ∈ M, i.e., pGp^{−1} = G. Then n_{p} stabilizes the Lie subalgebra L(\overline{G}) ⊂ L and n_{p}^{3} = 0.

Proof. Let ν = ν_{p} and n = n_{p}. As ν^{2} = 0, n^{2}(x) = −2νxν and n^{3}(x) = 0 for all x. Thus n^{3} = 0.

By Lie theory, D = e^{n} : L → L is the derivative at Id ∈ M of the group automorphism of M that sends m to exp(ν)m exp(−ν) = pmp^{−1}. This automorphism maps G to G and hence maps \overline{G} to \overline{G}. Thus D stabilizes L(\overline{G}).

Since n^{3} = 0, D = e^{n} = I + n + n^{2}/2. Squaring both sides of n + n^{2}/2 = D − I and using n^{3} = 0 yields n^{2} = (D − I)^{2}, which then is used to solve for n = (D − I) − (D − I)^{2}/2. Since D stabilizes L(\overline{G}), n must also do so. □

By Proposition 7.3, we see r normalizes G_{β}. Hence, the parabolic elements t = fr and u = rf also normalize G_{β}. By the lemma, n_{t} and n_{u} each stabilize the Lie algebra of G_{β}, which we denote L_{β} ⊂ L.

Let e_{ij} denote the 2 × 2 complex matrix whose (i, j) entry is 1 and whose other entries are 0, where 1 ≤ i, j ≤ 2. L has a basis over ℂ consisting of the matrices e_{12}, e_{21} and e_{11} − e_{22}. We find ν_{t} = βe_{12} and ν_{u} = βe_{21}.

We now estimate d_{β}. Since G_{β} is not discrete, L_{β} ≠ 0. Since n_{t}^{3} = 0, n_{t}(L_{β}) ⊂ L_{β}, and L_{β} ≠ 0, there is a nonzero element in ker(n_{t}) ∩ L_{β}. This element commutes with ν_{t} and so it must have the form ae_{12} for some nonzero a. By considering n_{u} we likewise find that be_{21} ∈ L_{β} for some nonzero b. Since L_{β} is a Lie subalgebra of L, L_{β} is closed under bracket. Thus ab(e_{11} − e_{22}) = [ae_{12}, be_{21}] ∈ L_{β}. This shows d_{β} ≥ 3.

Now n_{u}n_{t}(ae_{21}) = a′e_{21}, where a′ = 2β^{2}a. When β^{2} is not real, this shows L_{β} contains the complex line spanned by e_{21}. Likewise, by considering n_{t}n_{u}(ae_{12}), L_{β} contains the complex line spanned by e_{12}. Then L_{β} contains the bracket of these two complex lines and so d_{β} ≥ 6. □

Theorem 7.20 has simple consequences for Λ_{β}, since Λ_{β} = Λ(\overline{G_{β}}).

Corollary 7.22. When G_{β} is not discrete, as when |β| < 1, then Λ_{β} = ℝ̂ when β is real, Λ_{β} = iℝ̂ when β is imaginary, and Λ_{β} = ℂ̂ when β^{2} is not real.

Proof. When 0 < |β| < 1, Shimizu’s Lemma ([21] II.C.5) implies that the group H_{β} = ⟨t, r⟩ is not discrete. Since G_{β} is a subgroup of H_{β} of index at most two, G_{β} is not discrete either. □

8. Some J_{β} and Λ_{β}

We will determine whether or not G_{β} is discrete and find Λ_{β} and J_{β} whenever β^{2} is real or imaginary. The discreteness criteria and the limit sets are found using results of Hecke (when β^{2} is real) and of Klimenko and Kopteva (when β^{2} is imaginary).

We summarize our results in two related theorems and devote this section to their proofs.

Theorem 8.1. Suppose β^{2} is real.

1) G_{β} is discrete for |β| ≥ 2 and for |β| = 2 cos π/k, k ≥ 3, but for no other β [15]

2) \underline{For |β| > 2}: Λ_{β} is a Cantor set in βℝ̂. J_{β} and J_{β}′ are disjoint Cantor sets in Λ_{β}. J_{β}′ is a thick attractor for S_{β}.

3) \underline{For β = ±2 or ±2i}: Λ_{β} = βℝ̂. J_{β} and J_{β}′ are Cantor sets in Λ_{β} with J_{β} ∩ J_{β}′ = {β/2, −β/2}. S_{β} uniformly contracts J_{β}′ but S_{β} has no thick attractor.

4) \underline{For |β| < 2}: J_{β} = J_{β}′ = Λ_{β} = βℝ̂ and S_{β} has no attractor.

β  x

Figure 5. The picture shows the Julia set of S_{β} when −3 ≤ β ≤ −1.5. For each real value of β, the Julia set lies in the real axis. The Cantor sets are the horizontal sections of the curtains. The computer has a hard time filling the Julia sets of the semigroups when −2 < β < −1.5, where J_{β} = ℝ̂. See Remark 5.13.

Figure 6. Three gasket Julia sets that correspond to β = \sqrt{2i cos(π/k)} for k = 3, 4 and 5, respectively. These Julia sets are completely invariant. See Remark 5.13.

See Figure 5.

Theorem 8.2. Suppose β^{2} is imaginary.

1) G_{β} is discrete for |β^{2}| ≥ 2 and for |β^{2}| = 2 cos π/k, k ≥ 3, but for no other β [18]

2) \underline{For |β^{2}| > 2}: Λ_{β} is a Cantor set in ℂ̂. J_{β} and J_{β}′ are disjoint Cantor sets in Λ_{β}. J_{β}′ is a thick attractor for S_{β}.

3) \underline{For β = ±1 ± i}: Λ_{β} is a gasket. J_{β} and J_{β}′ are Cantor sets in Λ_{β} with J_{β}∩J_{β}′ = \{±1, ±i\}. S_{β} uniformly contracts J_{β}′ but S_{β} has no thick attractor.

4) \underline{For |β^{2}| < 2}: J_{β} = J_{β}′ = Λ_{β} and S_{β} has no attractor. Λ_{β} = ℂ̂ unless |β^{2}| = 2 cos π/k, k ≥ 3, in which case Λ_{β} is a gasket.

See Figures 6 and 4.

We recall the reflection group case of Poincare’s Polyhedron Theorem ([21], [27], [10]). An elliptic Möbius transformation is called primitive if it defines a rotation by an angle 2π/k for some integer k ≥ 2.

Theorem 8.3. Suppose P is a convex polyhedron in the compact ball H^{3} ∪ ℂ̂ bounded by finitely many planes. Let Γ be the group of hyperbolic isometries generated by the reflections ρ_{a} in these planes. Γ is discrete if and only if all the dihedral angles of P∩H^{3} are submultiples of π or, equivalently, when the product ρ_{a}ρ_{b} of any two generating reflections that fix an edge of P is either loxodromic, parabolic, or primitive elliptic of some order n_{ab} ≥ 2. Suppose Γ is discrete. Γ has a presentation with generators r_{a} and relations r_{a}^{2} = 1 and (r_{a}r_{b})^{n_{ab}} = 1. Also the interior of P ∩ ℂ̂ is a fundamental domain for the action of Γ on Ω(Γ).

Next we define a reflection group Γ_{β} whenever β^{2} is real or imaginary. Let μ = β/\overline{β} so μ = ±1 when β^{2} is real and μ = ±i when β^{2} is imaginary. Let ρ(z) = μ\overline{z}. We define σ = ρr if β is real, σ = ρrm if β is imaginary, and σ = ρu if β^{2} is imaginary. The following lemma is straightforward and its proof is omitted.

Lemma 8.4. ρ is the reflection that fixes βℝ̂, ρm is the reflection that fixes iβℝ̂, and ρtm is the reflection that fixes iβℝ̂ + β/2. σ is the reflection whose fixed circle is |z| = 1 if β^{2} is real and |βz + 1| = 1 if β^{2} is imaginary.

We define Γ_{β} = ⟨ρ, ρm, ρtm, σ⟩. The fixed circles of these generators bound fixed planes in H^{3} that bound a convex polygon P_{β} that contains ∞ (this specifies P_{β} when β^{2} is imaginary but there are two congruent choices if β^{2} is real). All the dihedral angles of P_{β} are right angles except for the dihedral angle made by the fixed planes of ρtm and σ, if these two planes intersect at all. By Theorem 8.3 Γ_{β} is discrete if and only if ρtmσ is either loxodromic, parabolic, or primitive elliptic of order k ≥ 3. Now Γ_{β} ∩ M = ⟨m, tm, ρσ⟩ is either H_{β}^{+}, if β^{2} is real, or P_{β}^{+}, if β^{2} is imaginary. Thus Γ_{β} ∩ M is commensurable with G_{β}.

Note that, by calculation, we have tr^{2}[ρtmσ] equals β^{2}, −β^{2}, or −β^{4} in the respective cases that β ∈ ℝ, β ∈ iℝ, or β^{2} ∈ iℝ. By Theorem 4.3.11 in [2], an elliptic map m of order k satisfies tr^{2}[m] = 4 cos^{2}(π/k) if and only if m is a rotation of angle ±2π/k. Parts (1) of Theorems 8.1 and 8.2 then follow (using Theorem 3.1).

Theorem 8.3 also determines the ordinary sets and hence the limit sets, as given in the above theorems. When β^{2} is imaginary, for instance, and ρtmσ is primitive elliptic of order k ≥ 3, P_{β} ∩ ℂ̂ consists of a bounded triangle T_{B} and an unbounded triangle T_{U} that are related by the involution rm, since conjugation by rm interchanges ρ with ρm and ρtm with σ. The sides of T_{B} are fixed by ρ, ρtm, and σ. Its vertex angles are 0 at the vertex 0, π/2 at the vertex β/2, and π/k at its 3rd vertex. The triangle reflection group ⟨ρ, ρtm, σ⟩ is Fuchsian (there is a Möbius change of variables taking T_{B} to a hyperbolic triangle in the upper halfplane with the same angles) and its ordinary set contributes a disc to Ω_{β}. In this way one sees that Ω_{β} is an infinite union of discs and so Λ_{β} is a gasket.

When ρtmσ is loxodromic, on the other hand, there is a gap between the fixed planes of ρtm and σ and so Λ_{β} is a Cantor set.

It remains to determine J_{β} and J_{β}′ in each case. Recall from Example 5.6 that S_{β} has a block when |ℜβ| + |ℑβ| > 2. Theorem 5.7 shows then that J_{β} and J_{β}′ are disjoint and that J_{β}′ is an attractor for S_{β}, which proves part 2) of each theorem.

Let W = (ρtmσ)^{2}. Now ρtmσ is fm, f, or fmf as β is real, β is imaginary, or β^{2} is imaginary. So W is respectively fg, f^{2}, or fg^{2}f, hence an element of S_{β}. Suppose ρtmσ is elliptic. Then W is elliptic so J_{β} = Λ_{β}, by Theorem 7.17, and J_{β}′ = Λ_{β}, by Proposition 7.13. Sources are dense in J_{β}′ by Theorem 2.4, so S_{β} does not uniformly contract J_{β}′. By Proposition 4.1, any S-invariant set uniformly contracted by S_{β} is a point. But G_{β} is a non-elementary Möbius group and so S_{β} has no fixed point. Thus S_{β} has no attractor. We now calculate Λ_{β}, using Corollary 7.22 when G_{β} is not discrete and using Theorem 8.3 as above when G_{β} is discrete. This proves part 4) of each theorem.

Assume β is as in part 3) of Theorem 8.2. Then W = fg^{2}f is parabolic so S_{β} has no thick attractor by Lemma 5.3.

The set {±1, ±i} consists of the fixed points of the parabolic elements fg^{2}f, g^{2}f^{2}, gf^{2}g, f^{2}g^{2} of S_{β} and so it is contained in J_{β} ∩ J_{β}′. Let N_{β} = N if β^{2} = 2i and N_{β} = k(N) if β^{2} = −2i, where N is as in Example 5.6. Then N_{β} is S_{β}-invariant so J_{β}′ ⊂ N_{β}. It follows that J_{β} ⊂ 1/N_{β} and so J_{β} ∩ J_{β}′ is contained in ∂N_{β}. But J_{β}′ ⊂ S_{β}^{2}(N_{β}) and S_{β}^{2}(N_{β}) ∩ ∂N_{β} = {±1, ±i}. This shows that J_{β} ∩ J_{β}′ = {±1, ±i}.

Let K_{β} = ∩S_{β}^{n}N_{β}. Each component of K_{β} is a point since S_{β} contracts the hyperbolic metric on Int N_{β} by the Theorem of Pick and since ∩W^{n}(N_{β}) is a point (recalling W = fg^{2}f is parabolic). Thus if we set h_{0} = f and h_{1} = g there is a homeomorphism h : K_{β} → {0, 1}^{ℕ} such that h(z) = (i_{1}, i_{2}, ...) if z ∈ h_{i_{1}}h_{i_{2}}...h_{i_{n}}(N_{β}), for each n ≥ 1. Each periodic sequence in {0, 1}^{ℕ} is h(z) for z the sink of some loxodromic element of S_{β} or z ∈ {±1, ±i}. Thus sinks are dense in K_{β} and so K_{β} ⊂ J_{β}′ by Theorem 2.4. But K_{β} is S_{β}-invariant and so J_{β}′ ⊂ K_{β} by Remark 2.20. Thus J_{β}′ = K_{β} is a Cantor set, so J_{β} = 1/J_{β}′ is also a Cantor set.

We define the inverse 2-shift to be the semigroup ≺ j_{0}, j_{1} ≻ on {0, 1}^{ℕ} where j_{a}(i_{1}, i_{2}, ...) = (a, i_{1}, i_{2}, ...). Let d be the usual metric on {0, 1}^{ℕ}, so d(x, y) = 2^{−N} if x_{N} ≠ y_{N} but x_{n} = y_{n} for all n < N. Then d(i_{a}(x), i_{a}(y)) = d(x, y)/2 so the inverse 2-shift is a CIFS.

Under the homeomorphism h, we can identify the action of S_{β} on J_{β}′ with the inverse 2-shift. Thus if we give J_{β}′ the metric h^{∗}d, S_{β} uniformly contracts J_{β}′.

This proves part 3) of Theorem 8.2. Part 3) of Theorem 8.1 can be proven in similar fashion, so both theorems are proven.

Remark 8.5. As in the preceding proof, the action of S_{β} on J_{β}′ can be identified with the inverse 2-shift for all β with |ℜβ| + |ℑβ| ≥ 2. Thus S_{β} uniformly contracts J_{β}′ for all such β.

References

[1] Michael F. Barnsley, John H. Elton, and Douglas P. Hardin. Recurrent iterated function systems. Constr. Approx., 5(1):3–31, 1989.

[2] Alan F. Beardon. The geometry of discrete groups, volume 91 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1983.

[3] Alan F. Beardon. Iteration of rational functions, volume 132 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991.

[4] Rainer Brück. Geometric properties of Julia sets of the composition of polynomials of the form z^{2} + c_{n}. Pacific J. Math., 198(2):347–372, 2001.

[5] Rainer Brück, Matthias Büger, and Stefan Reitz. Random iterations of polynomials of the form z^{2} + c_{n}: connectedness of Julia sets. Ergodic Theory Dynam. Systems, 19(5):1221–1231, 1999.

[6] Matthias Büger. Self-similarity of Julia sets of the composition of polynomials. Ergodic Theory Dynam. Systems, 17(6):1289–1297, 1997.

[7] Matthias Büger. On the composition of polynomials of the form z^{2} + c_{n}. Math. Ann., 310(4):661–683, [8] Horacio A. Caruso and Sebastian M. Marotta. Sequences of complex numbers resembling the fibonacci

series. Revista Ciências Exatas e Naturais, 2(1):49–59, 2000.

[9] John H. Elton. An ergodic theorem for iterated maps. Ergodic Theory Dynam. Systems, 7(4):481–488, 1987.

[10] David Epstein, Carlo Petronio. An exposition of Poincare’e polyhedron theorem. Enseign. Math. (2), 40:113-170, 1994.

[11] John Erik Fornæss and Nessim Sibony. Random iterations of rational functions. Ergodic Theory Dynam. Systems, 11(4):687–708, 1991.

[12] Leon Greenberg. Commensurable groups of Moebius transformations. In Discontinuous Groups and Riemann surfaces, volume 79 of Annals of Mathematics Studies, 227-237. Princeton University Press, Princeton, 1974.

[13] Tatsunori Harada. The dynamics of nearly abelian polynomial semigroups at infinity. Proc. Japan Acad. Ser. A Math. Sci., 73(3):36–37, 1997.

[14] Brian Hayes. The vibonacci numbers. American Scientist, 87(4):296–301, 1999.

[15] Erich Hecke. Dirichlet series. Planographed lecture notes, Princeton Institute for Advanced Study, Edwards Brothers, Ann Arbor, 1938.

[16] A. Hinkkanen and G. J. Martin. The dynamics of semigroups of rational functions I. Proc. London Math. Soc. (3), 73(2):358–384, 1996.

[17] John E. Hutchinson. Fractals and self-similarity. Indiana Univ. Math. J., 30(5):713–747, 1981.

[18] Elena Klimenko and Natalia Kopteva. Discrete RP groups with a parabolic generator. Sibirsk. Mat. Zh., 46(6):1324-1334, 2005.

[19] Linda Keen and Caroline Series. The Riley slice of Schottky space. Proc. London Math. Soc. (3), 69(1):72–90, 1994.

[20] Albert Marden. Outer circles: an introduction to hyperbolic 3-manifolds. Cambridge University Press, Cambridge, 2007.

[21] Bernard Maskit. Kleinian groups, volume 287 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1988.

[22] John Milnor. Dynamics in one complex variable, volume 160 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, third edition, 2006.

[23] David Mumford, Caroline Series, and David Wright. Indra’s pearls: the vision of Felix Klein. Cambridge University Press, New York, 2002.

[24] James R. Munkres. Topology: a first course. Prentice-Hall Inc., Englewood Cliffs, N.J., 1975.

[25] Jouni Parkkonen. The outside of the Teichmuller space of punctured tori in Maskit’s embedding. Ann. Acad. Sci. Fenn. Math. 24(2):305-342, 1999.

[26] Xavier Pennec. Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vision 25(1):127-154, 2006.

[27] John G. Ratcliffe. Foundations of hyperbolic manifolds, volume 149 of Graduate Texts in Mathematics. Springer, New York, second edition, 2006.

[28] Rich Stankewitz. Completely invariant Julia sets of rational semigroups. Ph.D. Thesis. University of Illinois, 1998.

[29] Rich Stankewitz. Completely invariant Julia sets of polynomial semigroups. Proc. Amer. Math. Soc., 127(10):2889–2898, 1999.

[30] Rich Stankewitz. Uniformly perfect sets, rational semigroups, Kleinian groups and IFS’s. Proc. Amer. Math. Soc., 128(9):2569–2575, 2000.

[31] Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup. J. Difference Equ. Appl., 16(5):763 – 771, 2010.

[32] Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Preprint, Ball State University, 2010.

[33] Rich Stankewitz and Hiroki Sumi. Dynamical properties and structure of julia sets of postcritically bounded polynomial semigroups. To appear in Transactions of the American Mathematical Society.

[34] Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups III: classification of semihyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles. Ergodic Theory Dynam. Systems, 30(6), 1869–1902, 2011.

[35] Hiroki Sumi. On dynamics of hyperbolic rational semigroups. J. Math. Kyoto Univ., 37(4):717–733, 1997.

[36] Hiroki Sumi. Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergodic Theory Dynam. Systems, 21(2):563–603, 2001.

[37] Hiroki Sumi. Random dynamics of polynomials and devil’s staircase-like functions in the complex plane. Appl. Math. Comput., 187:489–500, 2007.

[38] Hiroki Sumi. Dynamics of postcritically bounded polynomials semigroups I: Connected components of the julia sets. Discrete Cont. Dynam. Syst. 29(3):1205–1244, 2011.

[39] Yeshun Sun and Chung-Chun Yang. On the connectivity of the Julia set of a finitely generated rational semigroup. Proc. Amer. Math. Soc., 130(1):49–52 (electronic), 2001.

[40] Divakar Viswanath. Lyapunov Exponents from Random Fibonacci Sequences to the Lorenz Equations. Ph.D. Thesis. Cornell University, 1998.

[41] David J. Wright. Searching for the cusp. In Spaces of Kleinian groups and hyperbolic 3-manifolds, volume 329 of London Math. Soc. Lecture Notes, 301-336. Cambridge University Press, Cambridge, 2005.

(David Fried) Department of Mathematics, Boston University.

E-mail address: df@bu.edu

(Sebastian M. Marotta) Department of Mathematics, University of the Pacific.

  E-mail address: smarotta@pacific.edu

(Rich Stankewitz) Department of Mathematical Sciences, Ball State University.

  E-mail address: rstankewitz@bsu.edu