The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

We show that for each $d\in (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.


Introduction and main results
We will be concerned with the iteration of transcendental meromorphic functions f : C → C := C ∪ {∞}. The main objects studied here are the Fatou set F (f ), consisting of all z ∈ C for which the iterates f k of f are defined and form a normal family in some neighborhood of z, and the Julia set J(f ) := C \ F (f ). For an introduction to the dynamics of transcendental meromorphic functions we refer to [10].
The Speiser class S consists of all transcendental meromorphic functions f for which the set sing(f −1 ) of singularities of the inverse, i.e., the set of critical and asymptotic values of f , is finite. More precisely, if q denotes the cardinality of sing(f −1 ), then we write f ∈ S q . Equivalently, f ∈ S if there exist finitely many points a 1 , . . . , a q ∈ C such that f : C \ f −1 ({a 1 , . . . , a q }) → C \ {a 1 , . . . , a q } is a covering map. And if q is the minimal number with this property, then f ∈ S q . The monodromy theorem implies that we always have q ≥ 2. Both the Speiser and the Eremenko-Lyubich class play an important role in transcendental dynamics. The similarities and differences between these classes are addressed in a number of recent papers [2,12,13,14,21]. A survey of some of these as well as many other results concerning the dynamics of functions in S and B is given in [41].
Considerable attention has been paid to the Hausdorff dimension of Julia sets; see [30] and [45] for surveys. We will denote the Hausdorff dimension of a subset A of C by dim A.
Baker [4] showed that if f is a transcendental entire function, then J(f ) contains continua so that dim J(f ) ≥ 1. Stallard [44,Theorem 1.1] showed that for all d ∈ (1, 2) there exists a transcendental entire function f with dim J(f ) = d while Bishop [15] constructed an example with dim J(f ) = 1. Stallard's examples are actually in the Eremenko-Lyubich class B. Previously she had shown that dim J(f ) > 1 for entire f ∈ B. In particular, this is the case for entire functions in S.
Albrecht and Bishop [1] showed that given δ > 0 there exists an entire function f ∈ S such that dim J(f ) < 1 + δ. In fact, these were the first examples of entire functions in S for which the Julia set has dimension strictly less than 2. In their examples the inverse has three finite singularities. Since every non-constant entire function has the asymptotic value ∞ by Iversen's theorem, and since we include ∞ in sing(f −1 ), their examples are in S 4 .
For transcendental meromorphic functions f we have dim J(f ) > 0 by a result of Stallard [42]. On the other hand, she showed [43,Theorem 5] that for all d ∈ (0, 1) there exists a transcendental meromorphic function f such that dim J(f ) = d. Again her examples are in B. Together with her result covering the interval (1, 2) mentioned above, and since J(exp z) = C by a result of Misiurewicz [38] and J(tan z) = R, it follows that for all d ∈ (0, 2] there exists f ∈ B such that dim J(f ) = d.
We shall show that such examples also exist in the Speiser class S and in fact in S 3 .

Preliminary results
The escaping set I(f ) of a meromorphic function f is defined as the set of all z ∈ C for which f n (z) → ∞ as n → ∞. We always have I(f ) = ∅ and J(f ) = ∂I(f ). This was shown by Eremenko [22] for transcendental entire f and by Domínguez [17] for transcendental meromorphic f .
For  [2] where it was shown that for all d ∈ [0, 2] there exists a meromorphic function f ∈ S such that dim I(f ) = d.
An important concept in the theory of meromorphic functions is the order; see, e.g., [28]. The following result was proved in [11, Theorem 1.1].
Lemma 2.1. Let f ∈ B be of finite order ρ. Suppose that ∞ is not an asymptotic value of f and that there exists M ∈ N such that all but finitely many poles of f have multiplicity at most M. Then We will not actually use this lemma. Instead we will use the following closely related result, which can be proved by the same method.
Proof. We proceed as in [11]. Let (a j ) be the sequence of poles of f and let m j be the multiplicity of a j . Let b j ∈ C \ {0} be such that Let a λ j and b λ j the corresponding values for f λ . Then a λ j = a j /λ and b λ (2.3) As in [11] we choose R 0 > |f (0)| such that sing(f −1 λ ) = sing(f −1 ) ⊂ D(0, R 0 ) and R ≥ 2 M R 0 . Here and in the following D(a, r) denotes the open disk of radius r around a point a ∈ C. Choosing λ small we can achieve that We put B(R) := {z : |z| > R} ∪ {∞} and, as in [11, pp. 5376f.], consider Here diam χ (V ) denotes the spherical diameter of V . Together with (2.2) and (2.3) the last estimate yields that if λ is sufficiently small, then Hence dim J(f λ ) ≤ t for small λ.
It follows from a recent result of Mayer and Urbański [35] that Lemma 2.2 and (2.1) can be sharpened to In fact, their result says that dim I(f ) is the infimum of the set of all t > 0 for which (2.2) holds.
For entire functions, the following result can be found in [24, Section 3] and [21, Proposition 2.3]. Lemma 2.3. Let f, g ∈ S 3 and suppose that there exist homeomorphisms ψ : C → C and φ : Then there exist a fractional linear transformation α : C → C and an affine map β : Proof. Since f ∈ S 3 it follows from [21, Observation 1.10] that there exists a fractional linear transformation α : C → C which is isotopic to ψ relative to sing(f −1 ). We follow the argument in the proof of [21, Proposition 2.3(a)]. Let (ψ t ) t∈[0,1] be the isotopy between ψ 0 = ψ and ψ 1 = α. By the isotopy lifting property, there exists a unique isotopy (φ t ) t∈[0, 1] in 1]. It remains to show that φ t extends continuously to the preimages of the singular values of f and coincides with φ there for all t. Let is a proper map, with no critical points except possibly at z 0 . We may also assume that f (z) = v 0 for z ∈ D \ {z 0 }. It then suffices to show that φ t (z) ∈ φ(D) if z is sufficiently close to z 0 . By the continuity of f and the properties of an isotopy, we have if z is sufficiently close to z 0 . Thus φ t (z) can be obtained by analytic contin- If z is not a critical point of f , then β is of the form g −1 • ψ • f near z for some branch of the inverse of g. Hence β is holomorphic. Since β : C → C is a homeomorphism, this implies that β is affine.
The following result is due to Kotus and Urbański [29].
Lemma 2.4. Let f be an elliptic function. Let q be the maximal multiplicity of the poles of f . Then We shall need a variation of this result.
Lemma 2.5. Let f be as in Lemma 2.4. If (f n ) is a sequence of meromorphic functions which converges locally uniformly to f , then for all large n.
Lemma 2.5 can be deduced from the work of Kotus and Urbański; see Remark 2.1 below. But for the convenience of the reader we will include a proof of Lemma 2.5, thereby reproving Lemma 2.4. Here we will use the following result [25,Proposition 9.7].
Proof of Lemmas 2.4 and 2.5. Let (a j ) be the sequence of poles of multiplicity q. For sufficiently large R there is a neighborhood U j of a j such that is a covering map of degree q. There exists r 1 > 0 such that D(a j , r 1 ) ⊂ U j for all j. Let 0 < r 0 < r 1 . Then there exists M ∈ N such that is biholomorphic. Moreover, f 2 extends to a bijective map from W k to K := D(a M , r 0 ). Let S k : K → W k be the inverse function of f 2 : W k → K. Then S k extends to an injective map S k : D(a M , r 1 ) → C. Choosing r 0 ≤ (2− √ 3)r 1 we conclude that W k = S k (D(a M , r 0 )) is convex; see [19,Theorem 2.13].
In order to apply Lemma 2.6 we note that since f has a pole of multiplicity q at a j , there exists c 1 > 0 such that This implies that there exists c 2 > 0 such that |f ′ (z)| ≤ c 2 |a k | (q+1)/q if z ∈ D(a j , r 0 ) and f (z) ∈ D(a k , r 0 ).
With c 3 := c 2 2 |a M | (q+1)/q this yields that Since W k is convex this yields that It follows that (2.4) holds with It follows from Lemma 2.6 that the limit set of the iterated function system {S k : M ≤ k ≤ N} has Hausdorff dimension at least t. It is easily seen that this limit set is contained in the Julia set.
To prove Lemma 2.5 we note that for large n there are domains W n k and U n M close to W k and U M such that is biholomorphic, and the inverse function S n k satisfies (2.4) with a constant b n k instead of b k , with b n k → b k as n → ∞. The conclusion then follows again from Lemma 2.6.
Remark 2.1. The argument of Kotus and Urbański [29] is similar to the one used above, but they use an infinite iterated function system and apply results of Mauldin and Urbański [33] concerning such systems. However, it would suffice to consider a sufficiently large finite subsystem. This would yield Lemma 2.5 as above.
The proof actually yields that the hyperbolic dimension of f and f n , and not only the Hausdorff dimension of their Julia sets, have the given lower bound; see [9,39] for a discussion of the hyperbolic dimension of meromorphic functions.
Lemma 2.7. Let f ∈ S. Suppose that f has an attracting fixed point whose immediate attracting basin contains all finite singularities of f −1 . Suppose also that ∞ is not an asymptotic value of f and that there exists a uniform bound on the multiplicities of the poles of f . Then J(f ) is totally disconnected and F (f ) is connected.
There are several closely related results in the literature, see [ see also [26,Theorem 3.2]) do not require that the poles are simple, but they restrict to elliptic functions.
Our proof of Lemma 2.7 will use some ideas from the papers mentioned. A difference to the methods employed there, however, is that we will use the following consequence of the Grötzsch inequality; see [16,Section 5.2] and [36,Corollary A.7]. Here and in the following mod(A) denotes the modulus of an annulus A.
Lemma 2.8. Let (G k ) be a sequence of simply connected domains in C such that A k := G k \ G k+1 is an annulus for all k ∈ N. Suppose that is a compact subset of W . It is not difficult to see that there exist Jordan domains U and V such that Then A := V \ U is an annulus. Clearly, A ⊂ W . Let (a j ) be the sequence of poles of f and let m j denote the multiplicity of a j . By hypothesis, there exists M ∈ N such that m j ≤ M for all j. Let Y j be the component of the preimage of C \ U that contains a j . Then To prove that J(f ) is totally disconnected, let z ∈ J(f ). We want to show that the component of J(f ) containing z consists of the point z only. First we note that for all n ∈ N there exists j(n) ∈ N such that f n (z) ∈ Y j(n) . Let X n be the component of f −n (Y j(n) ) containing z. Note that Y j(n) and X n are Jordan domains. Since ∂U and hence ∂X n are contained in F (f ), the component of J(f ) containing z is contained in the intersection of the X n . It thus suffices to prove that this intersection consists of only one point. In order to do so we note that since P (f ) ⊂ U, the map f n : X n → Y j(n) is biholomorphic. This implies that is an annulus satisfying Moreover, X n is equal to the union of C n and the component of C \ C n that contains z.
Since the closure of A is a compact subset of the attracting basin W of ξ, there exists p ∈ N such that f p (A) ⊂ U. In particular, This implies that C n+p ∩ C n = ∅ for all n ∈ N. In fact, if w ∈ C n+p ∩ C n , then f n+p+1 (w) ∈ f p (A) ∩ A by (2.5), contradicting (2.6). It follows that X n+p ⊂ X n \ C n , and hence that mod X n \ X n+p ≥ mod(C n ) ≥ 1 M mod(A).
As already mentioned above it now follows from Lemma 2.8, applied with G k = X 1+pk , that J(f ) is totally disconnected. Of course, this yields that F (f ) is connected.
Extending this to the whole plane by reflections we obtain an elliptic function G. The critical values of G are 0, 1 and ∞ so that G ∈ S 3 . The zeros and poles of G have multiplicity 4 while the 1-points have multiplicity 2. We also note that G(R) = [0, 1]. We may express G in terms of the Weierstrass ℘-function with periods π and πi. The critical values of ℘ are e 1 , e 2 , e 3 and ∞, with e 2 = ℘((π + iπ)/2) = 0 and e 1 = ℘(π/2) = −e 3 = −℘(iπ/2).

It follows from this that
First we construct an example of a function f ∈ S 3 where the Julia set is the whole sphere and thus has Hausdorff dimension 2. To this end we consider the function f (z) := iG(πz/2). To construct functions in S 3 for which the Julia set has Hausdorff dimension d ∈ (0, 2), we consider, for p ∈ N and small η ∈ (0, π/2), the function H defined by H(z) := ηG(z) p . The critical values of H are 0, η and ∞. Thus H ∈ S 3 . We have H(0) = η and H(π/2) = 0. Also, H decreases in the interval (0, π/2). Choosing η sufficiently small we can achieve that H has an attracting fixed point ξ ∈ (0, π/2) whose attracting basin contains [0, η] and thus, since H(R) = [0, η], also contains R. Lemma 2.7 implies that this attracting basin coincides with F (H) and that J(H) is totally disconnected.
Let now m be a (large) odd integer. Then We fix such a value of m and, for λ ∈ (0, 1], we put f λ (z) := h m (λz) so that Since h m has order 0, Lemma 2.2 yields that We will show that the function λ → dim J(f λ ) is continuous in the interval (0, 1]. Since f 1 = h m it then follows from (3.1) and (3.2) that for all d ∈ (0, 8p/(4p + 1)] there exists λ ∈ (0, 1] such that dim J(f λ ) = d. Since p can be chosen arbitrarily large, this yields the conclusion.
It remains to prove that λ → dim J(f λ ) is continuous. Recalling that h m and h ′ m are decreasing in the interval [0, η] we can deduce that f λ has an attracting fixed point ζ λ ∈ (0, η) and that the multiplier is a decreasing function of λ in the interval (0, 1]. Note here that ζ 1 = ξ m and m 1 = h ′ m (ξ m ) < 0. As before it follows from Lemma 2.7 that the attracting basin of ζ λ is connected and coincides with the Fatou set of f λ .
Let λ ∈ (0, 1]. Koenigs' theorem [37,Theorem 8.2] yields that there exists a function g holomorphic and injective in some neighborhood U of ζ λ such that g(ζ λ ) = 0, g ′ (ζ λ ) = 1 and g(f λ (g −1 (z))) = m λ z for all z ∈ g(U). For κ ∈ (0, 1] we put for z ∈ φ(U). The maps h and φ are K-quasiconformal with For a detailed account of quasiconformal mappings, we refer to [32]. The complex dilatation µ(z) := φ z (z)/φ z (z) satisfies if z, f λ (z) ∈ U. We may use (3.5) to extend µ to C. More precisely, we put µ(z) = 0 for z ∈ J(f λ ) while for z ∈ F (f λ ) we define where n is chosen so large that f n λ (z) ∈ U. Using (3.5) it is easily seen that µ is well-defined, i.e., the definition does not depend on the value of n chosen in (3.6). We find that (3.5) holds for all z.
Let ψ : C → C be the solution of the Beltrami equation normalized by ψ(0) = 0 and ψ(η) = η. It follows from (3.5) that is meromorphic. Since f λ is symmetric with respect to the real axis, the same applies to g, φ, µ, ψ and k. By definition, f λ is even. In order to show that k is also even, we first note that since f λ is even, it follows from (3.5) that µ is even. This implies that ψ(z) and ψ(−z) have the same complex dilatation. Hence there exists an affine map L such that ψ(z) = L(ψ(−z)).
Since ψ(0) = 0 we have L(0) = 0 so that L has the form L(z) = az for some a ∈ C \ {0}. We also see that L is real on the real axis so that a ∈ R \ {0}.
Another function with a fixed point of multiplier m κ is f κ . We will show that k = f κ . In order to do so we note that both k and f κ are in S 3 , with critical values 0, η and ∞. It follows from Lemma 2.3 and (3.7) that there exist a fractional linear transformation α and an affine map β such that α • k = f λ • β. Since all poles of k and f κ have multiplicity 4p, all but two zeros of both functions have multiplicity 4p, and all η-points of both functions have multiplicity 2 or 1, we find that α(0) = 0, α(η) = η and α(∞) = ∞. Hence α(z) ≡ z so that k = f λ • β.
Since periodic functions have order at least 1 while f λ has order 0, this implies that β(0) = 0 so that β has the form β(z) = cz for some constant c. Thus k(z) = f λ (cz) = h m (λcz). As k has an attracting fixed point of multiplier m κ this yields that c = κ/λ so that k(z) = h m (κz) = f κ (z).
For our purposes, however, the weaker and simpler estimate (3.8) suffices.