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

Abstract

We show that for each d ∈ (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. * W. Cui expresses his gratitude to the Centre for Mathematical Sciences of Lund University for providing a nice working environment.

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

“…Combining the methods of [10] with those of the present paper we obtain the following generalization of Corollary 1.3.…”

“…A result of Kotus and Urbański [27] says that dim J(K) > 2M/(1 + M). It follows from [10, Lemma 2.5] that if N is sufficiently large, then we also have Using the same arguments as in [10] we find that the function…”

“…Our results can be seen as a contribution to recent efforts to understand the differences and similarities between the classes B and S. We refer to [2,3,10,15,16,17,21] and the survey [37] for results in this direction. As an example we mention the recent result of Albrecht and Bishop [1] who showed that given δ > 0 there exists an entire function f ∈ S such that 1 < dim J(f ) < 1 + δ.…”

“…It was shown in [10] that for all d ∈ (0, 2] there exists a function f ∈ S 3 with dim J(f ) = d. Note that one has I(f ) ⊂ J(f ) if f ∈ B, as shown by Eremenko and Lyubich [24, Theorem 1] for entire f and Rippon and Stallard [35, Theorem A] for meromorphic f . We also have dim J(f ) > 0 for every transcendental meromorphic function f by a result of Stallard [38].…”

“…and note that this defines a meromorphic function F ∈ S 3 . Examples of this type were previously considered in [41, p. 734], [5, Section 5], [30, Section 2], [23] and [10]. We note that the points z k := π/2 + ikπ/2 are critical points of H. It follows that the points…”