The Speiser class S consists of transcendental entire functions with only finitely many singular values. Albrecht and Bishop (2020) proved that the infimum of Hausdorff dimensions of Julia sets of Speiser-class functions equals 1, constructing, for every delta > 0, a function with three finite singular values whose Julia set has dimension at most 1 + delta; before their work no Speiser-class example with dimension below 2 was known. Their paper explicitly leaves open two linked questions: whether every dimension in (1, 2] is attained by some Speiser-class function, and whether dimensions arbitrarily close to 1 can be achieved with only two finite singular values. Subsequent literature (the meromorphic analogues of Bergweiler and Cui, the minimal-dimension constructions of Bishop and of Burkart and Lazebnik, and the 2025 survey of Bergweiler and Rempe) leaves both questions open in the entire setting: the meromorphic analogue of the spectrum was resolved with three singularities of the inverse, but no intermediate dimension is known to be attained exactly by an entire Speiser-class function, and no entire function with only two finite singular values is known to have Julia-set dimension below 2.

Background

The iteration of a transcendental entire function f:C→Cf:\mathbb{C}\to\mathbb{C} splits the plane into the Fatou set, where the iterates f,f2,f3,…f,f^2,f^3,\dots form a normal family (stable behavior), and its complement, the Julia set J(f)\mathcal{J}(f), the locus of chaotic dynamics. Baker proved in 1975 that the Julia set of a transcendental entire function always contains a nontrivial continuum, so its Hausdorff dimension satisfies dim⁡H(J(f))≥1\dim_H(\mathcal{J}(f))\ge 1; on the other hand the dimension can be maximal, since the exponential map — whose only finite singular value is 0 — has the whole plane as its Julia set, giving dimension 2. Constructing examples whose dimension is close to the theoretical minimum 1 is much harder, and is the transcendental counterpart of finding polynomial Julia sets with dimension near 2.

The fine structure of an entire function is controlled by its singular values: the critical values of ff together with its finite asymptotic values (limits of f(z)f(z) as z→∞z\to\infty along a curve). The Eremenko–Lyubich class B\mathcal{B} consists of transcendental entire functions whose set of singular values is bounded, and the Speiser class S⊂B\mathcal{S}\subset\mathcal{B} of those whose singular set is finite; Sq\mathcal{S}_q denotes the functions whose inverse has exactly qq singularities. (Conventions differ on whether the point ∞\infty is counted; since ∞\infty is always an asymptotic value of a transcendental entire function, the invariant datum is the number of finite singular values — the examples of Albrecht and Bishop below have three finite singular values 0,±10,\pm 1, which is S4\mathcal{S}_4 under the convention counting ∞\infty.) Comparing the classes B\mathcal{B} and S\mathcal{S} is a central theme of transcendental dynamics: for instance, functions in S\mathcal{S} cannot have wandering domains — Fatou components that are never mapped onto a periodic component — whereas functions in B\mathcal{B} can.

For the Hausdorff dimension of Julia sets in these classes the following is known. Stallard proved that dim⁡H(J(f))>1\dim_H(\mathcal{J}(f))>1 whenever f∈Bf\in\mathcal{B} — so the value 1, if attained at all, is attained only outside B\mathcal{B} — and that every value in (1,2)(1,2) is attained by some function in B\mathcal{B} (with 2 attained, for example, by the exponential map, which lies in S\mathcal{S}). Bishop constructed a transcendental entire Julia set of Hausdorff dimension exactly 1, but with an unbounded singular set, hence outside both classes; within B\mathcal{B} the packing dimension of the Julia set — a fractal dimension defined by packings of equal balls rather than coverings, always at least the Hausdorff dimension — is always 2, and any function in B\mathcal{B} of finite growth order (the classical order of an entire function, measuring the growth rate of its maximum modulus) has dim⁡H(J(f))=2\dim_H(\mathcal{J}(f))=2, so near-minimal examples must have infinite order.

Albrecht and Bishop then proved the breakthrough result that

inf⁡{dim⁡H(J(f)):f∈S}=1,\inf\{\dim_H(\mathcal{J}(f)) : f\in\mathcal{S}\}=1,

constructing, for each δ>0\delta>0, a Speiser-class function with dim⁡H(J(f))≤1+δ\dim_H(\mathcal{J}(f))\le 1+\delta — the first Speiser-class examples with dimension below 2. Their construction uses quasiconformal folding, a technique that builds entire functions with prescribed singular-value structure by folding a model map along an infinite planar graph; the examples have exactly three finite singular values, are of disjoint type (all singular values lie in the immediate basin of an attracting fixed point), have infinite order, and their Julia sets are Cantor bouquets — closed sets consisting of uncountably many pairwise disjoint arcs ('hairs') tending to infinity together with their endpoints, the standard topological model for disjoint-type maps.

What the paper does not determine is which dimensions are actually attained. The value 2 is attained in S\mathcal{S}, and dimensions arbitrarily close to 1 are attained, but no value strictly between 1 and 2 is known to occur exactly, and nothing is known about approaching 1 with fewer than three finite singular values. In the meromorphic category the analogous spectrum question was resolved by Bergweiler and Cui — every d∈(0,2]d\in(0,2] occurs as dim⁡H(J(f))\dim_H(\mathcal{J}(f)) for a meromorphic Speiser-class function whose inverse has three singularities — but their method glues functions along poles, a device with no counterpart for entire functions, and the 2025 survey of Bergweiler and Rempe records the entire-function questions as open.

Problem Statement

Let S\mathcal{S} denote the Speiser class of transcendental entire functions — entire functions whose set of singular values (critical values together with finite asymptotic values) is finite — and let J(f)\mathcal{J}(f) and dim⁡H\dim_H denote the Julia set and Hausdorff dimension. Albrecht and Bishop proved that

inf⁡{dim⁡H(J(f)):f∈S}=1,\inf\{\dim_H(\mathcal{J}(f)) : f\in\mathcal{S}\}=1,

realizing the infimum by functions with exactly three finite singular values, while the value 2 is attained in S\mathcal{S} (by the exponential map, whose only finite singular value is 0), and Stallard's theorem gives dim⁡H(J(f))>1\dim_H(\mathcal{J}(f))>1 for every f∈Sf\in\mathcal{S}. Two linked questions, posed explicitly at the end of their paper, remain open:

  1. (Spectrum) Is every intermediate dimension attained — for each d∈(1,2]d\in(1,2], does there exist f∈Sf\in\mathcal{S} with dim⁡H(J(f))=d\dim_H(\mathcal{J}(f))=d? Equivalently, is the set of attainable Hausdorff dimensions of entire Speiser-class Julia sets exactly the interval (1,2](1,2]?

  2. (Minimum number of singular values) Can the infimum be approached with only two finite singular values — for every δ>0\delta>0, does there exist a transcendental entire function with exactly two finite singular values whose Julia set satisfies dim⁡H(J(f))≤1+δ\dim_H(\mathcal{J}(f))\le 1+\delta?

The common objective is the determination of the attainable-dimension spectrum of Speiser-class Julia sets and of the minimum number of finite singular values needed to approach the theoretical lower bound. The two questions are linked, and either alone leaves the other open; since Theorem 1.1 of the source already yields dimensions arbitrarily close to 1 with three finite singular values, neither approximating a dimension rather than attaining it exactly nor using more singular values than stated settles either question. The same passage of the source poses two further companion questions — whether dim⁡H(J(g))\dim_H(\mathcal{J}(g)) can be non-constant, or constant at a value other than 2, on the quasiconformal equivalence manifold MfM_f, and whether sup⁡{dim⁡H(J(g)):g∈Mf}=2\sup\{\dim_H(\mathcal{J}(g)) : g\in M_f\}=2 for every ff — which remain open but are not part of this problem.

References

  1. Simon Albrecht and Christopher J. Bishop, "Speiser class Julia sets with dimension near one", Journal d'Analyse Mathématique 141 (2020), 49-98. DOI: 10.1007/s11854-020-0128-1. https://doi.org/10.1007/s11854-020-0128-1 (preprint arXiv:2007.07951, https://arxiv.org/abs/2007.07951)
  2. I. N. Baker, "The domains of normality of an entire function", Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 1 (1975), 277-283. DOI: 10.5186/aasfm.1975.0101. https://doi.org/10.5186/aasfm.1975.0101
  3. G. M. Stallard, "The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions. II", Ergodic Theory and Dynamical Systems 20 (2000), 895-910. DOI: 10.1017/s0143385700000481. https://doi.org/10.1017/s0143385700000481
  4. G. M. Stallard, "The Hausdorff dimension of Julia sets of entire functions. IV", Journal of the London Mathematical Society (2) 61 (2000), 471-488. DOI: 10.1112/s0024610799008601. https://doi.org/10.1112/s0024610799008601
  5. A. Eremenko and M. Lyubich, "Dynamical properties of some classes of entire functions", Annales de l'Institut Fourier 42 (1992), 989-1020. DOI: 10.5802/aif.1318. https://doi.org/10.5802/aif.1318
  6. Christopher J. Bishop, "A transcendental Julia set of dimension 1", Inventiones mathematicae 212 (2018), no. 2, 407-460. DOI: 10.1007/s00222-017-0770-0. https://doi.org/10.1007/s00222-017-0770-0
  7. Walter Bergweiler and Weiwei Cui, "The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class", Mathematische Zeitschrift 302 (2022), 2193-2205. DOI: 10.1007/s00209-022-03142-0. https://doi.org/10.1007/s00209-022-03142-0 (preprint arXiv:2105.00938, https://arxiv.org/abs/2105.00938)
  8. Walter Bergweiler and Weiwei Cui, "The Hausdorff Dimension of Escaping Sets of Meromorphic Functions in the Speiser Class", International Mathematics Research Notices (2024), no. 9, 7281-7306. DOI: 10.1093/imrn/rnad126. https://doi.org/10.1093/imrn/rnad126 (preprint arXiv:2209.06270, https://arxiv.org/abs/2209.06270)
  9. Magnus Aspenberg and Weiwei Cui, "Hausdorff dimension of escaping sets of meromorphic functions", Transactions of the American Mathematical Society 374 (2021), no. 9, 6145-6178. DOI: 10.1090/tran/8299. https://doi.org/10.1090/tran/8299 (preprint arXiv:2004.11283, https://arxiv.org/abs/2004.11283)
  10. Jack Burkart and Kirill Lazebnik, "Transcendental Julia sets of minimal Hausdorff dimension", Ergodic Theory and Dynamical Systems 45 (2025), no. 8, 2341-2413. DOI: 10.1017/etds.2024.124. https://doi.org/10.1017/etds.2024.124 (preprint arXiv:2109.05001, https://arxiv.org/abs/2109.05001)
  11. Walter Bergweiler and Lasse Rempe, "The Escaping Set in Transcendental Dynamics", Jahresbericht der Deutschen Mathematiker-Vereinigung 127 (2025), 1-141. DOI: 10.1365/s13291-025-00300-1. https://doi.org/10.1365/s13291-025-00300-1 (preprint arXiv:2507.11370, https://arxiv.org/abs/2507.11370)
  12. Christopher J. Bishop, "Models for the Speiser class", preprint (2025). arXiv:2501.01931, https://arxiv.org/abs/2501.01931