For a rational map of the Riemann sphere, the Assouad-type dimension theory of its Julia set — Assouad dimension, lower dimension, and the Assouad and lower spectra — is complete for hyperbolic maps (all these dimensions equal the Hausdorff dimension hh) and for parabolic maps (Fraser and Stuart proved dim⁡AJ=max⁡{1,h}\dim_\mathrm{A}J=\max\{1,h\}, dim⁡LJ=min⁡{1,h}\dim_\mathrm{L}J=\min\{1,h\} with explicit formulae for both spectra, and showed that a Cremer point forces dim⁡AJ=2\dim_\mathrm{A}J=2). The same authors state the continuation of this programme as an open problem: computing these dimensions when the map is neither hyperbolic nor parabolic, naming the Siegel-disk and Herman-ring cases explicitly. No formulae, exact values, or sharp bounds for any Assouad-type dimension of the Julia set are known in any of the remaining classes: maps with Siegel disks or Herman rings, infinitely renormalizable maps, positive-area Julia sets (where only the forced value dim⁡A=2\dim_\mathrm{A}=2 follows from positive planar measure), and Cremer maps beyond the Assouad dimension itself. The problem is open.

Background

For a rational map T:C^→C^T:\hat{\mathbb{C}}\to\hat{\mathbb{C}} of degree at least 2, the Julia set J(T)J(T) is the locus of chaotic dynamics: the closed set of points at which the family of iterates of TT fails to be normal (equicontinuous on neighbourhoods). It is a compact fractal subset of the sphere whose geometry encodes the dynamics, and its dimension theory is a central topic of one-dimensional complex dynamics. That geometry depends on the periodic orbits of TT. The map is hyperbolic if all critical points lie in attracting basins; it is parabolic if J(T)J(T) contains no critical point but contains a rationally indifferent (parabolic) periodic point, i.e. one at which the derivative of an iterate is a root of unity. An irrationally indifferent periodic point, with derivative e2πiαe^{2\pi i\alpha} for irrational α\alpha, is either analytically linearizable — then it is the centre of a Siegel disk, a rotation domain on which an iterate of TT is conjugate to the irrational rotation z↦e2πiαzz\mapsto e^{2\pi i\alpha}z — or non-linearizable, in which case it is a Cremer point. A Herman ring is a periodic annulus on which an iterate of TT is likewise conjugable to an irrational rotation. Finally, a polynomial is infinitely renormalizable if iterates of arbitrarily high order, restricted to nested neighbourhoods of a periodic point, remain conjugate to polynomials of uniformly bounded degree; the Feigenbaum quadratic, at the accumulation of period doubling, is the canonical example.

Several fractal dimensions quantify how J(T)J(T) fills space (see the monograph of Fraser for a systematic treatment). The Assouad dimension dim⁡AF\dim_\mathrm{A}F measures worst-case local thickness: dim⁡AF≤s\dim_\mathrm{A}F\le s if F∩B(x,R)F\cap B(x,R) can be covered by at most C(R/r)sC(R/r)^s balls of radius rr for all x∈Fx\in F and 0<r<R0<r<R, with CC uniform; it dominates the box and Hausdorff dimensions and is extremely sensitive to clustering at arbitrarily small scales. The lower dimension dim⁡LF\dim_\mathrm{L}F is the corresponding best-case notion and is dominated by the Hausdorff dimension. The Assouad spectrum dim⁡AθF\dim^\theta_\mathrm{A}F and lower spectrum dim⁡LθF\dim^\theta_\mathrm{L}F, 0<θ<10<\theta<1, refine these by constraining the ratio of the two scales: dim⁡AθF\dim^\theta_\mathrm{A}F tends to the upper box dimension as θ→0\theta\to0, dim⁡LθF\dim^\theta_\mathrm{L}F tends to the lower box dimension as θ→0\theta\to0 only under a strong form of dynamical invariance, and the limits of both spectra as θ→1\theta\to1 exist in general but need not equal the Assouad and lower dimensions, respectively. A set is porous if every ball centred on it contains a proportional hole; porous subsets of the plane have Assouad dimension strictly less than 2. The Sullivan dictionary — the running analogy between rational maps and Kleinian groups acting on hyperbolic 3-space, for whose limit sets Fraser and Stuart computed the Assouad spectrum in the geometrically finite case — pairs J(T)J(T) with limit sets and makes the comparison of their dimension theories a programme in its own right.

For hyperbolic maps, the unique hh-conformal measure (the probability measure mm satisfying m(T(A))=∫A∣T′∣h dmm(T(A))=\int_A\lvert T'\rvert^h\,dm on sets where TT is injective, with h=dim⁡HJ(T)h=\dim_\mathrm{H}J(T)) is Ahlfors regular, so every Assouad-type dimension of J(T)J(T) equals hh. For parabolic maps the box and packing dimensions equal hh and the conformal measure obeys a global measure formula; Fraser and Stuart completed the Assouad-type picture: dim⁡AJ(T)=max⁡{1,h}\dim_\mathrm{A}J(T)=\max\{1,h\}, dim⁡LJ(T)=min⁡{1,h}\dim_\mathrm{L}J(T)=\min\{1,h\}, with explicit phase-transition formulae for both spectra, and — beyond the parabolic class — a Cremer point in J(T)J(T) forces dim⁡AJ(T)=2\dim_\mathrm{A}J(T)=2, the weakest version of the conjecture, discussed by Cheraghi, DeZotti and Yang, that such Julia sets should even have Hausdorff dimension 2. In the remaining classes even the classical dimensions are only partially understood: McMullen proved that quadratic polynomials with Siegel disks can have porous (hence dim⁡A<2\dim_\mathrm{A}<2) Julia sets; Buff and Chéritat constructed quadratic Julia sets of positive planar Lebesgue measure; Avila and Lyubich established conformal measures for Feigenbaum (infinitely renormalizable) Julia sets. No Assouad-type formula is known for any of these classes, and Fraser and Stuart state the continuation in their paper as an explicit open programme, naming the Siegel-disk and Herman-ring cases.

Problem Statement

Let T:C^→C^T:\hat{\mathbb{C}}\to\hat{\mathbb{C}} be a rational map of degree at least 2 with Julia set J(T)J(T). The Assouad-type dimension theory of J(T)J(T) is known when TT is hyperbolic (all Assouad-type dimensions equal h=dim⁡HJ(T)h=\dim_\mathrm{H}J(T)) and when TT is parabolic (dim⁡AJ=max⁡{1,h}\dim_\mathrm{A}J=\max\{1,h\}, dim⁡LJ=min⁡{1,h}\dim_\mathrm{L}J=\min\{1,h\}, and explicit formulae for the Assouad and lower spectra), and a Cremer point forces dim⁡AJ=2\dim_\mathrm{A}J=2. Determine the Assouad dimension, lower dimension, and Assouad and lower spectra of J(T)J(T) for the remaining classes of rational maps — the open programme posed by Fraser and Stuart — specifically: (i) maps with a Siegel disk (analytically linearizable irrationally indifferent periodic points); (ii) maps with a Herman ring; (iii) infinitely renormalizable polynomials, such as the Feigenbaum quadratic; (iv) maps whose Julia set has positive planar Lebesgue measure, for the lower dimension and both spectra (the Assouad dimension is forced to be 2 by positive measure); and (v) maps with a Cremer point, for the lower dimension and both spectra (only dim⁡A=2\dim_\mathrm{A}=2 is known). Each of (i)–(v) is an independently submittable sub-question, so resolving any one of them settles that class, while a full resolution of the open programme requires all five. For each class the goal is to compute these dimensions in terms of dynamical invariants of TT — for example the Hausdorff dimension, the Diophantine class of the rotation number, or renormalization data. These results would jointly complete the Assouad-type dimension column of the Sullivan dictionary for rational maps. Any answer must be consistent with the pinned constraints dim⁡LJ≤dim⁡HJ≤dim⁡‾BJ≤dim⁡AJ\dim_\mathrm{L}J\le\dim_\mathrm{H}J\le\overline{\dim}_\mathrm{B}J\le\dim_\mathrm{A}J, dim⁡AJ<2\dim_\mathrm{A}J<2 for the porous Siegel-disk examples of McMullen, and dim⁡AJ=2\dim_\mathrm{A}J=2 whenever J(T)J(T) has positive area or contains a Cremer point.

References

  1. Jonathan M. Fraser and Liam Stuart, 'Assouad type dimensions of parabolic Julia sets', Mathematische Zeitschrift 312(4), Paper No. 108 (2026). DOI: 10.1007/s00209-026-03997-7; arXiv:2203.04943. https://doi.org/10.1007/s00209-026-03997-7
  2. Jonathan M. Fraser and Liam Stuart, 'A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra', Bulletin of the American Mathematical Society 61, 103-118 (2024). DOI: 10.1090/bull/1796; arXiv:2007.15493. https://doi.org/10.1090/bull/1796
  3. Jonathan M. Fraser and Liam Stuart, 'The Assouad spectrum of Kleinian limit sets and Patterson-Sullivan measure', Geometriae Dedicata 217, Paper No. 1 (2023). DOI: 10.1007/s10711-022-00734-2; arXiv:2203.04931. https://doi.org/10.1007/s10711-022-00734-2
  4. Amlan Banaji and Jonathan M. Fraser, 'Assouad type dimensions of infinitely generated self-conformal sets', Nonlinearity 37, 045004 (2024). DOI: 10.1088/1361-6544/ad2864; arXiv:2207.11611. https://doi.org/10.1088/1361-6544/ad2864
  5. Amlan Banaji, Jonathan M. Fraser, István Kolossváry and Alex Rutar, 'Assouad spectrum of Gatzouras-Lalley carpets', Advances in Mathematics 484, Article 110707 (2026). DOI: 10.1016/j.aim.2025.110707; arXiv:2401.07168. https://doi.org/10.1016/j.aim.2025.110707
  6. Jonathan M. Fraser and Yunlong Xu, 'Dimensions of orbital sets in complex dynamics', preprint arXiv:2510.19456 (2025). https://arxiv.org/abs/2510.19456
  7. C. T. McMullen, 'Self-similarity of Siegel disks and Hausdorff dimension of Julia sets', Acta Mathematica 180(2), 247-292 (1998). DOI: 10.1007/bf02392901. https://doi.org/10.1007/bf02392901
  8. Xavier Buff and Arnaud Chéritat, 'Quadratic Julia sets with positive area', Annals of Mathematics 176(2), 673-746 (2012). DOI: 10.4007/annals.2012.176.2.1. https://doi.org/10.4007/annals.2012.176.2.1
  9. Davoud Cheraghi, Alexandre DeZotti and Fei Yang, 'Dimension paradox of irrationally indifferent attractors', preprint arXiv:2003.12340 (2020). https://arxiv.org/abs/2003.12340
  10. Artur Avila and Mikhail Lyubich, 'Hausdorff dimension and conformal measures of Feigenbaum Julia sets', Journal of the American Mathematical Society 21(2), 305-363 (2008). DOI: 10.1090/s0894-0347-07-00583-8. https://doi.org/10.1090/s0894-0347-07-00583-8
  11. Jonathan M. Fraser, 'Assouad Dimension and Fractal Geometry', Cambridge University Press (2020). DOI: 10.1017/9781108778459. https://doi.org/10.1017/9781108778459
  12. Krzysztof Barański, Bogusława Karpińska, David Martí-Pete, Leticia Pardo-Simón and Anna Zdunik, 'On the dimension of the boundaries of attracting basins of entire maps', Journal of the London Mathematical Society 112(6), e70349 (2025). DOI: 10.1112/jlms.70349; arXiv:2504.11142. https://doi.org/10.1112/jlms.70349