Assouad type dimensions of parabolic Julia sets

Jonathan M. Fraser; Liam Stuart

doi:10.1007/s00209-026-03997-7

Assouad type dimensions of parabolic Julia sets

Jonathan M. Fraser^*, Liam Stuart

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that the Assouad dimension of a parabolic Julia set is max{1, h} where h is the Hausdorff dimension of the Julia set. Since h may be strictly less than 1, this provides examples where the Assouad and Hausdorff dimensions are distinct. The box and packing dimensions of the Julia set are also known to coincide with h and, moreover, h can be characterised by a topological pressure function. The distinctive behaviour of the Assouad dimension invites further analysis of the ‘Assouad type dimensions’, including the lower dimension and the Assouad and lower spectra. We derive formulae for all of the Assouad type dimensions for parabolic Julia sets and the associated h-conformal measure. Further, we show that if a Julia set has a Cremer point, then the Assouad dimension is 2.

Original language	English
Article number	108
Number of pages	29
Journal	Mathematische Zeitschrift
Volume	312
Issue number	4
Early online date	17 Mar 2026
DOIs	https://doi.org/10.1007/s00209-026-03997-7
Publication status	Published - 1 Apr 2026

Keywords

Rational map
Julia set
Conformal measure
Parabolicity
Assouad dimension
Assouad spectrum
Lower dimension
Lower spectrum
Sullivan dictionary

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10.1007/s00209-026-03997-7Licence: CC BY

Fraser_2026_MZ_Assouad-type-dimensions-parabolic-Julia-sets_CC
© The Author(s) 2026. This article is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
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New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
The Leverhulme Trust
1/09/19 → 31/01/23
Project: Standard
Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

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title = "Assouad type dimensions of parabolic Julia sets",

abstract = "We prove that the Assouad dimension of a parabolic Julia set is max\{1, h\} where h is the Hausdorff dimension of the Julia set. Since h may be strictly less than 1, this provides examples where the Assouad and Hausdorff dimensions are distinct. The box and packing dimensions of the Julia set are also known to coincide with h and, moreover, h can be characterised by a topological pressure function. The distinctive behaviour of the Assouad dimension invites further analysis of the {\textquoteleft}Assouad type dimensions{\textquoteright}, including the lower dimension and the Assouad and lower spectra. We derive formulae for all of the Assouad type dimensions for parabolic Julia sets and the associated h-conformal measure. Further, we show that if a Julia set has a Cremer point, then the Assouad dimension is 2.",

keywords = "Rational map, Julia set, Conformal measure, Parabolicity, Assouad dimension, Assouad spectrum, Lower dimension, Lower spectrum, Sullivan dictionary",

author = "Fraser, \{Jonathan M.\} and Liam Stuart",

note = "Funding: JMF was financially supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034). LS was financially supported by the University of St Andrews.",

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AU - Fraser, Jonathan M.

AU - Stuart, Liam

N1 - Funding: JMF was financially supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034). LS was financially supported by the University of St Andrews.

PY - 2026/4/1

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N2 - We prove that the Assouad dimension of a parabolic Julia set is max{1, h} where h is the Hausdorff dimension of the Julia set. Since h may be strictly less than 1, this provides examples where the Assouad and Hausdorff dimensions are distinct. The box and packing dimensions of the Julia set are also known to coincide with h and, moreover, h can be characterised by a topological pressure function. The distinctive behaviour of the Assouad dimension invites further analysis of the ‘Assouad type dimensions’, including the lower dimension and the Assouad and lower spectra. We derive formulae for all of the Assouad type dimensions for parabolic Julia sets and the associated h-conformal measure. Further, we show that if a Julia set has a Cremer point, then the Assouad dimension is 2.

AB - We prove that the Assouad dimension of a parabolic Julia set is max{1, h} where h is the Hausdorff dimension of the Julia set. Since h may be strictly less than 1, this provides examples where the Assouad and Hausdorff dimensions are distinct. The box and packing dimensions of the Julia set are also known to coincide with h and, moreover, h can be characterised by a topological pressure function. The distinctive behaviour of the Assouad dimension invites further analysis of the ‘Assouad type dimensions’, including the lower dimension and the Assouad and lower spectra. We derive formulae for all of the Assouad type dimensions for parabolic Julia sets and the associated h-conformal measure. Further, we show that if a Julia set has a Cremer point, then the Assouad dimension is 2.

KW - Rational map

KW - Julia set

KW - Conformal measure

KW - Parabolicity

KW - Assouad dimension

KW - Assouad spectrum

KW - Lower dimension

KW - Lower spectrum

KW - Sullivan dictionary

U2 - 10.1007/s00209-026-03997-7

DO - 10.1007/s00209-026-03997-7

M3 - Article

SN - 0025-5874

VL - 312

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 4

M1 - 108

ER -

Assouad type dimensions of parabolic Julia sets

Abstract

Keywords

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Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this