Critical exponents and dimension for generalised limit sets

Tianyi Feng; Jonathan M. Fraser

doi:10.1088/1361-6544/ae277c

Critical exponents and dimension for generalised limit sets

Tianyi Feng^*, Jonathan M. Fraser

^*Corresponding author for this work

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

Abstract

There is a beautiful and well-studied relationship between the Poincaré exponent and the fractal dimensions of the limit set of a Kleinian group. Motivated by this, given an arbitrary discrete subset of the unit ball we define a critical exponent and investigate how it relates to the fractal dimensions of the associated generalised limit set.

Original language	English
Article number	125018
Number of pages	22
Journal	Nonlinearity
Volume	38
Issue number	12
DOIs	https://doi.org/10.1088/1361-6544/ae277c
Publication status	Published - 30 Dec 2025

Keywords

Limit set
Critical exponent
Hausdorff dimension
Box dimension
Kleinian group
Poincare exponent

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10.1088/1361-6544/ae277cLicence: CC BY

Feng_2025_Nonlinearity_Critical-exponents-generalised-limit-sets_CC
© 2025 The Author(s). Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license (https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Final published version, 314 KBLicence: CC BY

Fourier analytic tachniques in finitie: Fourier analytic techniques in finite fields
Fraser, J. (PI)
EPSRC
1/05/24 → 30/04/25
Project: Standard
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

Cite this

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title = "Critical exponents and dimension for generalised limit sets",

abstract = "There is a beautiful and well-studied relationship between the Poincar{\'e} exponent and the fractal dimensions of the limit set of a Kleinian group. Motivated by this, given an arbitrary discrete subset of the unit ball we define a critical exponent and investigate how it relates to the fractal dimensions of the associated generalised limit set.",

keywords = "Limit set, Critical exponent, Hausdorff dimension, Box dimension, Kleinian group, Poincare exponent",

author = "Tianyi Feng and Fraser, \{Jonathan M.\}",

note = "Funding: JMF was financially supported by a Leverhulme Trust Research Project Grant (RPG-2019- 034) and an EPSRC Standard Grant (EP/Y029550/1).",

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doi = "10.1088/1361-6544/ae277c",

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AU - Fraser, Jonathan M.

N1 - Funding: JMF was financially supported by a Leverhulme Trust Research Project Grant (RPG-2019- 034) and an EPSRC Standard Grant (EP/Y029550/1).

PY - 2025/12/30

Y1 - 2025/12/30

N2 - There is a beautiful and well-studied relationship between the Poincaré exponent and the fractal dimensions of the limit set of a Kleinian group. Motivated by this, given an arbitrary discrete subset of the unit ball we define a critical exponent and investigate how it relates to the fractal dimensions of the associated generalised limit set.

AB - There is a beautiful and well-studied relationship between the Poincaré exponent and the fractal dimensions of the limit set of a Kleinian group. Motivated by this, given an arbitrary discrete subset of the unit ball we define a critical exponent and investigate how it relates to the fractal dimensions of the associated generalised limit set.

KW - Limit set

KW - Critical exponent

KW - Hausdorff dimension

KW - Box dimension

KW - Kleinian group

KW - Poincare exponent

U2 - 10.1088/1361-6544/ae277c

DO - 10.1088/1361-6544/ae277c

M3 - Article

SN - 0951-7715

VL - 38

JO - Nonlinearity

JF - Nonlinearity

IS - 12

M1 - 125018

ER -

Critical exponents and dimension for generalised limit sets

Abstract

Keywords

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Projects

Fourier analytic tachniques in finitie: Fourier analytic techniques in finite fields

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

Cite this