The Assouad spectrum of Kleinian limit sets and Patterson-Sullivan measure

Jonathan Fraser; Liam Stuart

doi:10.1007/s10711-022-00734-2

The Assouad spectrum of Kleinian limit sets and Patterson-Sullivan measure

Jonathan Fraser, Liam Stuart^*

^*Corresponding author for this work

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

Abstract

The Assouad dimension of the limit set of a geometrically finite Kleinian group with parabolics may exceed the Hausdorff and box dimensions. The Assouad spectrum is a continuously parametrised family of dimensions which ‘interpolates’ between the box and Assouad dimensions of a fractal set. It is designed to reveal more subtle geometric information than the box and Assouad dimensions considered in isolation. We conduct a detailed analysis of the Assouad spectrum of limit sets of geometrically finite Kleinian groups and the associated Patterson-Sullivan measure. Our analysis reveals several novel features, such as interplay between horoballs of different rank not seen the box or Assouad dimensions.

Original language	English
Article number	1
Number of pages	32
Journal	Geometriae Dedicata
Volume	217
DOIs	https://doi.org/10.1007/s10711-022-00734-2
Publication status	Published - 10 Oct 2022

Keywords

Kleinian group
Limit set
Patterson-Sullivan measure
Parabolic points
Assouad dimension
Assouad spectrum

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10.1007/s10711-022-00734-2Licence: CC BY

Fraser_2022_GD_Assouadspectrum_CC
Copyright © The Author(s) 2022. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, 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. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Final published version, 660 KBLicence: CC BY

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. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

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abstract = "The Assouad dimension of the limit set of a geometrically finite Kleinian group with parabolics may exceed the Hausdorff and box dimensions. The Assouad spectrum is a continuously parametrised family of dimensions which {\textquoteleft}interpolates{\textquoteright} between the box and Assouad dimensions of a fractal set. It is designed to reveal more subtle geometric information than the box and Assouad dimensions considered in isolation. We conduct a detailed analysis of the Assouad spectrum of limit sets of geometrically finite Kleinian groups and the associated Patterson-Sullivan measure. Our analysis reveals several novel features, such as interplay between horoballs of different rank not seen the box or Assouad dimensions.",

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author = "Jonathan Fraser and Liam Stuart",

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AB - The Assouad dimension of the limit set of a geometrically finite Kleinian group with parabolics may exceed the Hausdorff and box dimensions. The Assouad spectrum is a continuously parametrised family of dimensions which ‘interpolates’ between the box and Assouad dimensions of a fractal set. It is designed to reveal more subtle geometric information than the box and Assouad dimensions considered in isolation. We conduct a detailed analysis of the Assouad spectrum of limit sets of geometrically finite Kleinian groups and the associated Patterson-Sullivan measure. Our analysis reveals several novel features, such as interplay between horoballs of different rank not seen the box or Assouad dimensions.

KW - Kleinian group

KW - Limit set

KW - Patterson-Sullivan measure

KW - Parabolic points

KW - Assouad dimension

KW - Assouad spectrum

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VL - 217

JO - Geometriae Dedicata

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The Assouad spectrum of Kleinian limit sets and Patterson-Sullivan measure

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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