The Hausdorff dimension of self-projective sets

Argyris Christodoulou; Natalia Jurga

doi:10.1090/tran/9576

The Hausdorff dimension of self-projective sets

Argyris Christodoulou^*, Natalia Jurga

^*Corresponding author for this work

Pure Mathematics

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

Abstract

Given a finite set 𝒜 ⊆ SL(2, ℝ) we study the dimension of the attractor K_𝒜 of the iterated function system induced by the projective action of 𝒜. In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of K_𝒜 is given by the minimum of 1 and the critical exponent, under the assumption that A satisfies certain discreteness conditions and a Diophantine property. Our approach combines techniques from the theories of iterated function systems and Möbius semigroups, and allows us to discuss the continuity of the Hausdorff dimension, as well as the dimension of the support of the Furstenberg measure.

Original language	English
Pages (from-to)	1-32
Number of pages	32
Journal	Transactions of the American Mathematical Society
Volume	379
Issue number	1
Early online date	16 Oct 2025
DOIs	https://doi.org/10.1090/tran/9576
Publication status	Published - 1 Jan 2026

Keywords

Projective space
Iterated function system
Hausdorff dimension
Semigroups
Mobius transformations

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10.1090/tran/9576

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 = "The Hausdorff dimension of self-projective sets",

abstract = "Given a finite set 𝒜 ⊆ SL(2, ℝ) we study the dimension of the attractor K𝒜 of the iterated function system induced by the projective action of 𝒜. In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of K𝒜 is given by the minimum of 1 and the critical exponent, under the assumption that A satisfies certain discreteness conditions and a Diophantine property. Our approach combines techniques from the theories of iterated function systems and M{\"o}bius semigroups, and allows us to discuss the continuity of the Hausdorff dimension, as well as the dimension of the support of the Furstenberg measure.",

keywords = "Projective space, Iterated function system, Hausdorff dimension, Semigroups, Mobius transformations",

author = "Argyris Christodoulou and Natalia Jurga",

note = "Funding: Both authors were financially supported by the Leverhulme Trust (Research Project Grant number RPG-2016-194). The second author was also financially supported by the EPSRC (Standard Grant EP/R015104/1).",

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AU - Jurga, Natalia

N1 - Funding: Both authors were financially supported by the Leverhulme Trust (Research Project Grant number RPG-2016-194). The second author was also financially supported by the EPSRC (Standard Grant EP/R015104/1).

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N2 - Given a finite set 𝒜 ⊆ SL(2, ℝ) we study the dimension of the attractor K𝒜 of the iterated function system induced by the projective action of 𝒜. In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of K𝒜 is given by the minimum of 1 and the critical exponent, under the assumption that A satisfies certain discreteness conditions and a Diophantine property. Our approach combines techniques from the theories of iterated function systems and Möbius semigroups, and allows us to discuss the continuity of the Hausdorff dimension, as well as the dimension of the support of the Furstenberg measure.

AB - Given a finite set 𝒜 ⊆ SL(2, ℝ) we study the dimension of the attractor K𝒜 of the iterated function system induced by the projective action of 𝒜. In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of K𝒜 is given by the minimum of 1 and the critical exponent, under the assumption that A satisfies certain discreteness conditions and a Diophantine property. Our approach combines techniques from the theories of iterated function systems and Möbius semigroups, and allows us to discuss the continuity of the Hausdorff dimension, as well as the dimension of the support of the Furstenberg measure.

KW - Projective space

KW - Iterated function system

KW - Hausdorff dimension

KW - Semigroups

KW - Mobius transformations

U2 - 10.1090/tran/9576

DO - 10.1090/tran/9576

M3 - Article

SN - 0002-9947

VL - 379

SP - 1

EP - 32

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 1

ER -

The Hausdorff dimension of self-projective sets

Abstract

Keywords

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Projects

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this