Work

The Hausdorff dimension of self-projective sets

Year: 2025

Type: article

Abstract: Given a finite set A ⊆ S L ( 2 , R ) \mathcal {A}\subseteq \mathrm {SL}(2,\mathbb {R}) we study the dimension of the attractor K A K_\mathcal {A} of the iterated function system induced by the projective action of A \mathcal {A} . In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of K A K_\mathcal {A} is given by the minimum of 1 and the critical exponent, under the assumption that A \mathcal {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.

Source: Transactions of the American Mathematical Society

DOI: 10.1090/tran/9576

Authors: Argyrios Christodoulou, Natalia Jurga

Institutions: Queen Mary University of London, Aristotle University of Thessaloniki, University of St Andrews

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Leverhulme Trust, Engineering and Physical Sciences Research Council

Awards: Fourier analytic techniques in geometry and analysis, G373828484, G382211468, G6164852225