This research demonstrates the relationship between hausdorff dimension and critical exponent in self-projective sets, suggesting it is influenced by certain properties.
Given a finite set A ⊆ S L ( 2 , R ) A⊆ SL(2,R) we study the dimension of the attractor K A K_A of the iterated function system induced by the projective action of A A . In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of K A K_A is given by the minimum of 1 and the critical exponent, under the assumption that A 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.
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Christodoulou et al. (2025) studied this question.
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