Deformations of Lebesgue’s measure on the boundary of the Farey tree

Abstract

When presented as a boundary measure of the Farey tree, Lebesgue's measure possesses a subtle symmetry involving Dyer's outer automorphism of the group ${\mathsf{PGL}}_2(\mathbb{Z})$. In an unpublished manuscript of the second author, a census of some natural deformations of Lebesgue's measure that arise in this context was given. Here we study these measures to some depth and show that some of them are singular. These measures are potentially of arithmetic significance.