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

Uludağ A. M., Özkaraca I.

in: Essays in Geometry, Athanase Papadopoulos, Editor, european mathematical society publications, Aachen, pp.723-760, 2024

  • Publication Type: Book Chapter / Chapter Research Book
  • Publication Date: 2024
  • Publisher: european mathematical society publications
  • City: Aachen
  • Page Numbers: pp.723-760
  • Editors: Athanase Papadopoulos, Editor
  • Galatasaray University Affiliated: Yes

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 PGL2.Z/. In an unpub-

lished 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.