Abstract: Given a Farey-type map F with full branches in the extended Hecke group Gamma_m, its dual F_# results from constructing the natural extension of F, letting time go backwards, and projecting. Although numerical simulations may suggest otherwise, we show that the domain of F_# is always tame, that is, it always contains intervals. As a main technical tool we construct, for every m=3,4,5,..., a homeomorphism M_m that simultaneously linearizes all maps with branches in Gamma_m, and show that the resulting dual linearized iterated function system satisfies the strong open set condition. We explicitly compute the Holder exponent of every M_m, generalizing Salem's results for the Minkowski question mark function M_3.
16:15: Ayreena Bakhtawar (Scuola Normale Superiore di Pisa)
Title: Improvements to Dirichlet's theorem in Diophantine approximation
Abstract: Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. Dirichlet’s theorem (1842) is a fundamental result that gives an optimal approximation rate of any irrational number. The set of real numbers for which Dirichlet's theorem admits an improvement was originally studied by Davenport and Schmidt. In this talk I will discuss some new metrical results for the set of Dirichlet improvable numbers.