Dynamics and geometry on surfaces

Abstract

In [BdCH21], a continuous family of sphere homeomorphisms was constructed which is a mild quotient of the inverse limit of the tent family of interval endomorphisms. This family has the unimodal generalized pseudo-Anosov (ugpA) maps as a countable, dense subfamily, whose spheres of definition have well defined (and much studied) geometric structures. The purpose of the project is to prove that there are geometric structures on the spheres of definition of the entire family, making it the Gromov-Hausdorff completion of the ugpA family. The main tool to be used is a quasisymmetric uniformization theorem of Bonk-Kleiner [BK01] and the strategy is to prove that the ugpA family satisfies the conditions of their theorem uniformly. We also intend to study the relationship between the dynamics of a new class of surface homeomorphisms, the measurable pseudo-Anosov transformations introduced in [BdCH], and the geometry of the underlying spaces. The working examples are precisely the maps obtained in the completion mentioned before. (AU)