First Expansive Returns and Cross Sections

This talk explores the study of alpha-type families of continued fraction-like interval maps for countably infinite collections of Fuchsian triangle groups, presented at the Workshop on "Uniform Distribution of Sequences" at the Erwin Schrödinger International Institute. Discover how measure theoretic entropy defines a continuous function of the parameter for each family. Learn about the concept of 'first expansive power' of an interval map and follow the proof of a conjecture that the first expansive power for each map defines a system whose natural extension is given by a cross section to the geodesic flow on the unit tangent bundle of the hyperbolic surface uniformized by the corresponding triangle group. The presentation includes essential background and motivation for understanding these mathematical relationships between continued fractions, interval maps, and hyperbolic geometry.