Abstract: In a series of papers with Calta and Kraaikamp, we associate to each of an infinite number of hyperbolic triangles a one-parameter family of interval maps. We prove that the measure theoretic entropy value is continuous along each one-parameter family. We conjecture that the "first expansive power" of each of the maps arises as a factor of a section to the geodesic flow on the unit tangent space of the corresponding hyperbolic surface. In the talk, I will indicate the history of such investigations, explain basic notions, share some examples, and indicate the methods.