Circle Endomorphisms, Dual Circles and Thompson’s Group

We construct the dual Cantor set for a degree two expanding map f acting as cover of the circle T onto itself. Then we use the criterion for a continuous function on this Cantor set to be the scaling function of a uniformly asymptotically affine UAA expanding map to show that the scaling function for f descends to a continuous function on a dual circle T∗. We use this representation to view the Teichmüller space UAA as the set of scaling continuous functions on this dual circle and to construct a natural action of Thompson’s F -group as a group of geometrically realized biholomorphic isometries for UAA. Finally, we use the dual derivative D∗(f) for f defined on T∗ to obtain a generalized version of Rohlin’s formula for metric entropy where we take an integral over the dual circle.