A closed, oriented, compact surface of constant negative curvature and genus $/g/ \ge 2$ has an associated map on $/S/^1$ called the Bowen-Series boundary map. In 1991, Adler and Flatto described a particular dynamical cross-section (Poincaré section) for geodesic flow on such a surface using a "natural extension map" on $/S/^1 \times /S/^1$. Katok and Ugarcovici described an $(8/g/-4)$-parameter generalization of this map in 2016 and proved results for parameters satisfying the "short cycle property". The parameters corresponding to the original Bowen-Series map do not have this property but rather are "extremal" values in the parameter space.

In recent work, we have shown that (1) the Adler-Flatto construction is valid for all parameters choices that are either extremal or satisfy the short cycle property; (2) maps for extremal parameters have an associated dual system; and (3) within the Teichmüller space of the surface, the entropy of the boundary map with respect to its smooth invariant measure takes all values between zero and a maximum that is achieved on the surface that admits a regular fundamental polygon.