First return systems for some continued fraction maps

We prove a conjecture of Calta, Kraaikamp and the author: For all $n\ge 3$, each member of their one-parameter family of interval maps, denoted $T_{3,n,\alpha}$, has its `first expansive return map' of natural extension given by the first return map under the geodesic flow to a section of the unit tangent bundle of the hyperbolic surface uniformized by the underlying Fuchsian group $G_{3,n}$. To achieve the proof, we first prove the corresponding result for analogous one-parameter families related to the Hecke triangle Fuchsian group $G_{2,n}$. A direct comparison per $n$ of the $\alpha=1$ planar domains allows the Hecke group setting to provide sufficient information to prove the conjecture. We also give details about the entropy functions for the Hecke triangle Fuchsian group maps, $\alpha \mapsto h(T_{2,n,\alpha})$. Each is continuous on $(0,1)$, increasing on $(0,1/2)$, decreasing on $(1/2,1)$, with a central interval of constancy. We give precise formulas for the end points of the central intervals and also give precise formulas for the maximal entropy per family. For fixed $\alpha$, the entropy of $T_{2,n,\alpha}$ goes to zero as $n$ tends to infinity.