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We extend the results of our 2020 paper in the Annali della Scuola NormaleSuperiore di Pisa, Classe di Scienze. There, we associated to each of aninfinite family of triangle Fuchsian groups a one-parameter family of continuedfraction maps and showed that the matching (or, synchronization) intervals areof full measure. Here, we find planar extensions of each of the maps, and provethe continuity of the entropy function associated to each one-parameter family.  We also introduce a notion of "first pointwise expansive power" of aneventually expansive interval map. We prove that for every map in one of ourone-parameter families its first pointwise expansive power map has its naturalextension given by the first return of the geodesic flow to a cross section inthe unit tangent bundle of the hyperbolic orbifold uniformized by thecorresponding group. We conjecture that this holds for all of our maps. We givenumerical evidence for the conjecture.

Continuity of entropy for all $\alpha$-deformations of an infinite class of continued fraction transformations