  We study an infinite family of one-parameter deformations, so-called
$\alpha$-continued fractions, of interval maps associated to distinct triangle
Fuchsian groups. In general for such one-parameter deformations, the function
giving the entropy of the map indexed by $\alpha$ varies in a way directly
related to whether or not the orbits of the endpoints of the map synchronize.
For two cases of one-parameter deformations associated to the classical case of
the modular group $\text{PSL}_2(\mathbb Z)$, the set of $\alpha$ for which
synchronization occurs has been determined.
  Here, we explicitly determine the synchronization sets for each
$\alpha$-deformation in our infinite family. (In general, our Fuchsian groups
are not subgroups of the modular group, and hence the tool of relating
$\alpha$-expansions back to regular continued fraction expansions is not
available to us.) A curiosity here is that all of our synchronization sets can
be described in terms of a single tree of words. In a paper in preparation, we
identify the natural extensions of our maps, as well as the entropy functions
associated to each deformation.
