  In 1979, for each signature for Fuchsian groups of the first kind, Bowen and
Series constructed an explicit fundamental domain for one group of the
signature, and from this a function on $\mathbb S^1$ tightly associated with
this group. In general, their fundamental domain enjoys what has since been
called the `extension property'. We determine the exact set of signatures for
cocompact triangle groups for which this extension property can hold for any
convex fundamental domain, and verify that for this restricted set, the
Bowen-Series fundamental domain does have the property.
  To each Bowen-Series function in this corrected setting, we naturally
associate four continuous deformation families of circle functions. We show
that each of these functions is aperiodic if and only if it is surjective; and,
is finite Markov if and only if its natural parameter is a hyperbolic fixed
point of the triangle group at hand.
