Given a Farey-type map F with full branches in the extended Hecke group
Gamma_m, its dual F_# results from constructing the natural extension of F,
letting time go backwards, and projecting. Although numerical simulations may
suggest otherwise, we show that the domain of F_# is always tame, that is, it
always contains intervals. As a main technical tool we construct, for every
m=3,4,5,..., a homeomorphism M_m that simultaneously linearizes all maps with
branches in Gamma_m, and show that the resulting dual linearized iterated
function system satisfies the strong open set condition. We explicitly compute
the Holder exponent of every M_m, generalizing Salem's results for the
Minkowski question mark function M_3.
