  Triangle partition maps form a family that includes many, if not most,
well-known multidimensional continued fraction algorithms. This paper begins
the exploration of the functional analysis behind the transfer operator of each
of these maps. We show that triangle partition maps give rise to two classes of
transfer operators and present theorems regarding the origin of these classes;
we also present related theorems on the form of transfer operators arising from
compositions of triangle partition maps. In the next paper, Part II, we will
find eigenfunctions of eigenvalue 1 for transfer operators associated with
select triangle partition maps on specified Banach spaces and then proceed to
prove that the transfer operators, viewed as acting on one-dimensional families
of Hilbert spaces, associated with select triangle partition maps are nuclear
of trace class zero. We will finish in part II by deriving Gauss-Kuzmin
distributions associated with select triangle partition maps.
