  In this paper, we investigate a class of non-invertible piecewise isometries
on the upper half-plane known as Translated Cone Exchanges. These maps include
a simple interval exchange on a boundary we call the baseline. We provide a
geometric construction for the first return map to a neighbourhood of the
vertex of the middle cone for a large class of parameters, then we show a
recurrence in the first return map tied to Diophantine properties of the
parameters, and subsequently prove the infinite renormalizability of the first
return map for these parameters.
