Let G be a discrete group of isometries of the k-dimensional hyperbolic
space H^k, and let R be a representation of G into the group of the isometries
of H^n. We show that any R-equivariant map D from H^k to H^n weakly  extends 
to the boundary in the setting of Borel measures. The extension is
constructed using the family of so-called Patterson-Sullivan measures. As a
consequence we are able to extend a techinque of Besson Courtois and Gallot
for constructing volume-decreasing, R-equivariant maps, obtaining rigidity
results for representations. If we'll have the time, we'll show that, under
an additional hypothesis, the weak extension we construct is actually a
measurable map which extends the map D.