  In this paper we study the geodesic flow for a particular class of Riemannian
non-compact manifolds with variable pinched negative sectional curvature. For a
sequence of invariant measures we are able to prove results relating the loss
of mass and bounds on the measure entropies. We compute the entropy
contribution of the cusps. We develop and study the corresponding thermodynamic
formalism. We obtain certain regularity results for the pressure of a class of
potentials. We prove that the pressure is real analytic until it undergoes a
phase transition, after which it becomes constant. Our techniques are based on
the one side on symbolic methods and Markov partitions and on the other on
geometric techniques and approximation properties at level of groups.
