Structure and f-dependence of the a.c.i.m. for a unimodal map f of Misiurewicz type

By using a suitable Banach space on which we let the transfer operator act, we make a detailed study of the ergodic theory of a unimodal map $f$ of the interval in the Misiurewicz case. We show in particular that the absolutely continuous invariant measure $\rho$ can be written as the sum of 1/square root spikes along the critical orbit, plus a continuous background. We conclude by a discussion of the sense in which the map $f\mapsto\rho$ may be differentiable.