ABSTRACT:

(joint work with V. Baladi) We show the differentiability (in a weak sense) of the SBR measures on each topological class of piecewise expanding unimodal maps. This result is sharp, since we also show that the SBR measure is not in general differentiable in families transversal to the topological classes. We give an explicit formula for the derivative of SBR measures of a family where the topological dynamics does not change (the "smooth deformations") , which, under certain conditions, is the abelian limit of a formula conjectured by D. Ruelle. To show the richness of such theory, we characterize the smooth deformations as the families tangent to a continuous distribution of codimension-one subspaces (the ``horizontal" directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of smooth deformations tangent to every given horizontal direction.