  In this article we initially fix ourselves to smooth (C^r) expanding
dynamical systems. We prove the C^{r-1} differentiability of the topological
pressure, equilibrium states and their densities with respect to smooth
expanding dynamical systems and any smooth potential (C^{r-1}- linear response
formula wiyh respect to the dynamics, and analytical response formula with
respect to the potential). This is done by proving the regularity of the
dominant eigenvalue of the transfer operator with respect to dynamics and
potential. From that, we obtain strong consequences on the regularity of the
dynamical system statistical properties, that apply in more general contexts.
Indeed, we prove that the average and variance obtained from the central limit
theorem vary $C^{r-1}$ with respect to the $C^{r}-$expanding dynamics and
$C^{r}-$potential, and also, there is a large deviations principle with its
rate $C^{r-1}$ with respect to the dynamics and potential. An application for
multifractal analysis is given.
