Linear Response, and Consequences for Differentiability of Statistical Quantities and Multifractal Analysis

In this article we initially prove the differentiability of the topological pressure, equilibrium states and their densities with respect to smooth expanding dynamical systems and any smooth 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 exhibiting a $C^{r-1}$ rate with respect to the dynamics and the potential. An application for multifractal analysis is given. We also obtained asymptotic formulas for the derivatives of the topological pressure and other thermodynamical quantities.