On the smooth dependence of SRB measures for partially hyperbolic systems

In this paper, we study the differentiability of SRB measures for partially hyperbolic systems. We show that for any $s \geq 1$, for any integer $\ell \geq 2$, any sufficiently large $r$, any $\varphi \in C^{r}(\T, \R)$ such that the map $f : \T^2 \to \T^2, f(x,y) = (\ell x, y + \varphi(x))$ is $C^r-$stably ergodic, there exists an open neighbourhood of $f$ in $C^r(\T^2,\T^2)$ such that any map in this neighbourhood has a unique SRB measure with $C^{s-1}$ density, which depends on the dynamics in a $C^s$ fashion. We also construct a $C^{\infty}$ mostly contracting partially hyperbolic diffeomorphism $f: \T^3 \to \T^3$ such that all $f'$ in a $C^2$ open neighbourhood of $f$ possess a unique SRB measure $\mu_{f'}$ and the map $f' \mapsto \mu_{f'}$ is strictly H\"older at $f$, in particular, non-differentiable. This gives a partial answer to Dolgopyat's Question 13.3 in \cite{Do1}.