We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by $T(x,y) = (E (x), C(x,y))$, where E is an expanding map of $\mathbb {T}^u$ and C is a contracting map on each fiber. If $\inf |\!\det DT| \inf \| (D_yC)^{-1}\| ^{-2s}>1$ for some ${s<r-(({u+d})/{2}+1)}$, $r \geq 2$, and T satisfies a transversality condition between overlaps of iterates of T (a condition which we prove to be $C^r$-generic under mild assumptions), then the SRB measure $\mu _T$ of T is absolutely continuous and its density $h_T$ belongs to the Sobolev space $H^s({\mathbb {T}}^u\times {\mathbb {R}}^d)$. When $s> {u}/{2}$, it is also valid that the density $h_T$ is differentiable with respect to T. Similar results are proved for thermodynamical quantities for potentials close to the geometric potential.

We consider a robust class of random non-uniformly expanding local homeomorphisms and Hölder continuous potentials with small variation. For each element of this class we develop the thermodynamical formalism and prove the existence and uniqueness of equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.

We study the thermodynamic formalism of a $C^{\infty }$ non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a Hölder continuous potential with one additional condition, or geometric $t$-potential $\unicode[STIX]{x1D711}_{t}$ with $t<1$, the equilibrium state exists and is unique. We derive the level-2 large deviation principle for the equilibrium state of $\unicode[STIX]{x1D711}_{t}$. We study the multifractal spectra of the Katok map for the entropy and dimension of level sets of Lyapunov exponents.