We deal with all the mappings $f_\lambda(z)=\lambda e^z$ that have an attracting periodic orbit. We consider the set $J_r(f_\lambda)$ consisting of those points of the Julia set of $f_\lambda$ that do not escape to infinity under positive iterates of $f_\lambda$. Our ultimate result is that the function $\lambda\mapsto{\rm HD}(J_r(f_\lambda))$ is real-analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form $-t\log|F_\lambda'|$, where $F_\lambda$ is the natural map associated with $f_\lambda$ closely related to the corresponding map introduced in Urbański and Zdunik (2001). The formalism includes appropriately defined topological pressure, Perron–Frobenius operators, and geometric and invariant generalized conformal measures (Gibbs states). We show that our Perron–Frobenius operators are quasicompact and that they embed into a family of operators depending holomorphically on an appropriate parameter, and we obtain several other properties of these operators. We prove an appropriate version of Bowen's formula that the Hausdorff dimension of the set $J_r(f_\lambda)$ is equal to the unique zero of the pressure function. Since the formula for the topological pressure is independent of the set Jr(f), Bowen's formula also indicates that Jr(f) is the right set to deal with. We also study in detail the properties of quasiconformal conjugacies between the maps $f_\lambda$. As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the Central Limit Theorem and exponential decay of correlations.