Critically finite random maps of an interval

Given a finite collection \begin{document}$ {\mathcal{G}} $\end{document} of closed subintervals of the unit interval \begin{document}$ [0,1] $\end{document} with mutually empty interiors, we consider random multimodal \begin{document}$ C^3 $\end{document} maps with negative Schwarzian derivative, mapping each interval of \begin{document}$ {\mathcal{G}} $\end{document} onto the unit interval \begin{document}$ [0,1] $\end{document} . The randomness is governed by an invertible ergodic map \begin{document}$ {\theta}:{\Omega}\to{\Omega} $\end{document} preserving a probability measure \begin{document}$ m $\end{document} on some probability space \begin{document}$ {\Omega} $\end{document} . We denote the corresponding skew product map by \begin{document}$ T $\end{document} and call it a critically finite random map of an interval. We prove that there exists a subset \begin{document}$ AA(T) $\end{document} , defined in Definition 9.1, of \begin{document}$ [0,1] $\end{document} with the following properties: 1. For each \begin{document}$ t\in AA(T) $\end{document} a \begin{document}$ t $\end{document} –conformal random measure \begin{document}$ \nu_t $\end{document} exists. We denote by \begin{document}$ {\lambda}_{t,\nu_t,{\omega}} $\end{document} the corresponding generalized eigenvalues of the corresponding dual operators \begin{document}$ {\mathcal{L}}_{t,{\omega}}^* $\end{document} , \begin{document}$ {\omega}\in{\Omega} $\end{document} . 2. Given \begin{document}$ t\ge 0 $\end{document} any two \begin{document}$ t $\end{document} –conformal random measures are equivalent. 3. The expected topological pressure of the parameter \begin{document}$ t $\end{document} : \begin{document}$ \begin{align*} { {\mathcal{E}}{{\rm{P}}}}(t): = \int_{{\Omega}}\log {\lambda}_{t,\nu,{\omega}}dm({\omega}). \end{align*} $\end{document} is independent of the choice of a \begin{document}$ t $\end{document} –conformal random measure \begin{document}$ \nu $\end{document} . 4. The function \begin{document}$ AA(T)\ni t\longmapsto { {\mathcal{E}}{{\rm{P}}}}(t)\in\mathbb R $\end{document} is monotone decreasing and Lipschitz continuous. 5. With \begin{document}$ b_T $\end{document} being defined as the supremum of such parameters \begin{document}$ t\in AA(T) $\end{document} that \begin{document}$ { {\mathcal{E}}{{\rm{P}}}}(t)\ge 0 $\end{document} , it holds that \begin{document}$ { {\mathcal{E}}{{\rm{P}}}}(b_T) = 0 \ \ \ {\rm and} \ \ \ [0,b_T]{\subset} {{{\rm{Int}}}}(AA(T)). $\end{document} 6. \begin{document}$ {\rm{HD}}( {\mathcal{J}}_{\omega}(T)) = b_T $\end{document} for \begin{document}$ m $\end{document} –a.e \begin{document}$ {\omega}\in{\Omega} $\end{document} , where \begin{document}$ {\mathcal{J}}_{\omega}(T) $\end{document} , \begin{document}$ {\omega}\in{\Omega} $\end{document} , form the random closed set generated by the skew product map \begin{document}$ T $\end{document} . 7. \begin{document}$ b_T = 1 $\end{document} if and only if \begin{document}$ {\bigcup}_{ {\Delta}\in {\mathcal{G}}} {\Delta} = [0,1] $\end{document} , and then \begin{document}$ {\mathcal{J}}_{\omega}(T) = [0,1] $\end{document} for all \begin{document}$ {\omega}\in{\Omega} $\end{document} .