Critically Finite Random Maps of an Interval

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