Critically finite random maps of an interval

Given a finite collection $ {\mathcal{G}} $ of closed subintervals of the unit interval $ [0,1] $ with mutually empty interiors, we consider random multimodal $ C^3 $ maps with negative Schwarzian derivative, mapping each interval of $ {\mathcal{G}} $ onto the unit interval $ [0,1] $. The randomness is governed by an invertible ergodic map $ {\theta}:{\Omega}\to{\Omega} $ preserving a probability measure $ m $ on some probability space $ {\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) $, defined in Definition 9.1, 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 $:

$ \begin{align*} { {\mathcal{E}}{{\rm{P}}}}(t): = \int_{{\Omega}}\log {\lambda}_{t,\nu,{\omega}}dm({\omega}). \end{align*} $

is independent of the choice of a $ t $–conformal random measure $ \nu $.

4. The function

$ AA(T)\ni t\longmapsto { {\mathcal{E}}{{\rm{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}}{{\rm{P}}}}(t)\ge 0 $, it holds that

$ { {\mathcal{E}}{{\rm{P}}}}(b_T) = 0 \ \ \ {\rm and} \ \ \ [0,b_T]{\subset} {{{\rm{Int}}}}(AA(T)). $

6. $ {\rm{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} $.