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Thermodynamic Formalism for Random Interval Maps with Holes

We develop a quenched thermodynamic formalism for open random dynamical systems generated by finitely branched, piecewise-monotone mappings of the interval. The openness refers to the presence of holes in the interval, which terminate trajectories once they enter; the holes may also be random. Our random driving is generated by an invertible, ergodic, measure-preserving transformation $σ$ on a probability space $(Ω,\mathscr{F},m)$. For each $ω\inΩ$ we associate a piecewise-monotone, surjective map $T_ω:I\to I$, and a hole $H_ω\subset I$; the map $T_ω$, the random potential $φ_ω$, and the hole $H_ω$ generate the corresponding open transfer operator $\mathcal{L}_ω$. For a contracting potential, under a condition on the open random dynamics in the spirit of Liverani--Maume-Deschamps, we prove there exists a unique random probability measure $ν_ω$ supported on the survivor set ${X}_{ω,\infty}$ satisfying $ν_{σ(ω)}(\mathcal{L}_ωf)=λ_ων_ω(f)$. We also prove the existence of a unique random family of functions $q_ω$ that satisfy $\mathcal{L}_ωq_ω=λ_ωq_{σ(ω)}$. These yield an ergodic random invariant measure $μ=νq$ supported on the global survivor set, while $q$ combined with the random closed conformal measure yields a unique random absolutely continuous conditional invariant measure (RACCIM) $η$ supported on $I$. We prove quasi-compactness of the transfer operator cocycle and exponential decay of correlations for $μ$. Finally, the escape rates of the random closed conformal measure and the RACCIM $η$ coincide, and are given in terms of the expected pressure, as is the Hausdorff dimension of the surviving set $X_{ω,\infty}$. We provide examples of our general theory, including random $β$-transformations and random Lasota-Yorke maps.