Thermodynamic Formalism and Perturbation Formulae for Quenched Random Open Dynamical Systems

Abstract: 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. Our random driving is generated by an invertible, ergodic, measure-preserving transformation on a probability space $(\Omega,\mathscr{F},m)$. For each $\omega$ we associate a piecewise-monotone, surjective map $T_\omega$ and a hole $H_\omega\subset [0,1]$; the map and the hole generate the corresponding open transfer operator. In the first chapter we prove, for a contracting potential, that there exists a unique random conformal measure $\nu_\omega$ supported on the survivor set. We also prove the existence of a unique random invariant density $\phi_\omega$. These provide an ergodic random invariant measure $\mu=\nu \phi$ supported on the global survivor set. Further, we prove quasi-compactness of the transfer operator cocycle and exponential decay of correlations for $\mu$. The escape rate of $\nu$ is given by the difference of the expected pressures for the closed and open random systems. Finally, we prove that the Hausdorff dimension of the surviving set is equal to the unique zero of the expected pressure function for almost every fiber. In the second chapter we consider quasi-compact linear operator cocycles and their small perturbations. We prove an abstract fiberwise first-order formula for the leading Lyapunov multipliers. Our new machinery is deployed to create a spectral approach for a quenched extreme value theory that considers random dynamics and random observations. Finally, we prove quenched statistical limit theorems for random equilibrium states arising from contracting potentials. We illustrate the above theory with a variety of examples.