Invariant and conditionally invariant measures for random open interval maps with countably many branches

In this paper, building on previous work, we extend the thermodynamic formalism for random open dynamical systems generated by piecewise monotone interval maps with countably many branches. Under summable and contracting assumptions on the potential, we establish the Ruelle-Perron-Frobenius type theorem for the associated random open operator and prove exponential decay of correlations. In addition, we investigate the escape rate for the hole and conditionally invariant measure for the open system.