Perron–Frobenius spectrum for random maps and its approximation

To study the convergence to equilibrium in random maps, we develop the spectral theory of the corresponding transfer (Perron–Frobenius) operators acting in a certain Banach space of generalized functions (distributions). The random maps under study in a sense fill the gap between expanding and hyperbolic systems, since among their (deterministic) components there are both expanding and contracting ones. We prove the stochastic stability of the Perron–Frobenius spectrum and develop its finite rank operator approximations by means of a ‘stochastically smoothed’ Ulam approximation scheme. A counterexample to the original Ulam conjecture about the approximation of the SBR measure and the discussion of the instability of spectral approximations by means of the original Ulam scheme are presented as well.