Thermodynamic formalism for random transformations revisited

We return to the thermodynamic formalism constructions for random expanding in average transformations and for random subshifts of finite type with random rates of topological mixing, as well as to the Perron-Frobenius type theorem for certain random positive linear operators. Our previous expositions in [14, 19] and [21] were based on constructions which left some gaps and inaccuracies related to the measurability and uniqueness issues. Our approach here is based on Hilbert projective norms which were already applied in [5] for the thermodynamic formalism constructions for random subshifts of finite type but our method is somewhat different and more general so that it enables us to treat simultaneously both expanding and subshift cases.