In case of a random countable topologically mixing Markov shift, a relative version of the Ruelle--Perron--Frobenius theorem is formulated. The proof relies on a new construction of an equivariant family of fiber measures using Crauel's relative Prohorov theorem. Some properties of the relative Gibbs measures are discussed as well.

This is joint work with Manfred Denker and Yuri Kifer.