Stochastic matrix-valued cocycles and non-homogeneous Markov chains

We prove weak ergodicity theorems for non - homogeneous Markov chains $\{X_\nu\}_{\nu\geq 0}$ taking values in a finite state space $S=\{1,\cdots,n\}$ for which the family of transition matrices $\{g(x)\}_{x\in X}$ is generated from some underlying topological or measurable dynamical system $f:X\to X$. Using the projective metric of Hilbert on $\mathcal{S}=\{(x_1,\cdots,x_n)\in\mathbb R ^n : x_i\geq 0, x_1+\cdots+x_n=1\}$, the space of distributions, we form the skew-product $T:X\times\mathcal{S}\to X\times\mathcal{S}$ defined by $T(x,p)=(f(x),g(x)p)$ and show that, for continuous $g$ positive on some set, weak ergodicity for such processes is a result of the existence of a map $\gamma:X\to\mathcal{S}$ whose graph is attracting and invariant under $T$. Some results on random compositions of non-expansive maps are obtained on the way.