Uniqueness, ergodicity and unidimensionality of invariant measures under a Markov operator

Let X be a compact metric space and C(X) be the space of all continuous functions on X. In this article, the authors consider the Markov operator T: C(X)N → C(X)N defined by Tf=(∑j=1Np1jfjow1 j,…,∑j=1NpN jfjowN j,…)for any f =(f1,f2, …, fN), where (pij)isa N × N transition probability matrix and {wij} is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T*-invariant measures where T* is the adjoint operator of T .