Contractive Markov systems II

In this paper, we continue development of the theory of contractive Markov systems (CMSs) initiated in \cite{Wer1}. We extend some results from \cite{Wer1}, \cite{Wer3}, \cite{Wer5} and \cite{Wer6} to the case of contractive Markov systems with probabilities which have a square summable variation by using some ideas of A. Johansson and A. Oeberg \cite{JO}. In particular, we show that an irreducible CMS has a unique invariant Borel probability measure if the vertex sets form an open partition of the state space and the restrictions of the probability functions on their vertex sets have a square summable variation and are bounded away from zero.