The intervals of rotation of denumerable topological Markov chains with two classes of states

We consider finite and countable topological Markov chains whose states are partitioned into two classes; such Markov chais occur as symbolic description of systems like geometric Lorenz models. The rotation set for such a Markov chain is defined as the set of individual mean frequences of visiting the chosen class of the partition. We prove that for transitive topological Markov chain, the rotation set is a closed interval, which is nontrivial provided that the chain is topologically mixing. We also prove that for a finite transitive topological Markov chain, the endpoints of the rotation interval are rational and represent the rotation numbers of two periodic points.