Hey I have to find the equilibrium path in a dynamic directed stochastic game. The equilibrium path are the set of state space points with positive hit probability. I cannot calculate the hit probabilities themselves but for every state space point s' I can control whether there is positive prob. of transitioning to this state from any other state s. Also due to some other order aspects of the state space I think I can solve the problem if I could apply the following principle:

Principle formulated as question
Does it hold that a state space point s' has positive hit probability if and only if there exist some other state space point from which there with positive probability is transition p(s'|s) >0 and s itself has positive hit probability.

PS. The state space is finite.

I'm not necessarily looking for a theorem although that would be nice but educated guesses are just as welcome.
Also the process is really a controlled markov proces rather than simply markov.