Consider a Markov chain (e.g. 2x2 or 3x3) which is periodic, therefore there is no stationary distribution for the chain.
What is the way to find the fraction of time spent in each state by the system?
If it was possible to define a stationary distribution $\Pi$ I would have said that the fraction of time spent by the system in the state j is:$$f_j =\Pi_j$$
But in the case of periodic chain $\Pi$ is not defined, so what is the right way to get $f_j$?