Is it possible to construct a Markov chain having an infinite number of stationary distributions $\pi_i$? Maybe also with a finite set of states $S$?
Maybe someone can explain why the following Markov chain has an infinite number of stationary distrbutions (see comments) instead of two stationary distrbutions like $[1, 0]$ and $[0, 1]$.
Check the following, $[1,0]$ is a stationary distribution, $[0,1]$ is also a stationary distribution.
Now, consider their convex combination, that is
$$[\lambda, 1-\lambda]$$ where $\lambda \in [0,1]$ is also a stationary distribtuion. Since there are infinitely many choices for $\lambda$, we have infinite number of stationary distribution.