Let $P$ be transition matrix of a Markov chain with finite states that is not irreducible. Consider two initial distributions $a_0$ and $b_0$. Define sequences of distributions as
$$a_n = a_{n-1}P,\, b_n = b_{n-1}P$$
Can $\lim a_n$ and $\lim b_n$ exist? If they do, can they be the same?
Why I asked this: I have seen the following statement.
Main Result: Let $P$ be the transition matrix of an irreducible Markov chain. There exists a unique probability distribution $\pi$ satisfying $\pi = \pi P$.
This suggest that for arbitrary initial distribution $\mu_0$ that $\lim_n \mu_n$ exists (it doesn't have to exist even for irreducible chain. for example consider $2n$-cycle), the limit has to be $\pi$. But if $P$ is not irreducible, it is not clear to me whether limits of two different initial distributions can be the same? I can think of degenerate case where limit exists but are different. Namely consider graph with two points $A$ and $B$, with $P(A,A) = 1$ and $P(B,B) = 1$.
Sure they can be the same; consider a three state Markov chain with P(A,B) = P(B,A) = p, P(A,A) = P(B,B) = 1-p, and P(C,C) = 1 where 1 > p > 0. Now, let your initial distribution $a_0$ have weight 1 on A, and $b_0$ have weight 1 on B. Then, those will definitely mix and give you the same equilibrium limit.
More generally, a good way to interpret Markov chains is to interpret is as a flow of water along the chain. You can decompose a finite state Markov chain into a bunch of smaller irreducible chains, and if those chains happen to be aperiodic, then the "water" traveling around will balance out.