Let's consider the set $\mathcal{M}=\{\mu:|\mu(P-I)|_1\leq \epsilon\}$ where $\mu$ is a probability vector, $P$ is the transition matrix of a discrete homogeneous Markov chain, $I$ is the identity matrix, $|\cdot|_1$ is the total variation, and $\epsilon$ is a small positive constant.
In some sense, $\mathcal{M}$ is the set of probability vectors that are "almost" invariant with respect to the Markov chain given by $P$. I've tried to understand some property about this set.
Since it seems some sort of generalized eigenvector problem where one has some "bounded-error tolerance", I tried to look for some literature in numerical analysis about it, finding nothing.
First, any reference related to the object above would be really appreciated. Second, here is a specific question I tried to figure out: is $\mathcal{M}$ always the convex hull of a finite number of probability vectors? Checking that is convex is straightforward, but I didn't manage to prove that there is a finite number of "extremal" elements that generate $\mathcal{M}$ as their convex hull.