Consider a Markov chain, for simplicity let us consider time discrete chains.
The problem
We consider a TDMC (Time Discrete Markov Chain) $(X_t)_{t \geq 0}$ with $X \in \mathcal{X}$ (having $\mathcal{X}$ as the set of the chain states) under the assumption that the chain is ergodic.
Without going deeper into the reason that make me ask such a strange question, I would like to make the chain non-ergodic.
The question
Is there a way to make an ergodic chain non-ergodic?
For example by adding states preserving the old ones?
What approaches in literature, does anybody know?
In particular
I would like to preserve the environment of the network, I mean that given certain initial states, the chain should have a limit behavior that is coherent. If for example I use an absorption state, as I was suggested in one answer, should I use one absorption state connected to all states or an absorption state for every node?
Thankyou