Suppose I have a finite set say $C=\{c_0,c_1,\ldots,c_R\}$ which are some disjoint subsets of the space say $\{0,1\}^m$. The cardinality of the union of these subsets is much less than $2^m$ say.
If I start randomly from any subset of this class $C$ (say draw uniformly according to its size) then I move to the next state according to a matrix $P$ which describes probabilities of moving from any class to any other class inside $C$ or going to $\{0,1\}^m\setminus C$. If the probability of going from $\{0,1\}^m\setminus C$ to $C$ is zero is there any chance of having an invariant distribution for this model?
I find that reccurence and irreducibility of the graph ensure the existence of an invariant distribution. However, is there any chance of having an invariant distribution under this scenario?