I have found the following invariant distribution $\pi$, invariant w.r.t. my transition probability matrix: $$P = \begin{bmatrix} 0.8 & 0.2 \\ 0.5 & 0.5 \end{bmatrix}$$
$\pi = [0.7142857, 0.2857143]$
I now want to know - given that the initial state is drawn at random from $\pi$ - what is the joint probability that $X_{t} = 1$ and $X_{t+1} = 1$. Is it right to say that it is 0.7142857? Or am I not understanding what the meaning of invariant distribution is?
I also read that the joint ($X_t, X_{t+1}$) has to do with $X_{t}$ ~ $\pi$ and $X_{t+1}$|$X_{t}$ ~ $P$, but not sure of how to use this information.
$P(X_t=1,X_{t+1}=1)=P(X_{t+1}=1|X_t=1)P(X_t=1)=(0.8)(0.7142857)$ (assuming that your states are named $1$ and $2$). The distribution of $X_t$ is $\pi$ for all $t$ but $P(X_{t+1}=1|X_t=1)$ is given by the transition matrix.