I am wondering if $\pi=\begin{pmatrix} 1&0\\1&0\end{pmatrix}$ is an aperiodic Markov chain or not?
This chain is clearly not irreducible, but there is also no period for the chain since if you start in state 2, you will never return to it. The definition of aperiodicity I am using says that an irreducible chain with period $1$ is aperiodic, but I am not sure about this case since it is not irreducible and does not have a period.
Period is a concept defined for a state. Now if $x$ and $y$ are two states in the same irreducible component, then they have the same period $t$. Then we say $t$ is the period of this irreducible component. If the Markov Chain's state space is irreducible, we say the Markov Chain is irreducible, in this case the period of the Markov Chain is well defined, it is equal to the period of its only irreducible component.
In your example, $P = \begin{bmatrix}1&0\\1&0\end{bmatrix}$ (a side note here, we usually denote transition probability matrix with $P$, and donate stationary distribution with $\pi$), state 1 is irreducible and $\{1\}$ is an irreducible component with period 1. But the Markov Chain is not irreducible, therefore it doesn't make much sense to say anything about the period of this Markov Chain.
I hope this helps!
