# Periodic Markov chain in 3D state space

Does someone have an example for a discrete time (time-homogeneous) Markov chain in a three-dimensional state space that is characterized by a transition matrix resulting in periodic behaviour?

I am aware of the following (unique) example

• in two-dimensional state space: $$\hspace{2em} P=\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$$ and

• its analog in three-dimensional state space $$P=\begin{pmatrix} 0 & 1& 0\\ 1 & 0& 0\\ 0 & 0& 1\\ \end{pmatrix}$$.

Is there an intuitive way of constructing such transition matrices?