Suppose I have the following markov chain:

$\displaystyle \[ \left( \begin{array}{ccc}

0 & 0.5 & 0.5 \\

0.5 & 0 & 0.5 \\

0.5 & 0.5 & 0 \end{array} \right)\] $

it seems that in this case that a state is able to return to itself either by 2 steps or 3 steps. Then what is the period of any single state? (all three states have the same periodicity though)

By the way, does a transient state has a period too?