Let X be a finite set with a $\sigma$-algebra B = $2^X$ and let T : X → X be uniquely ergodic. The latter means that there exists the only one probabilistic invariant measure. Find all pairs of positive integers p,q such that for any points x, y $\in$ X there exists n = n(p, q, x, y, X) $\in \mathbb{Z}_+$ such that $T^{pn}x$ = $T^{qn}$y.
I have trouble finding anything I learnt useful to start. I think it maybe related to rotation.