Iterates of the Poincaré map $T$ and it's correspondence to the periodic orbits?

Question

Suppose the fixed point $P_{1}$ of the Poincaré map $T$ corresponds to a periodic orbit of the continuous system of the period $\tau$.

Then the iterates of the map $T$ that is say the $n$th iterate $T^n$, now the fixed point $P_{n}$ would correspond to a periodic orbit of period $n \tau$?

Is this true, any examples or counter examples, also how do I prove this any reference/hints?