Realizing Rotation Vectors for Torus Homeomorphisms

We consider the rotation set $\rho (F)$ for a lift $F$ of a homeomorphism $f:{T^2} \to {T^2}$, which is homotopic to the identity. Our main result is that if a vector $v$ lies in the interior of $\rho (F)$ and has both coordinates rational, then there is a periodic point $x \in {T^2}$ with the property that \[ \frac {{{F^q}({x_0}) - {x_0}}}{q} = v\] where ${x_0} \in {R^2}$ is any lift of $x$ and $q$ is the least period of $x$.