  In this paper, we discuss rotation number on the invariant curve of a one
parameter family of outer billiard tables. Given a convex polygon $\eta$, we
can construct an outer billiard table $T$ by cutting out a fixed area from the
interior of $\eta$. $T$ is piece-wise hyperbolic and the polygon $\eta$ is an
invariant curve of $T$ under the billiard map $\phi$. We will show that, if
$\beta $ is a periodic point under the outer billiard map with rational
rotation number $\tau = p / q$, then the $n$th iteration of the billiard map is
not the local identity at $\beta$. This proves that the rotation number $\tau$
as a function of the area parameter is a devil's staircase function.
