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In this paper, we discuss rotation number on the invariant curve of a oneparameter family of outer billiard tables. Given a convex polygon $\eta$, wecan construct an outer billiard table $T$ by cutting out a fixed area from theinterior of $\eta$. $T$ is piece-wise hyperbolic and the polygon $\eta$ is aninvariant curve of $T$ under the billiard map $\phi$. We will show that, if$\beta $ is a periodic point under the outer billiard map with rationalrotation number $\tau = p / q$, then the $n$th iteration of the billiard map isnot 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.

Devil's Staircase -- Rotation Number of Outer Billiard with Polygonal Invariant Curves