  We examine the operator $U_5$ defined on $L^2(\mu_{\frac14})$ where
$\mu_{\frac14}$ is the 1/4 Cantor measure. The operator $U_5$ scales the
elements of the canonical exponential spectrum for $L^2(\mu_{\frac14})$ by 5
--- that is, $Ue_{\gamma} = e_{5\gamma}$ where $e_{\gamma}(t) = e^{2\pi i
\gamma t}$. It is known that $U_5$ has a self-similar structure, which makes
its spectrum, which is currently unknown, of particular interest. In order to
better understand the spectrum of $U_5$, we demonstrate a decomposition of the
projection valued measures and scalar spectral measures associated with $U_5$.
We are also able to compute associated Radon-Nikodym derivatives between the
scalar measures. Our decomposition utilizes a system of operators which form a
representation of the Cuntz algebra $\mathcal{O}_2$.
