  The fundamental bijection is a bijection
$\theta:\mathcal{S}_n\to\mathcal{S}_n$ in which one uses the standard cycle
form of one permutation to obtain another permutation in one-line form. In this
paper, we enumerate the set of permutations $\pi \in \mathcal{S}_n$ that avoids
a pattern $\sigma \in \mathcal{S}_3$, whose image $\theta(\pi)$ also avoids
$\sigma$. We additionally consider what happens under repeated iterations of
$\theta$; in particular, we enumerate permutations $\pi \in \mathcal{S}_n$ that
have the property that $\pi$ and its first $k$ iterations under $\theta$ all
avoid a pattern $\sigma$. Finally, we consider permutations with the property
that $\pi=\theta^2(\pi)$ that avoid a given pattern $\sigma$, and end the paper
with some directions for future study.
