  We study degree preserving maps over the set of irreducible polynomials over
a finite field. In particular, we show that every permutation of the set of
irreducible polynomials of degree $k$ over $\mathbb{F}_q$ is induced by an
action from a permutation polynomial of $\mathbb{F}_{q^k}$ with coefficients in
$\mathbb{F}_q$. The dynamics of these permutations of irreducible polynomials
of degree $k$ over $\mathbb{F}_q$, such as fixed points and cycle lengths, are
studied. As an application, we also generate irreducible polynomials of the
same degree by an iterative method.
