  This note aims to study the iteration theory of noncommutative self-maps of
bounded matrix convex domains. We prove a version of the Denjoy-Wolff theorem
for the row ball and the maximal quantization of the unit ball of
$\mathbb{C}^d$. For more general bounded matrix convex sets, we prove a version
of Wolff's theorem inspired by the results of Abate. Lastly, we use iteration
and fixed point theory to generalize the commutative results of Davidson,
Ramsey, and Shalit to quotients of the free semigroup algebra by WOT closed
ideals.
