  Let $F=(\phi, \psi):\mathbb{D}^2\to\mathbb{D}^2$ denote a holomorphic
self-map of the bidisk without interior fixed points. It is well-known that,
unlike the case with self-maps of the disk, the sequence of iterates
$$\{F^n:=F\circ F\circ \cdots \circ F\}$$ needn't converge. The cluster set of
$\{F^n\}$ was described in a classical 1954 paper of Herv\'{e}. Motivated by
Herv\'{e}'s work and the Hilbert space perspective of Agler, McCarthy and Young
on boundary regularity, we propose a new approach to boundary points of
Denjoy-Wolff type for the coordinate maps $\phi, \psi.$ We establish several
equivalent descriptions of our Denjoy-Wolff points, some of which only involve
checking specific directional derivatives and are particularly convenient for
applications. Using these tools, we are able to refine Herv\'{e}'s theorem and
show that, under the extra assumption of $\phi$ and $\psi$ possessing
Denjoy-Wolff points with certain regularity properties, one can draw much
stronger conclusions regarding the behavior of $\{F^n\}.$
