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Let $F=(\phi, \psi):\mathbb{D}^2\to\mathbb{D}^2$ denote a holomorphicself-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 byHerv\'{e}'s work and the Hilbert space perspective of Agler, McCarthy and Youngon boundary regularity, we propose a new approach to boundary points ofDenjoy-Wolff type for the coordinate maps $\phi, \psi.$ We establish severalequivalent descriptions of our Denjoy-Wolff points, some of which only involvechecking specific directional derivatives and are particularly convenient forapplications. Using these tools, we are able to refine Herv\'{e}'s theorem andshow that, under the extra assumption of $\phi$ and $\psi$ possessingDenjoy-Wolff points with certain regularity properties, one can draw muchstronger conclusions regarding the behavior of $\{F^n\}.$

Denjoy-Wolff points on the bidisk via models