In this paper, we extend one of the main results from our joint work [12] with Hone and Mase, in which we studied a deformed type \(D_{4}\) map, to the general case of type \(D_{2N}\) for \(N\ge 3\) . This can be achieved through a “local expansion" operation, introduced in our joint work [7] with Grabowski and Hone. This operation involves inserting a specific subquiver into the quiver arising from the Laurentification of the deformed type \(D_{4}\) map. This insertion yields a new quiver, obtained through the Laurentification of the deformed type \(D_{6}\) map and thus enables systematic generalization to higher ranks \(D_{2N}\) . We also study the degree growth of the deformed type \(D_{2N}\) map via the tropical method and conjecture that, for each N, the deformed map is integrable, as indicated by the algebraic entropy test, a criterion for detecting integrability in discrete dynamical systems.
