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 D₄ map, to the general case of type D₂₍ for N 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 D₄ map. This insertion yields a new quiver, obtained through the Laurentification of the deformed type D₆ map and thus enables systematic generalization to higher ranks D₂₍. We also study the degree growth of the deformed type D₂₍ 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.
Wookyung Kim (Thu,) studied this question.