Integrable deformations of cluster maps of type $D_{2N}$

In this paper, we extend one of the main results from our joint work with Hone and Mase, in which we studied a deformed type $D_4$ map, to the general case of the type $D_{2N}$ for $N\ge 3$. This can be achieved through a ``local expansion" operation, introduced in our joint work 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 deformed type $D_{2N}$ map via the tropical method and conjecture that, for each $N$, the deformed map is an integrable, as indicated by the algebraic entropy test, the criterion for detecting integrability in the discrete dynamical systems.