Integrable Deformations of Cluster Maps of Type $$D_{2N}$$

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.