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

TL;DR

This paper generalizes the integrability analysis of deformed type D maps from D4 to D2N, introducing a local expansion method and studying degree growth to support conjectures of integrability.

Contribution

It extends the known results on deformed type D maps to higher ranks D2N using a local expansion technique and analyzes their integrability through algebraic entropy.

Findings

The deformed type D maps are conjectured to be integrable for all N.

The local expansion operation systematically constructs higher-rank maps.

Degree growth analysis supports the integrability conjecture.

Abstract

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 $N\geq3$. 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.