Commuting integrable maps from a deformed D$_4$ cluster algebra

TL;DR

This paper explores integrable maps derived from a deformed D4 cluster algebra, revealing their geometric structure, constructing commuting maps, and analyzing their degree growth through algebraic and geometric methods.

Contribution

It introduces a new integrable map from a deformed D4 cluster algebra and constructs a commuting map, linking cluster mutations to elliptic surfaces and Painlevé equations.

Findings

The map has a rational first integral defining an elliptic surface.

Constructed a second integrable map that commutes with the first.

Degree growth analyzed via tropical equations and blow-up techniques.

Abstract

In this paper we revisit an integrable map of the plane which we obtained recently as a two-parameter family of deformed mutations in the cluster algebra of type D$_4$. The rational first integral for this map defines an invariant foliation of the plane by level curves, and we explain how this corresponds to a rational elliptic surface of rank 2. This leads us to construct another (independent) integrable map, commuting with the first, such that both maps lift to compositions of mutations in an enlarged cluster algebra, whose underlying quiver is equivalent to the one found by Okubo for the $q$-Painlev\'e VI equation. The degree growth of the two commuting maps is calculated in two different ways: firstly, from the tropical (max-plus) equations for the d-vectors of the cluster variables; and secondly, by constructing the minimal space of initial conditions for the two maps, via blowing up $\mathbb{P}^1 \times \mathbb{P}^1$.