Earthquake theorem for cluster algebras of finite type

TL;DR

This paper introduces a cluster algebraic generalization of Thurston's earthquake map for finite type cluster algebras, establishing a homeomorphism between certain tropical and positive spaces and connecting to classical earthquake maps.

Contribution

It defines the cluster earthquake map, proves an earthquake theorem for finite type cluster algebras, and explores its asymptotic behavior and relation to classical earthquake maps.

Findings

The cluster earthquake map is a homeomorphism between tropical and positive points of the cluster extit{X}-variety.

For types A_n and D_n, the map recovers classical earthquake maps for specific surfaces.

Asymptotic analysis reveals continuous deformations of the Fock--Goncharov fan.

Abstract

We introduce a cluster algebraic generalization of Thurston's earthquake map for the cluster algebras of finite type, which we call the \emph{cluster earthquake map}. It is defined by gluing exponential maps, which is modeled after the earthquakes along ideal arcs. We prove an analogue of the earthquake theorem, which states that the cluster earthquake map gives a homeomorphism between the spaces of $\mathbb{R}^\mathrm{trop}$- and $\mathbb{R}_{>0}$-valued points of the cluster $\mathcal{X}$-variety. For those of type $A_n$ and $D_n$, the cluster earthquake map indeed recovers the earthquake maps for marked disks and once-punctured marked disks, respectively. Moreover, we investigate certain asymptotic behaviors of the cluster earthquake map, which give rise to "continuous deformations" of the Fock--Goncharov fan.