We introduce a cluster algebraic generalization of Thurston’s earthquake map for the cluster algebras of finite type, which we call the 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 $\mathbb {R}^{\textrm {trop}}$- 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.

中文翻译：

---

有限型簇代数的地震定理

我们引入了有限型簇代数的瑟斯顿地震图的簇代数推广，我们称之为簇地震图。它是通过粘合指数图来定义的，该指数图是沿着理想弧线对地震进行建模的。我们证明了地震定理的类比，该定理指出，集群地震图给出了 $\mathbb {R}^{\textrm {trop}}$- 和 $\mathbb {R}_{> 空间之间的同胚。簇 $\mathcal {X}$ 种类的 0}$ 值点。对于$A_{n}$和$D_{n}$类型，簇地震图确实分别恢复了标记盘和一次刺穿标记盘的地震图。此外，我们还研究了集群地震图的某些渐近行为，这些行为导致了福克-冈恰洛夫扇的“连续变形”。