In the theory of cluster algebras, a mutation loop induces discrete dynamical systems via its actions on the cluster \({\mathcal {A}}\)- and \({\mathcal {X}}\)-varieties. In this paper, we introduce a property of mutation loops, called the sign stability, with a focus on the asymptotic behavior of the iteration of the tropical \({\mathcal {X}}\)-transformation. The sign stability can be thought of as a cluster algebraic analogue of the pseudo-Anosov property of a mapping class on a surface. A sign-stable mutation loop has a numerical invariant which we call the cluster stretch factor, in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster \({\mathcal {A}}\)- and \({\mathcal {X}}\)-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This gives a cluster algebraic analogue of the classical theorem which relates the topological entropy of a pseudo-Anosov mapping class with its stretch factor.

在簇代数理论中，突变环通过其对簇（{{\ mathcal {A}} \） -和\（{\ mathcal {X}} \）-变量的作用而诱导出离散的动力学系统。在本文中，我们介绍了一种称为符号稳定性的突变环属性，重点是热带\（{{mathcal {X}} \）变换的迭代的渐近行为。可以将符号稳定性视为表面上映射类的拟Anosov性质的簇代数类似物。符号稳定的突变环具有数值不变量，我们称其为簇拉伸因子，类似于标记表面上伪Anosov映射类的拉伸因子。我们计算符号稳定突变环引起的簇\（{{mathcal {A}} \） -和\（{\ mathcal {X}} \）-变换的代数熵，并得出结论，这两个与集群拉伸因子的对数。这给出了经典定理的簇代数类似物，其将伪Anosov映射类的拓扑熵与其拉伸因子相关联。