Algebraic entropy of sign-stable mutation loops

Abstract

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.