Algebraic entropy of sign-stable mutation loops

Abstract

We introduce a property of mutation loops, called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropical $\mathcal{X}$-transformation. A sign-stable mutation loop has a numerical invariant which we call the cluster stretch factor, in analogy with that 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.

Final version. Section 3.2 and 3.3 are added. The proof of Theorem 3.12 is Corrected. To appear in Geometriae Dedicata. 45 pages, 6 figures