Algebraic entropy of sign-stable mutation loops

TL;DR

This paper introduces the concept of sign stability in mutation loops, linking algebraic entropy with a new invariant called the cluster stretch factor, and computes these entropies for certain transformations.

Contribution

It defines sign stability for mutation loops and establishes a connection between algebraic entropy and the cluster stretch factor, a novel invariant.

Findings

Algebraic entropies of cluster transformations are computed.

Algebraic entropies coincide with the logarithm of the cluster stretch factor.

Sign stability relates to asymptotic behavior of tropical transformations.

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.