Algebraic entropy of sign-stable mutation loops

We introduce a property of mutation loops, called the sign stability, with focus on an asymptotic behavior iteration tropical $\mathcal{X}$-transformation. A sign-stable loop has numerical invariant which we call cluster stretch factor, in analogy that pseudo-Anosov mapping class marked surface. compute algebraic entropies $\mathcal{A}$- and $\mathcal{X}$-transformations induced by loop, conclude these two coincide logarithm factor.