  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.
