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arXiv · 1911.07587

Algebraic entropy of sign-stable mutation loops

Tsukasa Ishibashi·Shunsuke Kano
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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.

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Tsukasa Ishibashi, Shunsuke Kano. 2019-11-18. Algebraic entropy of sign-stable mutation loops. https://doi.org/10.1007/s10711-021-00606-1

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