Algebraic entropy of sign-stable mutation loops

Tsukasa Ishibashi, Shunsuke Kano

Research output: Contribution to journalArticlepeer-review

Abstract

In the theory of cluster algebras, a mutation loop induces discrete dynamical systems via its actions on the cluster A- and X-varieties. In this paper, we introduce a property of mutation loops, called the sign stability, with a focus on the asymptotic behavior of the iteration of the tropical X-transformation. The sign stability can be thought of as a cluster algebraic analogue of the pseudo-Anosov property of a mapping class on a surface. A sign-stable mutation loop has a numerical invariant which we call the cluster stretch factor, in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This gives a cluster algebraic analogue of the classical theorem which relates the topological entropy of a pseudo-Anosov mapping class with its stretch factor.

Original languageEnglish
Pages (from-to)79-118
Number of pages40
JournalGeometriae Dedicata
Volume214
Issue number1
DOIs
Publication statusPublished - 2021 Oct
Externally publishedYes

Keywords

  • Algebraic entropy
  • Cluster algebra
  • Mutation loop
  • Pseudo-Anosov mapping class

ASJC Scopus subject areas

  • Geometry and Topology

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