Categorical dynamical systems arising from sign-stable mutation loops

TL;DR

This paper explores categorical dynamics from mutation loops in algebraic geometry, linking entropy and pseudo-Anosov properties to cluster algebra structures.

Contribution

It introduces an autoequivalence for derived categories associated with sign-stable mutation loops and computes their categorical entropies.

Findings

Categorical entropies equal the logarithm of the cluster stretch factor.

The autoequivalence preserves sign stability in the derived category.

Connections to pseudo-Anosov dynamics are discussed.

Abstract

We give an autoequivalence of the derived category of the Ginzburg dg algebra for a mutation loop satisfying the sign stability introduced in [IK21]. We compute the categorical entropies of their restrictions to some subcategories and conclude that they are both given by the logarithm of the cluster stretch factor. Moreover, we discuss the pseudo-Anosovness of them in the sense of [FFHKL19] and [DHKK14].