Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops

TL;DR

This paper calculates the entropy of the Serre functor in partially wrapped Fukaya categories of surfaces with stops, linking it to winding numbers and boundary data, and relates it to algebraic invariants in gentle algebras.

Contribution

It provides an explicit formula for the entropy of the Serre functor in these categories and establishes a Gromov-Yomdin-like relation for gentle algebras.

Findings

Entropy formula involving boundary winding numbers

Upper and lower Serre dimensions expressed via boundary data

Gromov-Yomdin-like equality for gentle algebras

Abstract

We prove that the entropy of the Serre functor $\mathbb{S}$ in the partially wrapped Fukaya category of a graded surface $\Sigma$ with stops is given by the function sending $t \in \mathbb{R}$ to $ h_t(\mathbb{S}) = (1-\min \Omega)t$, for $t\geq 0$, and to $h_t(\mathbb{S})=(1-\max \Omega)t$, for $t\leq 0$, where $\Omega = \{\frac{\omega_1}{m_1} \ldots, \frac{\omega_b}{m_b},0\}$, and $\omega_i$ is the winding number of the $i$th boundary component $\partial_i\Sigma$ of the surface with $b$ boundary components and $m_i$ stops on $\partial_i \Sigma$. It then follows that the upper and lower Serre dimensions are given by $1-\min \Omega$ and $1-\max \Omega$, respectively. Furthermore, in the case of a finite dimensional gentle algebra $A$, we show that a Gromov-Yomdin-like equality holds by relating the categorical entropy of the Serre functor of the perfect derived category of $A$ to the logarithm of the spectral radius of the Coxeter transformation.