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

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.