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 $t\ge 0$, and to $h_t(\mathbb{S})=(1-\max \Omega )t$, for $t\le 0$, where $\Omega =\{\frac{{\omega }_1}{m_1}\dots ,\frac{{\omega }_b}{m_b},0\}$, 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.

New version in which we merged the paper with the concurrent paper of Alexey Elagin on the same subject. Exposition and content changed, in particular, the paper now includes a section with explicit calculations of the entropy and the upper and lower Serre dimensions for many well-known examples of gentle algebras