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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.14860 |
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| _version_ | 1866910027050123264 |
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| author | Chang, Wen Elagin, Alexey Schroll, Sibylle |
| author_facet | Chang, Wen Elagin, Alexey Schroll, Sibylle |
| contents | We prove that the entropy of the Serre functor $\mathbb{S}$ in the partially wrapped Fukaya category of a graded surface $Σ$ with stops is given by the function sending $t \in \mathbb{R}$ to $ h_t(\mathbb{S}) = (1-\min Ω)t$, for $t\geq 0$, and to $h_t(\mathbb{S})=(1-\max Ω)t$, for $t\leq 0$, where $Ω= \{\frac{ω_1}{m_1} \ldots, \frac{ω_b}{m_b},0\}$, and $ω_i$ is the winding number of the $i$th boundary component $\partial_iΣ$ of the surface with $b$ boundary components and $m_i$ stops on $\partial_i Σ$. It then follows that the upper and lower Serre dimensions are given by $1-\min Ω$ and $1-\max Ω$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14860 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops Chang, Wen Elagin, Alexey Schroll, Sibylle Representation Theory We prove that the entropy of the Serre functor $\mathbb{S}$ in the partially wrapped Fukaya category of a graded surface $Σ$ with stops is given by the function sending $t \in \mathbb{R}$ to $ h_t(\mathbb{S}) = (1-\min Ω)t$, for $t\geq 0$, and to $h_t(\mathbb{S})=(1-\max Ω)t$, for $t\leq 0$, where $Ω= \{\frac{ω_1}{m_1} \ldots, \frac{ω_b}{m_b},0\}$, and $ω_i$ is the winding number of the $i$th boundary component $\partial_iΣ$ of the surface with $b$ boundary components and $m_i$ stops on $\partial_i Σ$. It then follows that the upper and lower Serre dimensions are given by $1-\min Ω$ and $1-\max Ω$, 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. |
| title | Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2508.14860 |