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

TL;DR

This work computes the entropy of the Serre functor for partially wrapped Fukaya categories of graded surfaces with stops, showing the entropy H_t(S) is piecewise linear with slopes (1−minΩ) for t≥0 and (1−maxΩ) for t≤0, where Ω encodes boundary winding data via ω_i/m_i. The results translate the geometric data into graded gentle algebras, enabling explicit Serre-dimension calculations: upper and lower Serre dimensions are 1−minΩ and 1−maxΩ, respectively. In the finite-dimensional graded gentle algebra setting, a Gromov–Yomdin-type equality is established by relating the categorical entropy of the Serre functor to the logarithm of the spectral radius of the Coxeter transformation, with h_0(S)=log ρ([S]). The paper also develops the algebraic-geometric dictionary via AG-invariants and provides a broad suite of examples (including discs and annuli) to illustrate the theory and its implications for derived-discrete and affine-type cases. Altogether, it advances the understanding of how surface topology and stop data control dynamical invariants of derived categories and their Fukaya counterparts. $H_t(S)$, $Ω$, and Serre dimensions are expressed using precise surface-algebra data, connecting categorical entropy to explicit numerical invariants.$

Abstract

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.

Figures (8)

• Figure 1: Two types of oriented intersections between $\color{black}{\circ}$-arcs
• Figure 2: Choice of the arcs $\ell^*_1$ and $\ell^*_2$ for the definition of the degree of an oriented intersection
• Figure 3: Auslander--Reiten translate of an arc
• Figure 4: Intersections of $\ell$ and $^{[s_i]}\ell^{[s_j]}$ when $\partial_i$ and $\partial_j$ are distinct (for $s_i=4$ and $s_j=3$)
• Figure 5: Intersections of $\ell$ and $^{[s_i]}\ell^{[s_i]}$ when both endpoints of $\ell$ are on the same boundary component $\partial_i$ (for $s_i=2$)

Theorems & Definitions (68)

• Theorem 1: Theorem \ref{['thm:main1gentle']}
• Theorem 1$'$
• Theorem 2: Theorem \ref{['thm:Main2Section2']}
• Theorem 2$'$
• Remark 1
• Corollary 1
• Theorem 3: Theorem \ref{['thm:Main3Section2']}
• Theorem 4: Theorem \ref{['thm:categorical vs topological entropy']}
• Definition 1.1
• Theorem 1.2