Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops · arXivDesk
2508.14860Aug 20, 2025New 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
Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops
We prove that the entropy of the Serre functor S in the partially wrapped Fukaya category of a graded surface Σ with stops is given by the function sending t∈R to ht(S)=(1−minΩ)t
Nearby in the stack
, for
t≥0
, and to
ht(S)=(1−maxΩ)t
, for
t≤0
, where
Ω={m1ω1…,mbωb,0}
, and
ωi
is the winding number of the
i
th boundary component
∂iΣ
of the surface with
b
boundary components and
mi
stops on
∂iΣ
. It then follows that the upper and lower Serre dimensions are given by
1−minΩ
and
1−maxΩ
, 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.