A Note on Categorical Entropy of Bielliptic Surfaces and Enriques Surfaces

Abstract: In this note, we show that there exists an autoequivalence of positive categorical entropy on the derived category of certain bielliptic surfaces. This gives the first example of a surface admitting positive categorical entropy in the absence of both positive topological entropy and any spherical objects. Moreover, we prove a Gromov-Yomdin type equality for the categorical entropy of autoequivalences on bielliptic surfaces and give a counterexample of this equality on Enriques surfaces.