A Note on Categorical Entropy of Bielliptic Surfaces and Enriques Surfaces

In this note, we show that there exists an autoequivalence of positive categorical entropy on the derived category of 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 to this equality on Enriques surfaces.