A Note on Categorical Entropy of Bielliptic Surfaces and Enriques Surfaces

TL;DR

This paper demonstrates the existence of autoequivalences with positive categorical entropy on bielliptic surfaces, providing new insights into the relationship between categorical and topological entropy.

Contribution

It presents the first example of a surface with positive categorical entropy without positive topological entropy or spherical objects, and explores entropy equality on bielliptic and Enriques surfaces.

Findings

Existence of autoequivalence with positive categorical entropy on bielliptic surfaces

Counterexample to Gromov-Yomdin type equality on Enriques surfaces

First example of a surface with positive categorical entropy absent of positive topological entropy

Abstract

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.