On the categorical entropy and the topological entropy

TL;DR

This paper explores the relationship between categorical and topological entropy, proposing a conjecture that generalizes Gromov-Yomdin's theorem and confirming the equality of these entropies for certain geometric transformations.

Contribution

It introduces a conjecture linking categorical and topological entropy and proves their equality for surjective endomorphisms of smooth projective varieties.

Findings

Categorical entropy of surjective endomorphisms equals their topological entropy.

Computed entropy for autoequivalences related to ample canonical or anti-canonical sheaves.

Proposed a natural generalization of Gromov-Yomdin's theorem.

Abstract

To an exact endofunctor of a triangulated category with a split-generator, the notion of entropy is given by Dimitrov-Haiden-Katzarkov-Kontsevich, which is a (possibly negative infinite) real-valued function of a real variable. In this paper, we propose a conjecture which naturally generalizes the theorem by Gromov-Yomdin, and show that the categorical entropy of a surjective endomorphism of a smooth projective variety is equal to its topological entropy. Moreover, we compute the entropy of autoequivalences of the derived category in the case of the ample canonical or anti-canonical sheaf.