On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity

TL;DR

This paper develops a categorical analogue of holomorphic automorphisms and proves a Gromov--Yomdin type theorem in this new categorical setting, linking topological entropy with spectral radius.

Contribution

It introduces a novel categorical framework for holomorphic automorphisms and establishes a Gromov--Yomdin type theorem within this context.

Findings

Categorical analogue of holomorphic automorphisms proposed

Gromov--Yomdin type theorem proven for the categorical automorphisms

Links between entropy and spectral radius established in the categorical setting

Abstract

In topological dynamics, the Gromov--Yomdin theorem states that the topological entropy of a holomorphic automorphism $f$ of a smooth projective variety is equal to the logarithm of the spectral radius of the induced map $f^*$. In order to establish a categorical analogue of the Gromov--Yomdin theorem, one first needs to find a categorical analogue of a holomorphic automorphism. In this paper, we propose a categorical analogue of a holomorphic automorphism and prove that the Gromov--Yomdin type theorem holds for them.