On entropy for autoequivalences of the derived category of curves

TL;DR

This paper investigates the entropy at zero of autoequivalences in the derived category of smooth projective curves, establishing its equality with the logarithm of the spectral radius of the induced automorphism.

Contribution

It proves that the entropy at zero for autoequivalences of derived categories of curves equals the natural logarithm of the spectral radius of the induced automorphism.

Findings

Entropy at zero matches the log of spectral radius.

Established a link between categorical entropy and automorphism spectral properties.

Provides a precise calculation method for entropy in this context.

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. It is important to evaluate the value of the entropy at zero in relation to the topological entropy. In this paper, we study the entropy at zero of an exact autoequivalence of the derived category of a complex smooth projective curve, and prove that it coincides with the natural logarithm of the spectral radius of the induced automorphism on its numerical Grothendieck group.