# A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines

## TL;DR

This paper investigates the entropy of auto-equivalences in categorical dynamics, establishing a lower bound generally and proving equality for orbifold projective lines, thus advancing understanding of entropy-spectral radius relations.

## Contribution

It provides a general lower bound for the entropy-spectral radius equality and confirms this equality specifically for orbifold projective lines.

## Key findings

- Established a lower bound for entropy in categorical dynamics.
- Proved the entropy equals spectral radius for orbifold projective lines.
- Enhanced understanding of entropy behavior in algebraic geometry contexts.

## Abstract

Entropy of categorical dynamics is defined by Dmitrov-Haiden-Katzarkov-Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov-Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.