Hochschild entropy and Categorical entropy

TL;DR

This paper introduces Hochschild entropy as an invariant of dg categories, establishes its lower bound relation to categorical entropy, and explores examples of positive entropy in symplectic mapping classes, connecting homological mirror symmetry and Floer theory.

Contribution

It defines Hochschild entropy, proves it bounds categorical entropy from below, and demonstrates the existence of symplectic mapping classes with positive entropy.

Findings

Hochschild entropy is a lower bound for categorical entropy.

Existence of symplectic Torelli mapping class with positive categorical entropy.

Connections established between categorical, Hochschild, and Floer-theoretic entropies.

Abstract

We study the categorical entropy and counterexamples to Gromov-Yomdin type conjecture via homological mirror symmetry of K3 surfaces established by Sheridan-Smith. We introduce asymptotic invariants of quasi-endofunctors of dg categories, called the Hochschild entropy. It is proved that the categorical entropy is lower bounded by the Hochschild entropy. Furthermore, motivated by Thurston's classical result, we prove the existence of a symplectic Torelli mapping class of positive categorical entropy. We also consider relations to the Floer-theoretic entropy.