Mass growth of objects and categorical entropy

TL;DR

This paper explores the relationship between categorical entropy and mass growth in triangulated categories, establishing fundamental properties and conditions under which they coincide, thus linking dynamical systems and categorical stability.

Contribution

It clarifies the connection between categorical entropy and mass growth, providing fundamental properties and conditions for their equivalence in triangulated categories.

Findings

Mass growth rate properties established

Conditions for entropy and mass growth to coincide identified

Enhanced understanding of categorical dynamics achieved

Abstract

In the pioneer work by Dimitrov-Haiden-Katzarkov-Kontsevich, they introduced various categorical analogies from classical theory of dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. In the connection between categorical theory and classical theory, a stability condition on a triangulated category plays the role of a measured foliation so that one can measure the "volume" of objects, called the mass, via the stability condition. The aim of this paper is to establish fundamental properties of the growth rate of mass of objects under the mapping by the endofunctor and to clarify the relationship between the entropy and that. We also show that they coincide under a certain condition.