On Categorical Entropy from the viewpoint of Symplectic Topology

TL;DR

This paper investigates categorical entropy within symplectic topology, establishing relations with localization, providing computational methods for Fukaya categories, and exploring duality properties that enable entropy calculation from morphism spaces.

Contribution

It introduces new relations between categorical entropy and localization, offers a method to compute entropy in Fukaya categories, and demonstrates entropy calculation via duality in symplectic manifolds.

Findings

Relation between categorical entropy and localization established

Method for calculating categorical entropy in Fukaya categories provided

Existence of duality in Fukaya categories enabling entropy computation from morphisms

Abstract

In this paper, motivated by symplectic topology, we explore categorical entropy and present two main results. The first result establishes a relation between categorical entropies of functors on a category and its localization. Additionally, it demonstrates analogies between the notions of topological and categorical entropy. This result is then applied to symplectic topology, where we provide a method for calculating the categorical entropy of a functor on a (partially) wrapped Fukaya category, assuming that the functor is induced by a compactly supported symplectic automorphism. For the second main result of the paper, we observe the existence of natural examples of symplectic manifolds whose Fukaya categories satisfy a type of Floer-theoretic duality. Motivated by this observation, we prove that categorical entropy can be computed from the morphism spaces under the assumption of duality. The formula is similar to the result of [DHKK14], which is proven for the case of smooth and proper categories.