Dynamical systems and categories

TL;DR

This paper explores the intersection of dynamical systems and category theory, defining entropy for functors, relating it to classical cases, and analyzing phase density in stability conditions within triangulated categories.

Contribution

It introduces a framework for entropy of endofunctors, connects it to classical dynamical entropy, and characterizes phase density in Bridgeland stability conditions for derived categories.

Findings

Entropy computed for various functors, matching classical entropy for pseudo-Anosov maps.

Complete characterization of phase density in derived categories of quivers.

Construction of stability conditions with dense phases using Kronecker pairs.

Abstract

We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity categories. First, entropy is defined for exact endofunctors and computed in a variety of examples. In particular, the classical entropy of a pseudo-Anosov map is recovered from the induced functor on the Fukaya category. Second, the density of the set of phases of a Bridgeland stability condition is studied and a complete answer is given in the case of bounded derived categories of quivers. Certain exceptional pairs in triangulated categories, which we call Kronecker pairs, are used to construct stability conditions with density of phases. Some open questions and further directions are outlined as well.