Asymptotic shifting numbers in triangulated categories

We study invariants, called shifting numbers, that measure the asymptotic amount by which an autoequivalence of a triangu-lated category translates inside the category. The invariants are analogous to Poincare translation numbers that are widely used in dynamical systems. We additionally establish that in some examples the shifting numbers provide a quasimorphism on the group of autoequivalences. Additionally, the shifting numbers are related to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich, as well as the phase functions of Bridgeland stability conditions.& COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

This article studies invariants called shifting numbers to measure the asymptotic translation of an autoequivalence inside the triangulated category. These invariants are similar to Poincare translation numbers widely used in dynamical systems. Furthermore, in some examples, the shifting numbers are shown to provide a quasimorphism on the group of autoequivalences. Additionally, the shifting numbers are related to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich, as well as the phase functions of Bridgeland stability conditions.