On pseudo-Anosov autoequivalences

TL;DR

This paper introduces a new framework for understanding autoequivalences of triangulated categories through filtrations derived from stability conditions, proposes a broader definition of pseudo-Anosov autoequivalences, and provides new examples and dynamical properties.

Contribution

It generalizes the concept of pseudo-Anosov autoequivalences, constructs new examples in Calabi-Yau categories, and links their action to hyperbolic dynamics on stability spaces.

Findings

Filtration of categories based on autoequivalence growth rates

A broader, more inclusive definition of pseudo-Anosov autoequivalences

Pseudo-Anosov autoequivalences act hyperbolically on stability condition spaces

Abstract

Motivated by results of Thurston, we prove that any autoequivalence of a triangulated category induces a filtration by triangulated subcategories, provided the existence of Bridgeland stability conditions. The filtration is given by the exponential growth rate of masses under iterates of the autoequivalence, and only depends on the choice of a connected component of the stability manifold. We then propose a new definition of pseudo-Anosov autoequivalences, and prove that our definition is more general than the one previously proposed by Dimitrov, Haiden, Katzarkov, and Kontsevich. We construct new examples of pseudo-Anosov autoequivalences on the derived categories of quintic Calabi-Yau threefolds and quiver Calabi-Yau categories. Finally, we prove that certain pseudo-Anosov autoequivalences on quiver 3-Calabi-Yau categories act hyperbolically on the space of Bridgeland stability conditions.