  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.
