Categorical polynomial entropy

For classical dynamical systems, the polynomial entropy serves as a refined invariant of topological entropy. In setting categorical that is, triangulated categories endowed with an endofunctor, we develop theory entropy, refining defined by Dimitrov-Haiden-Katzarkov-Kontsevich. We justify this notion showing for automorphism smooth projective variety, pullback functor on derived category coincides growth rate induced action cohomology. also establish in general Yomdin-type lower bound endofunctor terms endomorphism numerical Grothendieck group category. As examples, compute some standard functors like shifts, Serre functors, tensoring line bundles, automorphisms, spherical twists, P-twists, and so on, illustrating clearly how refines study enables us to phenomenon trichotomy. A parallel mass is developed presence Bridgeland stability conditions.