Categorical polynomial entropy

Abstract

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

Final version. Published in Advances in Mathematics

References in corpus (12)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Calabi-Yau algebras
• Flat surfaces and stability structures
• Relative dynamical degrees of correspondences over a field of arbitrary characteristic
• Stability conditions and the $A_2$ quiver
• A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines
• On entropy for autoequivalences of the derived category of curves
• On the categorical entropy and the topological entropy
• Some aspects of explicit birational geometry inspired by complex dynamics
• Polynomial growth of volume of balls for zero-entropy geodesic systems
• Dynamical systems and categories
• On the cohomological action of automorphisms of compact Kähler threefolds

Cited by in corpus (5)

• Asymptotic shifting numbers in triangulated categories
• Stability conditions and Teichmüller space
• An upper bound for polynomial volume growth of automorphisms of zero entropy
• Polynomial volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang)
• On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity