期刊
ADVANCES IN MATHEMATICS
卷 383, 期 -, 页码 -出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.aim.2021.107655
关键词
Dynamical systems; Triangulated categories; Categorical entropy; Topological entropy; Polynomial entropy; Automorphisms; Stability conditions
类别
资金
- Radboud Excellence Initiative program of Radboud University
- Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) in RIKEN
- JSPS KAKENHI [19K14520]
- Grants-in-Aid for Scientific Research [19K14520] Funding Source: KAKEN
The study introduces the theory of categorical polynomial entropy, refining the categorical entropy defined by previous scholars. By computing the categorical polynomial entropy for standard functors, the research illustrates how this concept refines the study of categorical entropy and enables the investigation of the categorical trichotomy phenomenon.
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, Katzar-kov, and 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. (c) 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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