论文标题
分类多项式熵
Categorical polynomial entropy
论文作者
论文摘要
对于经典的动力学系统，多项式熵是拓扑熵的精致不变。在分类动力学系统的设置中，即带有内型函数的三角类别的类别，我们发展了分类多项式熵的理论，完善了由Dimitrov-Haiden-Katzarkarkov-Kontsevich定义的分类熵。我们通过表明平滑投射品种的自动形态（在派生类别上的分类多项式熵）与诱导诱导的共同体作用的多项式生长速率相吻合。我们一般还建立了Yomdin型的下限，以根据该类别的数值Grothendieck组的诱导内态性内态性质的分类多项式熵。 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.多项式质量生长速率的平行理论是在存在Bridgeland稳定性条件下发展的。
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.
