Categorical dynamics on stable module categories

Abstract

Let $ A $ be a finite connected graded cocommutative Hopf algebra over a field $ k $. There is an endofunctor $ \mathsf{tw} $ on the stable module category $ \mathrm{StMod}_A $ of $ A $ which twists the grading by $ 1 $. We show the categorical entropy of $ \mathsf{tw} $ is zero. We provide a lower bound for the categorical polynomial entropy of $ \mathsf{tw} $ in terms of the Krull dimension of the cohomology of $ A $, and an upper bound in terms of the existence of finite resolutions of $ A $-modules of a particular form. We employ these tools to compute the categorical polynomial entropy of the twist functor for examples of finite graded Hopf algebras over $ \mathbb{F}_2 $.