Categorical dynamics on stable module categories

Topological entropy is a measure of the complexity of a continuous self-map of a compact topological space. Categorical entropy generalizes this measure to exact endofunctors of triangulated categories. In this work we ask: How can categorical entropy serve to quantify growth in stable homotopy theory, and how can the methods of stable homotopy theory aid in understanding and computing categorical entropy? We show that the categorical polynomial entropy of a twist functor on the stable module category of a finite connected graded Hopf algebra A over a field k recovers one less than the Krull dimension of H*(A;k), generalizing a computation of Fan, Fu, and Ouchi. Along the way, we will see how a stable homotopy theoretic-perspective permits us to make this refinement.