Entropies of Serre functors for higher hereditary algebras

Abstract: For a higher hereditary algebra, we calculate its upper (lower) Serre dimension, the entropy and polynomial entropy of Serre functor, and the Hochschild (co)homology entropy of Serre quasi-functor. These invariants are given by its Calabi-Yau dimension for a higher representation-finite algebra, and by its global dimension and the spectral radius and polynomial growth rate of its Coxeter matrix for a higher representation-infinite algebra. For this, we prove the Yomdin type inequality on Hochschild homology entropy for a finite dimensional elementary algebra of finite global dimension. Our calculations imply that the Kikuta and Ouchi's question on relations between entropy and Hochschild (co)homology entropy has positive answer, and the Gromov-Yomdin type equalities on entropy and Hochschild (co)homology entropy hold, for the Serre functor on perfect derived category and the Serre quasi-functor on perfect dg module category of an indecomposable elementary higher hereditary algebra.