  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.
