Entropy of Monomial Algebras and Derived Categories

Let A be a finitely presented associative monomial algebra. We study the category qgr(A), which is a quotient of the category of graded finitely presented A-modules by the finite-dimensional ones. As this category plays a role of the category of coherent sheaves on the corresponding noncommutative variety, we consider its bounded derived category D-b (qgr(A)). We calculate the categorical entropy of the Serre twist functor on D-b (qgr(A)) and show that it is equal to the (natural) logarithm of the entropy of the algebra A itself. Moreover, we relate these two kinds of entropy with the topological entropy of the Ufnarovski graph of A and the entropy of the path algebra of the graph. If A is a path algebra of some quiver, the categorical entropy is equal to the logarithm of the spectral radius of the quiver's adjacency matrix.