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Let   $A$   be a finitely presented associative monomial algebra. We study the category   $\textsf{qgr}(A)$  , 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   $\textbf{D}^b(\textsf{qgr}(A))$  . We calculate the categorical entropy of the Serre twist functor on   $\textbf{D}^b(\textsf{qgr}(A))$   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.

Entropy of Monomial Algebras and Derived Categories