On the Categorical Entropy and the Topological Entropy

To an exact endofunctor of a triangulated category with a split generator, the notion of entropy is given by Dimitrov-Haiden-Katzarkov-Kontsevich, which is a (possibly negative infinite) real-valued function of a real variable. In this article, we propose a conjecture which naturally generalizes the theorem by Gromov-Yomdin, and show that the categorical entropy of a surjective endomorphism of a smooth projective variety is equal to its topological entropy. Moreover, we compute the entropy of autoequivalences of the derived category in the case of the ample canonical or anti-canonical sheaf.