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

On the categorical entropy and the topological entropy