On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity

In topological dynamics, the Gromov–Yomdin theorem states that the topological entropy of a holomorphic automorphism $f$ of a smooth projective variety is equal to the logarithm of the spectral radius of the induced map $f^{\ast }$. In order to establish a categorical analogue of the Gromov–Yomdin theorem, one first needs to find a categorical analogue of a holomorphic automorphism. In this paper, we propose a categorical analogue of a holomorphic automorphism and prove that the Gromov–Yomdin type theorem holds for them.