On entropies of autoequivalences on smooth projective varieties
Entropy for endofunctors on triangulated categories is defined by Dmitrov-Haiden-Katzarkov-Kontsevich. Based on the joint work with Kohei Kikuta, one of my students, I show that the categorical entropy of an automorphism of a complex smooth projective variety is equal to the topological entropy, which is done by DHKK under a certain technical condition.
It is natural to expect a generalization of the fundamental theorem by Gromov-Yomdin; the entropy of an autoeuqivalence on a complex smooth projective variety should be given by the logarithm of the spectral radius of the induced map on the numerical Grothendieck group. I also show that this conjecture holds for elliptic curves (Kikuta's result) and if the canonical or anti-canonical sheaf is ample.