On entropy for autoequivalences of the derived category of curves

Abstract

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. It is important to evaluate the value of the entropy at zero in relation to the topological entropy. In this paper, we study the entropy at zero of an exact autoequivalence of the derived category of a complex smooth projective curve, and prove that it coincides with the natural logarithm of the spectral radius of the induced automorphism on its numerical Grothendieck group.