  We study the categorical entropy and counterexamples to Gromov-Yomdin type
conjecture via homological mirror symmetry of K3 surfaces established by
Sheridan-Smith. We introduce asymptotic invariants of quasi-endofunctors of dg
categories, called the Hochschild entropy. It is proved that the categorical
entropy is lower bounded by the Hochschild entropy. Furthermore, motivated by
Thurston's classical result, we prove the existence of a symplectic Torelli
mapping class of positive categorical entropy. We also consider relations to
the Floer-theoretic entropy.
