One can regard a self-map on a space as a dynamical system, and study its
long-term behavior under larger iterations. In this talk, we will introduce
the categorical version of such dynamical systems, and establish invariants
that one can associate to endofunctors of categories from the dynamical
perspective. We will also explain some results on categorical dynamical
systems that are related to holomorphic dynamics, symplectic dynamics,
Teichmuller theory, and Poincare rotation numbers. No background in
dynamical systems is assumed for this talk.