This thesis consists of four chapters on the author's research in algebraic dynamics. The first chapter introduces the main results. The second chapter is an expository introduction to algebraic dynamics, moduli spaces, and integrability. The third chapter is a detailed study of the algebraic dynamics of the pentagram map, a discrete integrable system defined on moduli spaces of polygons, over algebraically closed fields. The fourth chapter constructs a moduli space of linear maps on projective space with marked points via geometric invariant theory.
