This monograph by a prolific author adds one significant entry to the growing list of publications dealing with the arithmetical/algebraic side of dynamical systems. It can be viewed as an extension of the volume [The arithmetic of dynamical systems. New York, NY: Springer (2007; Zbl 1130.37001)], by the same author. The idea of the so-called moduli spaces is developed, which is concerned with the study of the parameter space in parametrised families of algebraic dynamical systems. Often these families have the structure of a manifold, or an algebraic variety, in which an equivalence relation is introduced through the action of some group. A moduli space is the corresponding quotient space.
The maps under study are rational maps of the \(n\)-dimensional projective space over an arbitrary field. Equivalence is conjugacy by automorphisms. Some topics (dynatomic polynomials and their modular curves, theory of heights) are also covered in [loc. cit.], but many are new, such as the construction of dynamical moduli spaces via geometric invariant theory (part of which is reviewed), and a study of post-critically finite maps (all critical points have finite orbit) in moduli space. The exposition is succinct. The reader should be prepared to consult other sources (particularly [loc. cit.]) for the necessary background.