Families of Dynamical Systems

Abstract

Most of our work in previous chapters has focused on a single rational map φ(z) and the effect its iterates have on different initial values. We now shift focus and consider the effect of varying the rational map φ(z). In order to do this, we study the set of all rational maps. This set turns out to have a natural structure as an algebraic variety, as does the set of rational maps modulo the equivalence relation defined by PGL₂ conjugation. There are many threads to this story. The specific topics that we touch upon in this chapter include:

• 1. Dynatomic polynomials and fields generated by periodic points.

• 2. The space of quadratic polynomials (the simplest nontrivial case).

• 3. Rational maps that are PGL2-equivalent over $\bar K$, but not over K (twists).

• 4. Field of defintion versus field of moduli for rational maps.

• 5. Minimal models for rational maps.

• 6. Moduli spaces of rational maps (with marked periodic points).