Dynamics Associated to Algebraic Groups

Abstract

In the forest of untamed rational maps live a select few whose additional structure allows them to be more easily domesticated. They are the power maps, Chebyshev polynomials, and Lattes maps, whose complex dynamics were briefly discussed in Section 1.6. The underlying structure that they possess comes from an algebraic group, namely the multiplicative group for the power maps and Chebyshev polynomials and elliptic curves for the Lattes maps. Although such maps are special in many ways, they yet provide important examples, testing grounds, and boundary conditions for general results in dynamics. In this chapter we investigate some of the algebraic and arithmetic properties of the rational maps associated to algebraic groups.