Rational Maps on Elliptic Surfaces

We will discuss rational self-maps on surfaces which respect (but do not necessarily fix) an elliptic fibration on the surface i.e. fibres are mapped to fibres. Such elliptic surfaces arguably form the largest swathe of the classification of surfaces, and these maps, elliptic skew products, likewise form a significant class of rational maps. However, little has been said about their dynamics. First I will show that these maps fall into two clean categories. Intriguing algebraic and arithmetic questions follow, which I will share and answer to a greater extent. Using a mix of algebraic, geometric, arithmetic, and dynamical arguments we will work toward a descriptive classification. Joint work in progress with Giacomo Mezzedimi (Bonn).

Go to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for more information