Number Theory Seminar: Alon Levy

Berkovich's rigid analytic spaces are path-connected, Hausdorff, locally compact spaces that generalize non-archimedean fields in a way that allows conducting analysis. We use them to prove non-archimedean analogs of results in complex dynamics.

 
It is a classical result that over the complex numbers, whenever a rational function φ has a fixed point that is attracting but not superattracting, that is a fixed point z with 0 < |φ'(z)| < 1, there is a critical point of φ whose orbit is attracted to z. We show that a similar, but not identical, result holds over non-archimedean fields, with applications to both global and local non-archimedean dynamics.