A family of wildly ramified dynamical systems

Abstract: Dynamical systems come naturally equipped with an algebraic invariant called the (profinite) iterated monodromy group. In this talk, we introduce a dynamical analogue of the lifting problem for Galois covers, considering lifts of a dynamical system which preserve its iterated monodromy group. We compute the iterated monodromy group of all additive, separable polynomials defined over $\overline{\mathbb{F}}_p$ and explore barriers to the resulting group arising in characteristic zero. We compare the degree $p$ case with a $\mathbb{Z}/p\mathbb{Z}$-lift explicitly constructed by Green and Matignon, and find that no lift which preserves the geometric iterated monodromy group can exist.

number theory

Audience: researchers in the topic

Five College Number Theory Seminar