A Dynamical Lifting Problem For Additive Polynomials

Abstract

We introduce a dynamical analogue of the lifting problem for Galois covers of algebraic curves and find a negative solution for the collection of separable additive and semi-additive polynomials over ${\overline{\mathbb{F}}}_p$. We also explicitly compute the dimension of the space of linear conjugacy classes in ${\mathcal{M}}_{p^m}({\overline{\mathbb{F}}}_p)$ which contain additive or semi-additive polynomials providing explicit counterexamples to Thurston Rigidity in positive characteristic.

arXiv commentV2: reorganized for efficiency and clarity, added a discussion of a second family of wildly ramified maps -- the semi-additive polynomials