Galois equivariant functions on Galois orbits in large $p$-adic fields

Abstract

Given a prime number $p$ let ${\mathbb{C}}_p$ be the topological completion of the algebraic closure of the field of $p$-adic numbers. Let $O(T)$ be the Galois orbit of a transcendental element $T$ of ${\mathbb{C}}_p$ with respect to the absolute Galois group. Our aim is to study the class of Galois equivariant functions defined on $O(T)$ with values in ${\mathbb{C}}_p$. We show that each function from this class is continuous and we characterize the class of Lipschitz functions, respectively the class of differentiable functions, with respect to a new orthonormal basis. Then we discuss some aspects related to analytic continuation for the functions of this class.