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

Given a prime number p let Cp 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 Cp 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 Cp. 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. Mathematics Subject Classification (2020). Primary: 11S99; Secondary: 11S20, 11S80.