Transition Semi–groups on a Local Field Induced by Galois Group and their Representation

Abstract

For a given map $\phi: \mathbb{Q}_{p} \longrightarrow \mathbb{Q}_{p}$ defined on the field $\mathbb{Q}_{p}$ of p-adic numbers satisfying

$$ \parallel x - y\parallel_p \leqslant p^r \Rightarrow \parallel \phi (x) - \phi (y)\parallel_p \leqslant p^r ,\quad \forall x,y \in \mathbb{Q}_{p}, $$

for some integer r, a Markov process on $\mathbb{Q}_{p}$ induced by the map ϕ is constructed in (Kaneko and Zhao (1994) Forum Math. J. 16, 69). This approach can still be our choice in constructing a Markov process on finite algebraic extension of $\mathbb{Q}_{p}$. We will give an answer to the question as to how Markov process driven by set of maps will be addressed. Especially, we will focus on case the maps are given by the elements of Galois group of the extension.