Hausdorff dimension and Galois orbits

Abstract

The aim of this paper is to investigate fractals and their Hausdorff dimensions in a non-Archimedean setting. We are interested in fractals like equilibrated and fundamental sets of the Tate fields and, in particular, in fractals like Galois orbits of generic elements of infinite Galois extensions of $p$-adic fields. Also, we study the invariance of the Hausdorff dimension of Galois orbits defined by integral transcendental elements of ${\mathbb{C}}_p$.