Continued Fractions and the modular ray on a Berkovich space

In this work, we study a continued fractions theory for the topological completion of the field of Puiseux series. As in the classical case, any element in the completion can be uniquely written as a continued fractions, and this approximation is optimal. In this work, we interpret the preceding results in terms of the action of a suitable arithmetic subgroup of the special linear group on the Berkovich space defined over said completion. The quotient space plays a significant role in such description. We also explore the connections between points of type IV of the Berkovich space in terms of some “non-convergent” or “undefined” continued fractions, in a sense that we make precise in the text.