A Continued Fractions Theory for the completion of the Puiseux field

In this work, we study a continued fractions theory for the topological completion of the field of Puiseux series. As usual, we prove that any element in the completion can be developed as a unique continued fractions, whose coefficients are polynomials in roots of the variable, and that this approximation is the best ''rational'' Diophantine approximation of such element. Then, we interpret the preceding result in terms of the action of a suitable arithmetic subgroup of the special linear group on the Berkovich space defined over the said completion. 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.