Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields

Let F-q be a finite field of order a positive power q of a prime number. We study the nonarchimedean quadratic Lagrange spectrum defined by Parkkonen and Paulin by considering the approximation by elements of the orbit of a given quadratic power series in F-q ((Y-1)), for the action by homographies and anti-homographies of PGL(2)(F-q[Y]) on F-q((Y-1)) boolean OR {infinity}. While Parkkonen and Paulin's approach used geometric methods of group actions on Bruhat-Tits trees, ours is based on the theory of continued fractions in power series fields.