Continued fractions for hyperquadratic power series over a finite field

An irrational power series over a finite field F_q of characteristic p is called hyperquadratic if it satisfies an algebraic equation x=f(x^r) ,where r is a power of p and f is a linear fractional transformation with polynomial coefficients in F_q[T]. These algebraic power series are analogues of quadratic real numbers. This analogy makes their continued fraction expansions specific