 By applying the theory of linear fractional transformations to

where "A""k"-1, "B""k"-1, "A""k", and "B""k" are the numerators and denominators of the "k"-1st and "k"th convergents of the infinite periodic continued fraction "x", it can be shown that "x" converges to one of the fixed points of "s"("w") if it converges at all.Specifically, let "r"1 and "r"2 be the roots of the quadratic equation

These roots are the fixed points of "s"("w").If "r"1 and "r"2 are finite then the infinite periodic continued fraction "x" converges if and only if

If the denominator "B""k"-1 is equal to zero then an infinite number of the denominators "B""nk"-1 also vanish, and the continued fraction does not converge to a finite value.