at infinity which is mapped into an arbitrarily small neighborhood of . So if the continued fraction converges the transformation maps both very small and very large into an arbitrarily small neighborhood of , the value of the continued fraction, as gets larger and larger. For intermediate values of , since the successive convergents are getting closer together we must have where is a constant, introduced for convenience. But then, by substituting in the expression for we obtain so that even the intermediate values of (except when ) are mapped into an arbitrarily small neighborhood of , the value of the continued fraction, as gets larger and larger. Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point. Notice that the sequence lies within the automorphism group of the extended complex plane, since each is a linear fractional transformation for which . And every member of that automorphism group maps the extended complex plane into itself: not one of the can possibly map the plane into a single point. Yet in the limit the sequence defines an infinite continued fraction which (if it converges) represents a single point in the complex plane. When an infinite continued fraction converges, the corresponding sequence of LFTs "focuses" the plane in the direction of , the value of the continued fraction. At each stage of the process a larger and larger region of the plane is mapped into a neighborhood of , and the smaller and smaller region of the plane that's left over is stretched out ever more thinly to cover everything outside that neighborhood. For divergent continued fractions, we can distinguish three cases: The two sequences and might themselves define two convergent continued fractions that have two different values, and . In this case the continued fraction defined by the sequence diverges by oscillation between two distinct limit points. And in fact this idea can be generalized: sequences can be constructed that oscillate among three, or four, or indeed any number of limit points. Interesting instances of this case arise when the sequence constitutes a subgroup of finite order within the group of automorphisms over the extended complex plane. The sequence may produce an infinite number of zero denominators while also producing a subsequence of finite convergents. These finite convergents may not repeat themselves or fall into a recognizable oscillating pattern. Or