are the fixed points of . If the discriminant is zero the LFT fixes a single point; otherwise it has two fixed points. If the LFT is an invertible conformal mapping of the extended complex plane onto itself. In other words, this LFT has an inverse function such that for every point in the extended complex plane, and both and preserve angles and shapes at vanishingly small scales. From the form of we see that is also an LFT. The composition of two different LFTs for which is itself an LFT for which . In other words, the set of all LFTs for which is closed under composition of functions. The collection of all such LFTs, together with the "group operation" composition of functions, is known as the automorphism group of the extended complex plane. If the LFT reduces to which is a very simple meromorphic function of with one simple pole (at ) and a residue equal to . (See also Laurent series.) The continued fraction as a composition of LFTs Consider a sequence of simple linear fractional transformations Here we use to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbol to represent the composition of transformations ; that is, and so forth. By direct substitution from the first set of expressions into the second we see that and, in general, where the last partial denominator in the finite continued fraction is understood to be . And, since , the image of the point under the iterated LFT is indeed the value of the finite continued fraction with partial numerators: A geometric interpretation Defining a finite continued fraction as the image of a point under the iterated linear functional transformation leads to an intuitively appealing geometric interpretation of infinite continued fractions. The relationship can be understood by rewriting and in terms of the fundamental recurrence formulas: In the first of these equations the ratio tends toward as tends toward zero. In the second, the ratio tends toward as tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents are eventually arbitrarily close together. Since the linear fractional transformation is a continuous mapping, there must be a neighborhood of that is mapped into an arbitrarily small neighborhood of . Similarly, there must be a neighborhood of the point