Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 315–327

THE HYPERBOLIC GEOMETRY OF CONTINUED FRACTIONS K(1 | b_{n})

Ian Short

4, Park Close, Harrow HA3 6EX, United Kingdom; ims25@cam.ac.uk

Abstract. The Stern–Stolz theorem states that if the infinite series ∑ |b_{n}| converges, then the continued fraction K(1 | b_{n}) diverges. On p. 33 of [7], H. S. Wall asks whether just convergence, rather than absolute convergence of ∑ b_{n} is sufficient for the divergence of K(1 | b_{n}). We investigate the relationship between ∑ |b_{n}| and K(1 | b_{n}) with hyperbolic geometry and use this geometry to construct a sequence b_{n} of real numbers for which both ∑ b_{n} and K(1 | b_{n}) converge, thereby answering Wall’s question.

1. Introduction

An infinite complex continued fraction is a formal expression

\frac{a_{1}}{b_{1} + \frac{a_{2}}{b_{2} + \frac{a_{3}}{b_{3} + ⋯}}} ,

where the a_{i} and b_{j} are complex numbers and no a_{i} is equal to 0. This continued fraction will be denoted by K(a_{n} | b_{n}). We define Möbius transformations t_{n}(z) = a_{n}/(b_{n} + z), for n = 1, 2, …, and let T_{n} = t_{1} ∘ ⋯ ∘ t_{n}. The continued fraction is said to converge classically if the sequence T_{1}(0), T_{2}(0), … converges, else it is said to diverge classically. Each of the Möbius transformations T_{n} acts on the extended complex plane C_{∞} and, through identifying C_{∞} with the boundary {(x, y, t) : t = 0} ∪ {∞} of three-dimensional hyperbolic space H^{3} = {(x, y, t) : t > 0}, the T_{n} may also be considered to act on H^{3} (see Section 2 for details). In this paper we restrict to continued fractions K(1 | b_{n}) for which every a_{n} = 1, and examine the relationship between divergence of the series ∑ |b_{n}|, classical convergence of T_{n} and convergence of T_{n} within H^{3} . Throughout the rest of this exposition, t_{n}(z) = 1/(b_{n} + z) and T_{n} = t_{1} ∘ ⋯ ∘ t_{n}, although the ∘ symbol will in future be omitted from all functional compositions.

The chordal metric σ of C_{∞} is defined for distinct points z and w in C by

σ(z, w) = \frac{2|z − w|}{\sqrt{1 + |z|^{2}} \sqrt{1 + |w|^{2}}} , σ(z, ∞) = \frac{2}{\sqrt{1 + |z|^{2}}} .

2000 Mathematics Subject Classification: Primary 40A15, 30B70; Secondary 30F45, 11J70.

The author wishes to acknowledge the contribution of A. F. Beardon to this paper. Many of the ideas present here were conceived as a result of our informal discussions.

This definition is also valid for points z and w in H^{3} . Let

σ_{0}(f, g) = sup_{z∈C_{∞}} σ(f(z), g(z)),

for Möbius transformations f and g. This is the metric of uniform convergence on C_{∞}, with respect to σ.

In Section 3, we completely classify the behaviour of T_{n} when ∑ |b_{n}| converges. This section consists of joint work with A. F. Beardon. The following general theorem is the main result of Section 3.

Theorem 1.1. Suppose that the series ∑_{n=1}^{∞} σ_{0}(s_{n}, s) converges, where s, s_{1}, s_{2}, … are Möbius transformations such that s^{q} is the identity map for some q ∈ N. Then there is another Möbius map f such that s_{1}s_{2} ⋯ s_{nq+r} → fs^{r} as n → ∞, where 0 ⩽ r < q and the convergence is uniform with respect to the chordal metric.

We can apply Theorem 1.1 to gain information about continued fractions K(1 | b_{n}). Let s_{n}(z) = t_{n}(z) = 1/(b_{n} + z) and s(z) = ι(z) = 1/z. The map ι is an isometry in the chordal metric. Therefore

σ(s_{n}(z), s(z)) = σ(b_{n} + z, z) ≤ 2|b_{n}|.

Thus if the series ∑ |b_{n}| converges then Theorem 1.1 may be applied with q = 2 to show that T_{2n} → f for some Möbius transformation f, and T_{2n+1} → fι. Thus T_{n} converges only at the fixed points of ι. The behaviour of T_{n} is thereby fully understood in the instance when ∑ |b_{n}| < ∞.

One important aspect of Theorem 1.1 is that it contains a proof of the following well-known classical theorem.

The Stern–Stolz Theorem. Let b_{1}, b_{2}, … be complex numbers. If the infinite series ∑_{n=1}^{∞} |b_{n}| converges then the continued fraction K(1 | b_{n}) diverges classically.

Using the notation of Theorem 1.1, T_{2n}(0) → f(0) and T_{2n+1}(0) → f(∞) ≠ f(0), therefore T_{n}(0) does not converge, so that the Stern–Stolz theorem is proven. H. S. Wall asks on p. 33 of [7], ‘whether or not the simple convergence of the series ∑ b_{p} is sufficient for the divergence of the continued fraction’. In Section 4, we answer this question by providing an example, motivated by hyperbolic geometry, of a sequence of real numbers b_{1}, b_{2}, … for which both ∑ b_{n} and K(1 | b_{n}) converge. The reason Wall’s simple question has gone so long unanswered is not because the algebra necessary to produce an example is difficult, rather it is because without the guidance provided by hyperbolic geometry, it is not clear how one should proceed in constructing such an example.

The hyperbolic geometry of continued fractions K(1 | b_{n}) 317

Whilst the example of Section 4 demonstrates that the converse to the Stern– Stolz theorem does not hold even for real numbers b_{n}, the Seidel–Stern theorem says that the converse does hold when the b_{n} are all positive numbers. In Section 5 we investigate the Seidel–Stern theorem in terms of hyperbolic geometry. The geometry uncovers more than the classical algebraic analyses and we prove the next more general theorem (in which j = (0, 0, 1), H = {(x, 0, t) : t > 0} and the usual notation t_{n}(x) = 1/(b_{n} + x) and T_{n} = t_{1} ⋯ t_{n} are assumed).

Theorem 1.2. Let b_{1}, b_{2}, … be non-negative real numbers. The sequence T_{1}(j), T_{2}(j), … converges in the chordal metric to a value ζ in \overline{H} (closure taken in R_{∞}^{3} ). Moreover, the following are equivalent,

(i) ∑_{n=1}^{∞} b_{n} converges; (ii) ζ ∈ H; (iii) K(1 | b_{n}) diverges.

If ζ ∈ ∂H and T_{n}(j) → ζ as n → ∞, the Möbius sequence T_{n} is said to converge generally to ζ (see [2] for a discussion of general convergence in a geometric context). Preservation of hyperbolic distance ensures that the choice of the point j ∈ H in the definition of general convergence to ζ is not important.

Our proof of Theorem 1.2 is not necessarily shorter or simpler than existing proofs of the Seidel–Stern theorem, but the hyperbolic geometry provides us with a deeper understanding of the sequence T_{n} and its orbits, whereas the algebra conceals the general picture. Theorem 1.2 is an improvement on the Seidel–Stern theorem in that we also derive conclusions about the orbit of j under T_{n}. Moreover, the geometric techniques we utilise are not dependent on remaining in two dimensions; thus our result can easily be generalised to many dimensions. Note also that we only require the b_{n} not to be negative; they need not necessarily be positive.

We conclude this introduction by briefly discussing the role of the hyperbolic plane H (a more detailed treatment is supplied in Section 2). Real Möbius maps f(x) = (ax + b)/(cx + d) fix the upper half-plane H = {z ∈ C : Im[z] > 0} if and only if ad − bc > 0 (if f does not fix H, it interchanges H with {z : Im[z] < 0}). We are working with Möbius maps of the form t(x) = 1/(x + b), which do not satisfy this criterion, hence we cannot exploit the two-dimensional hyperbolic geometry of H. To circumvent this problem, we extend the action of the real Möbius map t, not to C_{∞}, but to a half-plane H = {(x, 0, t) : t > 0} in R_{∞}^{3} that is perpendicular to C. With this alternative two-dimensional action, the half-plane H is fixed, because the transformation ι(x) = 1/x fixes j, and it also fixes H. Since t is a Möbius map, the action on H is isometric with respect to the hyperbolic metric on H. In fact, the action of t can be extended to the entire plane Π ∪ {∞} that contains H, where Π = {(x, 0, t) : x, t ∈ R}. We write points (x, 0, t) of R in the form x + tj to correspond with the usual real and imaginary part notation for elements of C. This plane Π may be viewed as another model of the complex plane, and we manipulate it accordingly.

2. Möbius transformations and hyperbolic geometry

We quickly describe the geometric machinery and notation necessary to implement our methods. Rigorous details can be found in either [1] or [6].

The complex plane may be embedded as the t = 0 plane of R^{3} . Through this embedding, known as the Poincaré extension, the Möbius transformation group ℳ acts on both the extended complex plane C_{∞} and three-dimensional upper half-space, H^{3} = {(x, y, t) : t > 0}. The Möbius action on the latter space is isometric with respect to the hyperbolic metric ϱ_{H^{3}} of H^{3} . We denote the hyperbolic metric of any suitable space D by ϱ_{D}.

Let ϕ: R^{3} → R^{3} be inversion in the sphere centred on j = (0, 0, 1) of radius \sqrt{2} . This inversion maps C_{∞} to the unit sphere S^{2} and maps S^{2} to C_{∞}. Furthermore, ϕ is its own inverse and satisfies ϕ(0) = −j and ϕ(∞) = j. If f is a Möbius map fixing C_{∞}, then f^{∗} = ϕfϕ is a Möbius map fixing S^{2} . Likewise we use the notation 0^{∗} = ϕ(0), and more generally z^{∗} = ϕ(z), to denote images of points under ϕ (this notation is only used at the end of Section 4). The Euclidean metric restricted to S^{2} can be transferred via ϕ to a metric σ on C_{∞} defined by σ(z, w) := |ϕ(z) − ϕ(w)|. This is the chordal metric introduced in Section 1.

The supremum metric σ_{0} was also defined in the introduction. The space (ℳ , σ) is complete. The metric σ_{0} is right-invariant and satisfies

σ_{0}(hf, hg) ≤ L(h)σ_{0}(f, g),

for a certain positive constant L(h) = exp ϱ_{H^{3}}(j, h(j)), although we do not make use of the exact value of L(h). If f_{n} → f uniformly then,

σ_{0}(f_{n}^{−1}, f^{−1}) = σ_{0}(I, f^{−1}f_{n}) ≤ L(f^{−1})σ_{0}(f, f_{n}),

so that f_{n}^{−1} → f^{−1} uniformly. If also g_{n} → g uniformly then

σ_{0}(f_{n}g_{n}, fg) ≤ σ_{0}(f_{n}g_{n}, fg_{n}) + σ_{0}(fg_{n}, fg) ≤ σ_{0}(f_{n}, f) + L(f)σ_{0}(g_{n}, g),

so that f_{n}g_{n} → fg uniformly.

Let f be a real Möbius transformation (a Möbius transformation with real coefficients) acting on C_{∞}. This map f fixes the extended real axis R_{∞}. Now consider f to act on H^{3} . It fixes the half-plane H = {(x, 0, t) : t > 0} and this action is isometric with respect to the two-dimensional hyperbolic metric on H. Let D = {x + tj : x^{2} + t^{2} < 1} so that ϕ maps H to D. In fact, ϕ is an isometry from (H, ϱ_{H}) to (D, ϱ_{D}), so that f^{∗} = ϕfϕ is an isometry of (D, ϱ_{D}). We now have three distinct actions of f on R_{∞}, H and D, and we make use of all three of them. The hyperbolic metric on H satisfies the formula,

(2.1)  cosh ϱ_{H}(z, w) = 1 + 2 sinh^{2}(\frac{1}{2}ϱ_{H}(z, w)) = 1 + \frac{|z − w|^{2}}{2(z · j)(w · j)}.

This formula can be found in [1], as can the following lemma in hyperbolic geometry.

Lemma 2.1. Choose two points a and b on the boundary of the unit disc D, separated by an angle θ ⩽ π. Denote the hyperbolic line joining a and b by δ. Then cosh ϱ_{D}(0, δ) sin(θ/2) = 1.

Notice that |a − b| = 2 sin(θ/2).

3. Continued fractions K(1 | b_{n}) for which ∑_{n=1}^{∞} |b_{n}| < ∞

The crux of Theorem 1.1 can be found in [2, Theorem 3.7] in which A. F. Beardon credits an earlier source [5]. The identity map is denoted by I.

Theorem 3.7 from [2]. Let u_{n} be a sequence of Möbius transformations for which ∑ σ_{0}(u_{n}, I) converges. Then u_{1} ⋯ u_{n} converges uniformly on C_{∞} to a Möbius transformation.

The proof of this theorem from [2] is sufficiently short that we reproduce it here.

Proof of Theorem 3.7 from [2]. Right-invariance of σ_{0} ensures that for m < n,

σ_{0}(u_{n}^{−1} ⋯ u_{1}^{−1}, u_{m}^{−1} ⋯ u_{1}^{−1}) = σ_{0}(u_{n}^{−1} ⋯ u_{m+1}^{−1}, I).

This latter distance is equal to or less than

σ_{0}(u_{n}^{−1} ⋯ u_{m+1}^{−1}, u_{n−1}^{−1} ⋯ u_{m+1}^{−1})
+ σ_{0}(u_{n−1}^{−1} ⋯ u_{m+1}^{−1}, u_{n−2}^{−1} ⋯ u_{m+1}^{−1})
+ ⋯
+ σ_{0}(u_{m+1}^{−1}, I)

which is equal to

σ_{0}(u_{n}^{−1}, I) + σ_{0}(u_{n−1}^{−1}, I) + ⋯ + σ_{0}(u_{m+1}^{−1}, I).

Since σ_{0}(u_{k}^{−1}, I) = σ_{0}(I, u_{k}), this sum can be forced to be arbitrarily small, provided m and n are restricted to being sufficiently large. Hence the sequence (u_{1} ⋯ u_{n})^{−1} = u_{n}^{−1} ⋯ u_{1}^{−1} is Cauchy with respect to the σ_{0} metric, therefore it converges and hence so does u_{1} ⋯ u_{n}.

We are now in a position to prove Theorem 1.1.

Proof of Theorem 1.1. Define u_{n} = s_{(n−1)q+1}s_{(n−1)q+2} ⋯ s_{nq} then, again using right-invariance of σ_{0} ,

σ_{0}(u_{n}, I) = σ_{0}(u_{n}, s^{q})
≤ σ_{0}(u_{n}, ss_{(n−1)q+2} ⋯ s_{nq}) + σ_{0}(ss_{(n−1)q+2} ⋯ s_{nq}, s^{q})
≤ σ_{0}(s_{(n−1)q+1}, s) + L(s)σ_{0}(s_{(n−1)q+2} ⋯ s_{nq}, s^{q−1}).

This argument may be repeated q−2 times with s_{i} ⋯ s_{nq} replacing s_{(n−1)q+1} ⋯ s_{nq} for (n − 1)q + 1 < i < nq, to obtain

σ_{0}(u_{n}, I) ≤ max{1, L(s)^{q−1}}(σ_{0}(s_{(n−1)q+1}, s) + ⋯ + σ_{0}(s_{nq}, s)).

Therefore ∑ σ_{0}(u_{n}, I) converges so that [2, Theorem 3.7] applies to show that s_{1} ⋯ s_{nq} = u_{1} ⋯ u_{n} → f for some Möbius map f. Since s_{n} → s uniformly we also see that s_{1} ⋯ s_{nq+r} → fs^{r}.

It was demonstrated in the introduction that when s_{n}(z) = t_{n}(z) = 1/(b_{n}+z) and ∑ |b_{n}| < ∞, Theorem 1.1 shows that

T_{2n} → f, T_{2n+1} → fι,

where ι(z) = 1/z. Therefore

T_{2n}^{−1} → f^{−1}, T_{2n+1}^{−1} → ιf^{−1}.

In particular, this accounts for the well-known result that

T_{n−1}^{−1}(∞)T_{n}^{−1}(∞) → 1

as n → ∞.

4. Examples to answer Wall’s question

For each n = 1, 2, …, let
(4.1)  b̂_{2n−1} = (−1)^{n}(\sqrt{n} − \sqrt{n − 1} ), b̂_{2n} = \frac{−2(−1)^{n}\sqrt{n}}{n + 1}.

This sequence shows that absolute convergence of the series ∑ b_{n} is strictly necessary in the Stern–Stolz theorem, as for this particular sequence, ∑ b̂_{n} converges and so does K(1 | b̂_{n}). That ∑ b̂_{n} converges is clear. The remainder of this section consists of a proof that K(1 | b_{n}) converges. Our proof uses the hyperbolic geometry of H = {x + tj : t > 0} as the real Möbius transformations t_{n} and T_{n} are isometries of this plane.

Let γ = {tj : t > 0} denote the hyperbolic line in H that joins 0 and ∞. It will be seen that K(1 | b_{n}) converges for any real sequence b_{1}, b_{2}, … that satisfies the two conditions,

(i) b_{1} < 0, b_{2} > 0, b_{3} > 0, b_{4} < 0, b_{5} < 0, b_{6} > 0, b_{7} > 0, …; (ii) ϱ_{H}(T_{n}^{−1}(j), γ) is a positive unbounded increasing sequence.

Our particular sequence b̂_{n} was chosen to satisfy conditions (i) and (ii). That it satisfies (i) is evident. Before proving rigorously that b̂_{n} gives rise to a sequence T_{n} that satisfies condition (ii), we supply a brief geometric explanation of how the b̂_{n} were chosen so that ϱ_{H}(T_{n}^{−1}(j), γ) → ∞.

A horocircle 𝒞 in H is a line or circle in \overline{H} that is tangent to the boundary ∂H of H at one point ζ. As a point z in H approaches ζ along 𝒞 , the hyperbolic distance ϱ_{H}(z, δ) → ∞, where δ is any hyperbolic line with one end point at ζ. We define two particular horocircles in H. The first is a Euclidean line l = {x + j : x ∈ R} (tangent to ∂H at ∞) and the second is a Euclidean circle C = \{x+tj : |x+(t− \frac{1}{2})j| = \frac{1}{2}\} (tangent to ∂H at 0). The inversion ι(x) = 1/x maps l to C and C to l. We choose b̂_{n} so that T_{n}^{−1}(j) remains on l ∪ C. Notice that t_{n}^{−1}(z) = −b̂_{n} + 1/z and T_{n}^{−1}(j) = t_{n}^{−1}T_{n−1}^{−1}(j). Thus if T_{n−1}^{−1}(j) ∈ l then ι(T_{n−1}^{−1}(j)) ∈ C, and b̂_{n} is chosen to be the unique non-zero real number such that T_{n}^{−1}(j) = −b̂_{n} + ι(T_{n−1}^{−1}(j)) also lies on C. Similarly, if T_{n−1}^{−1}(j) ∈ C then T_{n}^{−1}(j) ∈ l. The b̂_{n} are suitably fashioned so that the backwards orbits T_{n}^{−1}(j) converge to ∞ on l and converge to 0 on C. Since l and C are horocircles and γ is a hyperbolic line connecting 0 and ∞, we see that ϱ_{H}(T_{n}^{−1}(j), γ) → ∞ as n → ∞.

Condition (ii) will now be verified rigorously for b_{n} = b̂_{n}. It is easily proven by induction that for this choice of b_{n},

T_{2n−1}^{−1}(j) = −(−1)^{n}\sqrt{n} + j, T_{2n}^{−1}(j) = \frac{(−1)^{n}\sqrt{n}}{n + 1} + \frac{j}{n + 1}

for every n. The hyperbolic line (Euclidean half-circle) through a point z ∈ H that is symmetric about the j axis and hence orthogonal to γ, intersects γ at |z|j. This is the point in γ of least hyperbolic distance from z, thus ϱ_{H}(z, γ) = ϱ_{H}(z, |z|j). Using equation (2.1), it is a simple matter of algebra to prove that

cosh ϱ_{H}(T_{n}^{−1}(j), γ) = cosh ϱ_{H}(T_{n}^{−1}(j), |T_{n}^{−1}(j)|j) = \sqrt{n + 1} ,

for every n. Condition (ii) has thereby been established for b̂_{1}, b̂_{2}, ….

It remains to demonstrate that K(1 | b_{n}) converges for any sequence b_{1}, b_{2}, … satisfying conditions (i) and (ii). A preliminary lemma is required.

Lemma 4.1. Let T_{1}, T_{2}, … be a sequence of Möbius transformations arising from a continued fraction K(1 | b_{n}) that satisfies the above condition (i). The set R_{∞}\{T_{n−1}(0), T_{n}(0)} consists of two open components. The points T_{n−2}(0) and T_{n+1}(0) each lie in one of these open components, and they lie in the same component if and only if b_{n} and b_{n+1} differ in sign.

Proof. Observe that t_{n}(∞) = 0, t_{n}^{−1}(∞) = −b_{n} and t_{n+1}(0) = 1/b_{n+1} from which it follows that

(a) T_{n}(∞) = T_{n−1}(0); (b) T_{n}(−b_{n}) = T_{n−2}(0) and T_{n}(1/b_{n+1}) = T_{n+1}(0).

Equation (a) shows that the open components separated by T_{n−1}(0) and T_{n}(0) are T_{n}(R^{+}) and T_{n}(R^{−}). Equations (b) show that T_{n−2}(0) and T_{n+1}(0) both lie in a component (since the b_{i} are non-zero) and that they lie in the same component if and only if b_{n} and b_{n+1} differ in sign.

For the remainder of this section, we make frequent use of the unit disc model of hyperbolic space described at the end of Section 2. Let θ_{n} ⩽ π be the acute angle between T_{n−1}^{∗}(0^{∗}) and T_{n}^{∗}(0^{∗}) (where θ_{1} < π is the angle between 0^{∗} = e^{−πj/2} and T_{1}^{∗}(0^{∗}) = (1/b_{1})^{∗}). Using Lemma 2.1 and preservation of hyperbolic distance under T_{n}, we see that,

sin(θ_{n}/2) = 1/ cosh ϱ_{H}(T_{n}^{−1}(j), γ) = 1/ cosh ϱ_{H}(j, T_{n}(γ)),

thus θ_{1}, θ_{2}, … is a sequence in (0, π) that decreases towards 0.

We encapsulate the convergence of T_{n}(0) in a theorem and even obtain an explicit series formula for T_{n}(0).

Theorem 4.2. Let T_{1}, T_{2}, … be a sequence of Möbius maps arising from a continued fraction K(1 | b_{n}) satisfying the above conditions (i) and (ii). Then

ϕ(T_{n}(0)) = T_{n}^{∗}(0^{∗}) = exp[(−\frac{π}{2} + ∑_{k=1}^{n} ε_{k}θ_{k})j],

where θ_{n} < π is the acute angle between T_{n−1}^{∗}(0^{∗}) ∈ ∂D and T_{n}^{∗}(0^{∗}) ∈ ∂D, and

ε_{k} = \{ \begin{matrix}−1 & if k = 1, 2 (mod 4), \\ 1 & if k = 3, 4 (mod 4). \end{matrix}

Hence T_{1}(0), T_{2}(0), … converges.

Proof. Let z_{0} = −j (where −j = e^{−jπ/2} = 0^{∗}) and z_{n} = T_{n}^{∗}(0^{∗}), for n ⩾ 1. Since all z_{n} lie on ∂D and the angle between z_{n−1} and z_{n} is θ_{n} < π, there is a unique sequence η_{1}, η_{2}, η_{3}, …, where η_{k} ∈ {−1, 1}, such that for every n,

z_{n} = exp[(−\frac{π}{2} + ∑_{k=1}^{n} η_{k}θ_{k})j].

Recall that θ_{n} decreases to 0.

We denote the open segment of ∂D of angle θ_{n} < π between z_{n−1} and z_{n} by I_{n} and the other open component of ∂D\{z_{n−1}, z_{n}} by J_{n}. Notice that were z_{n−2} to lie in I_{n}, then I_{n−1} ⊊ I_{n}, which is impossible as θ_{n−1} ⩾ θ_{n}. Therefore z_{n−2} ∈ J_{n}. Lemma 4.1 shows that if n is odd then z_{n+1} ∈ J_{n} (since b_{n} and b_{n+1} differ in sign) and if n is even then z_{n+1} ∈ I_{n} (since b_{n} and b_{n+1} share the same sign).

We have that

I_{n} = {z_{n}e^{−jη_{n}t} : 0 < t < θ_{n}},

z_{n+1} = z_{n}e^{jη_{n+1}θ_{n+1}} and θ_{n+1} ⩽ θ_{n} < π. Therefore the point z_{n+1} lies in I_{n} if and only if η_{n+1} = −η_{n}. Thus

η_{n+1} = \{ \begin{matrix}η_{n} & if n is odd, \\ −η_{n} & if n is even. \end{matrix}

It is easy to check that η_{1} = η_{2} = −1, therefore by induction η_{n} = ε_{n} for every n.

Finally, the series ∑_{k=1}^{∞} ε_{k}θ_{k} must converge, hence T_{1}(0), T_{2}(0), … converges also.

5. The proof of Theorem 1.2

The following observation of A. F. Beardon was used in [2] to prove the Stern– Stolz theorem. The numbers b_{n} may be complex, although we shortly lose this generality after Corollary 5.2.

Lemma 5.1. Suppose that b_{n} → 0 as n → ∞. Given ε > 0 there is an integer N such that if n ⩾ N then

(1 − ε)|b_{n}| ⩽ ϱ_{H^{3}}(j, t_{n}(j)) ⩽ (1 + ε)|b_{n}|.

Proof. Since ι(z) = 1/z is a hyperbolic isometry of H^{3} that fixes j,

sinh^{2}(\frac{1}{2}ϱ_{H^{3}}(j, t_{n}(j))) = sinh^{2}(\frac{1}{2}ϱ_{H^{3}}(j, b_{n} + j)) = \frac{|b_{n}|^{2}}{4} ,

using equation (2.1). The result follows as sinh x and x are asymptotic as x → 0.

Lemma 5.1 has the following corollary, from which the Stern–Stolz theorem is easily derived (see [2] for details).

Corollary 5.2. The two series ∑ |b_{n}| and ∑ ϱ_{H^{3}}(j, t_{n}(j)) either both converge or both diverge.

Observe that ϱ_{H^{3}}(T_{n−1}(j), T_{n}(j)) = ϱ_{H^{3}}(j, t_{n}(j)), therefore if b_{n} → 0, we can use Lemma 5.1 to estimate the hyperbolic distance between the terms T_{n−1}(j) and T_{n}(j). In this section we focus on the significance in terms of hyperbolic geometry of the restraint that each b_{n} is not negative. We shall see that this condition ensures that ϱ_{H}(j, T_{n}(j)) is an increasing sequence (Theorem 5.4). We then show that the limit of this sequence is finite if and only if ∑ ϱ_{H}(j, t_{n}(j)) converges (Theorem 5.5). Only after these theorems do we focus on classical convergence and prove Theorem 5.7, which says that for continued fractions K(1 | b_{n}) with b_{n} ⩾ 0, classical and general convergence are equivalent. Finally, Theorem 1.2 is proven. All results other than those conclusions present in the Seidel–Stern theorem, and all proofs are due to the author (see [4] for an account of the Seidel–Stern theorem).

For the rest of this section we retain the notation t_{n}(z) = 1/(b_{n} +z), but now all the b_{n} are required to be non-negative real numbers. None of the theorems in this section are true when the b_{n} are allowed to be negative. Let L = {x + tj ∈ H : x ≤ 0} and let T_{n}^{−1}(j) = x_{n} + τ_{n}j. We also write ϱ_{n} for ϱ_{H}(j, T_{n}(j)). The following lemma contains the essential hyperbolic geometry behind Theorem 5.4.

Lemma 5.3. Let b ⩾ 0. If z = u + tj ∈ L then z − b ∈ L and

cosh ϱ(z − b, j) − cosh ϱ(z, j) ⩾ \frac{−u}{t}b.

Proof. That z − b ∈ L is clear. Using (2.1),

cosh ϱ(z − b, j) − cosh ϱ(z, j) = \frac{|z − b − j|^{2}}{2t} − \frac{|z − j|^{2}}{2t}

= \frac{b^{2} − 2bu}{2t}

⩾ \frac{−u}{t}b.

Theorem 5.4. Each point T_{n}^{−1}(j) is contained within L and the sequence ϱ_{n} = ϱ_{H}(j, T_{n}(j)) is increasing.

Proof. The first claim is true by induction as certainly j ∈ L, and if x_{n}+τ_{n}j = T_{n}^{−1}(j) ∈ L then T_{n+1}^{−1}(j) = −b_{n+1} + 1/T_{n}^{−1}(j) ∈ L. To prove the second claim, observe that

ϱ_{n} = ϱ_{H}(T_{n}^{−1}(j), j) = ϱ_{H}(ι(T_{n}^{−1}(j)), j)

and

ϱ_{n+1} = ϱ_{H}(T_{n+1}^{−1}(j), j) = ϱ_{H}(−b_{n+1} + ι(T_{n}^{−1}(j)), j),

then apply Lemma 5.3 to see that

(5.1)  cosh ϱ_{n+1} − cosh ϱ_{n} ⩾ \frac{−x_{n}}{τ_{n}} b_{n+1} ⩾ 0,

from which the claim follows.

Either ϱ_{n} → ∞, in which case T_{n}(j) accumulates only on R_{∞}, or ϱ_{n} → k where k < ∞, in which case T_{n}(j) remains within a compact subset of H. The next theorem relates the convergence of T_{n}(j) to convergence of the series ∑ ϱ_{H}(j, t_{n}(j)).

Theorem 5.5. If b_{n} ⩾ 0, then from Theorem 5.4, ϱ_{H}(j, T_{n}(j)) ↗ k, where k ∈ (0, ∞]. The sum ∑ ϱ_{H}(j, t_{n}(j)) converges if and only if k < ∞.

Proof. Suppose that K = ∑ ϱ_{H}(j, t_{n}(j)) is finite. Since T_{n−1} is a ϱ_{H} isometry, ϱ_{H}(j, t_{n}(j)) = ϱ_{H}(T_{n−1}(j), T_{n}(j)). Therefore

ϱ_{H}(j, T_{n}(j)) ≤ ϱ_{H}(j, T_{1}(j)) + ϱ_{H}(T_{1}(j), T_{2}(j)) + ⋯ + ϱ_{H}(T_{n−1}(j), T_{n}(j)),

and this latter sum is equal to or less than K. Now suppose that k < ∞. The points x_{n}+τ_{n}j = T_{n}^{−1}(j) all lie within a compact subset of H so there is a positive constant M such that |x_{n}/τ_{n}| > M for every n. If ϱ_{n} = ϱ_{n−1} then we see from equation (5.1) that b_{n} = 0. If ϱ_{n} ≠ ϱ_{n−1} , apply the mean value theorem with the function cosh and points ϱ_{n−1} and ϱ_{n} to see that there is a value q_{n} between ϱ_{n−1} and ϱ_{n} ≤ k such that

ϱ_{n} − ϱ_{n−1} = \frac{cosh ϱ_{n} − cosh ϱ_{n−1}}{sinh q_{n}} .

Therefore using equation (5.1), whether or whether not ϱ_{n} is equal to ϱ_{n−1} ,

ϱ_{n} − ϱ_{n−1} ⩾ \frac{−x_{n−1}}{τ_{n−1} sinh q_{n}} b_{n} ⩾ \frac{M}{sinh k} b_{n}.

Summing these equations for values of n from 1 to m we obtain,

ϱ_{m} ⩾ \frac{M}{sinh k} ∑_{n=1}^{m} b_{n},

therefore ∑ b_{n} converges. Corollary 5.2 shows that ∑ ϱ_{H}(j, t_{n}(j)) converges.

We must now relate convergence of T_{n}(j) to convergence of T_{n}(0). The following theorem is well known, although the proof is our own.

Theorem 5.6. The sequence T_{1}(0), T_{3}(0), … decreases to β ⩾ 0. The sequence T_{2}(0), T_{4}(0), … increases to α ⩾ 0, where α ⩽ β.

Proof. Let l = [0, ∞] denote the closure of the positive real axis within R_{∞}. Notice that t_{n}(l) ⊆ l for every n. Hence

l ⊇ T_{1}(l) ⊇ T_{2}(l) ⊇ · · · .

Write T_{n}(l) = [α_{n}, β_{n}], then α_{n} increases to a limit α and β_{n} decreases to a limit β, where α ⩽ β. Since α_{n} and β_{n} are the end-points of T_{n}(l), either α_{n} = T_{n}(0) and β_{n} = T_{n}(∞), or β_{n} = T_{n}(0) and α_{n} = T_{n}(∞). Suppose the former situation occurs. Then

T_{n+1}(∞) = T_{n}(0) = α_{n} ⩽ α_{n+1} < β_{n+1},

therefore β_{n+1} = T_{n+1}(0) and α_{n+1} = T_{n+1}(∞). Similarly, if β_{n} = T_{n}(0) and α_{n} = T_{n}(∞) then α_{n+1} = T_{n+1}(0) and β_{n+1} = T_{n+1}(∞). Since β_{1} = T_{1}(0) and α_{1} = T_{1}(∞), the result follows by induction.

If α = β then the continued fraction K(1 | b_{n}) converges classically, otherwise it diverges classically.

Let γ denote the hyperbolic line in H from 0 to ∞. Then T_{n}(γ) is the hyperbolic line joining T_{n}(0) and T_{n}(∞). Lemma 2.1 shows that

σ(T_{n}(0), T_{n}(∞)) = 2/ cosh ϱ_{H}(j, T_{n}(γ)).

Either α = β, in which case T_{n} converges classically to α, or α ≠ β. In the former case, T_{n}(0) → α and T_{n}(∞) = T_{n−1}(0) → α therefore also T_{n}(j) → α, as T_{n}(j) lies on the hyperbolic line T_{n}(γ) joining T_{n}(0) and T_{n}(∞), which is a semi-circle of diminishing radius. In the latter case, σ(T_{n}(0), T_{n}(∞)) decreases towards σ(α, β) > 0. We subsequently show that in this case ϱ_{H}(j, T_{n}(j)) is bounded so that T_{n} does not converge generally. The next theorem encapsulates certain aspects of this information.

Theorem 5.7. The sequence T_{n} converges generally if and only if it converges classically.

Proof. We have seen that classical convergence entails general convergence (this is true of any Möbius sequence associated with a continued fraction). Suppose that T_{n} diverges classically. Theorem 5.6 ensures that the sequence with nth term σ(T_{n}(0), T_{n}(∞)) decreases to a positive constant. Hence cosh ϱ_{H}(T_{n}^{−1}(j), γ) = 2/σ(T_{n}(0), T_{n}(∞)) increases to a positive constant k > 1. Let z_{n} = x_{n} + τ_{n}j = T_{n}^{−1}(j). The closest point on γ to z_{n} is |z_{n}|j, therefore

cosh ϱ_{H}(z_{n}, γ) = 1 + \frac{|z_{n} − |z_{n}|j|^{2}}{2τ_{n}|z_{n}|} = \frac{|z_{n}|}{τ_{n}},

by equation (2.1). Therefore

\frac{x_{n}^{2}}{τ_{n}^{2}} = \frac{x_{n}^{2} + τ_{n}^{2}}{τ_{n}^{2}} − 1 ⩽ k − 1.

This shows that |x_{n}/τ_{n}| ⩽ K for every n, where K = \sqrt{k − 1} . Recall from Theorem 5.4 that all z_{n} ∈ L, therefore we have shown that all z_{n} lie in the sector S = {x + tj ∈ H : x ⩽ 0, |x| ⩽ Kt}. We now show that no τ_{n} has value greater than 1. This is certainly true of τ_{1} and τ_{2} . Suppose that τ_{n} is the smallest counterexample to this posit, for some n > 2. A short computation shows that

z_{n−2} = \frac{κ_{n} + τ_{n}j}{κ_{n}^{2} + τ_{n}^{2}} = \frac{κ_{n} + τ_{n}j}{1 + 2b_{n}κ_{n} − b_{n}^{2}((x_{n} + b_{n})^{2} + τ_{n}^{2})},

where κ_{n} = (x_{n} + b_{n}) + ((x_{n} + b_{n})^{2} + τ_{n}^{2})b_{n}. Since x_{n−2} ⩽ 0, also κ_{n} ⩽ 0, hence

τ_{n−2} = \frac{τ_{n}}{1 + 2b_{n}κ_{n} − b_{n}^{2}((x_{n} + b_{n})^{2} + τ_{n}^{2})} ⩾ τ_{n} > 1,

which is a contradiction. Hence all τ_{n} ≤ 1. On the other hand, if τ_{n} < 1/(1+K^{2}) then

τ_{n+1} = \frac{τ_{n}}{x_{n}^{2} + τ_{n}^{2}} ⩾ \frac{τ_{n}}{K^{2}τ_{n}^{2} + τ_{n}^{2}} = \frac{1}{(K^{2} + 1)τ_{n}} > 1,

which is again a contradiction. Hence z_{n} is restricted to that portion of S with j component between 1/(K^{2} + 1) and 1 (a compact set). Thus ϱ_{H}(j, T_{n}^{−1}(j)) and hence ϱ_{H}(T_{n}(j), j) are bounded sequences, therefore T_{n}(j) does not converge generally.

We remark that the above proof shows that when T_{n} diverges classically, the sequence ϱ_{H}(T_{n}^{−1}(j), j) is bounded. Theorem 5.5 and Corollary 5.2 then demonstrate that ∑ |b_{n}| converges, hence the work of Section 3 applies to show that T_{2n}^{−1} → g and T_{2n+1}^{−1} → ιg, for some Möbius map g. Thus, not only is T_{n}^{−1}(j) restricted to a compact subset of H, in fact T_{2n}^{−1}(j) → ξ and T_{2n+1}^{−1}(j) → ι(ξ) for some ξ ∈ H.

It remains to supply a proof of Theorem 1.2.

Proof of Theorem 1.2. The comments preceding this proof along with the work of Section 3 show that when T_{n} does not converge generally, we have T_{2n} → f and T_{2n+1} → fι, for some Möbius map f. Since j is a fixed point of ι, the sequence T_{n}(j) converges. Hence we have proved the statement that T_{n}(j) → ζ, for some ζ ∈ \overline{H} . Theorem 5.5 demonstrates the equivalence of (i) and (ii) and Theorem 5.7 demonstrates the equivalence of (ii) and (iii).

References

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Received 22 February 2005