Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 22, 1997, 245–254

THE EXPONENT OF CONVERGENCE OF A FINITE BLASCHKE PRODUCT

Gavin L. Jones

Liverpool University, Department of Mathematical Sciences P.O. Box 147, Liverpool L69 3BX, UK; gljones@liv.ac.uk

Abstract. This paper describes the iteration theory of finite Blaschke products using the techniques of Fuchsian groups. In particular the Julia set is shown to decompose into points of conical approach and parabolic points. We also use the hyperbolic geometry of the unit disc to describe the construction and properties of conformal measures on the Julia set.

1. Introduction

It is a commonplace to remark the similarities between the theory of Kleinian groups and the iteration of rational maps. Here we restrict our attention to finite Blaschke products, those rational maps which fix a disc and examine their iteration theory from the point of view of Fuchsian groups. The techniques will be derived from the study of hyperbolic geometry and its influence on the geometry and ergodic theory of Fuchsian groups, both classically and from the work of Patterson and Sullivan [11], [13]. Though the results on conformal measures on S^{1} we describe occur as special cases of the investigations of Denker and Urbanski [6], [5], [1] and Sullivan [12], the hyperbolic approach gives a different insight.

2. Blaschke products

Let B be a finite Blaschke product of degree d > 1. It is easy to show that J_{B}, the set on which the iterates B^{n} fail to be normal on any neighbourhood is either S^{1} , the unit circle, or a Cantor subset [2]. The Denjoy–Wolff theorem shows that the iterates of B converge locally uniformly to a unique point z_{0} in the closure of △, the unit disc.

If z_{0} is contained in the open unit disc then it is an attracting fixed point for B and |B′(z_{0})| < 1. Otherwise z_{0} ∈ S^{1} and either |B′(z_{0})| < 1 or B′(z_{0}) = 1 and is a parabolic fixed point [2].

Each of the possible cases occur. If we write f_{r}(z) = (z + r)/(1 + rz) and B_{r} = f_{r}^{2} . One checks easily that B_{0} has J_{B_{0}} = S^{1} , 0 fixed. B_{1/3} has J_{B_{1/3}} = S^{1} parabolic point at 1, and B_{2/3} has 1 as an attractive fixed point and Julia set a Cantor subset of S^{1} . If we consider B(z) = z/(1 + z − z^{2}), we obtain a map conjugate to a Blaschke product, with parabolic fixed point at the origin and J_{B} a Cantor subset of the extended real line [9].

1991 Mathematics Subject Classification: Primary 30D05.

Currently supported by EPSRC grant GR/K/75453.

We call those Blaschke products B with J_{B} = S^{1} of the first kind by analogy with Fuchsian groups, and those with J_{B} ≠ S^{1} of the second kind. A Blaschke product B is parabolic if the fixed point z_{0} of the Denjoy–Wolff theorem satisfies B′(z_{0}) = 1.

3. Exponent of convergence

Write P_{c} for the closure of the forward orbit of the critical points and △_{c} for △ \ P_{c}. If z ∈ △_{c} then there is a disc D, containing z, free of the images of critical points of B^{n}. By the monodromy theorem we have well defined inverses to B^{n} on D, and by shrinking D if necessary we may ensure that all pre-images of D are disjoint. Denote by B_{n} the family of univalent inverses of B^{n} defined on D and O^{−}(z) for ⋃_{n} ⋃_{f∈B_{n}} f(z), O^{−}(z) accumulates only at J_{B} [2], for z not fixed. Given z ∈ △_{c} we form

S_{B,z,α} = ∑_{n=1}^{∞} ∑_{f∈B_{n}} |f′(z)|^{α}.

Define the exponent of convergence as

δ_{B,z} = inf_{α}{S_{B,z,α} < ∞}.

Lemma 3.1. Let B be a finite Blaschke product, then δ_{B,z} is constant, for z ∈ △_{c}.

Proof. For such z there is a disc \overline{D(z, ϱ)} ⊂ △_{c}, so all inverse branches of all iterates of B are defined on D. Let f be the inverse branch of some iterate of B, and let w be some point in D. By the distortion theorem there is a constant M dependent on D, z, w but not on f for which 1/M|f′(w)| < |f′(z)| < M|f′(w)|. Thus the series S_{B,z,α} converges independently of the choice of z in D. So δ_{B,z,α} is constant off P_{c}.

We now use univalent function theory to replace our current definition of δ_{B} with one which emphasises the non-Euclidean geometry.

Lemma 3.2. For z ∈ △_{c} let
δ′_{B} = inf_{α}{∑_{n} ∑_{f∈B_{n}}(1 − |f(z)|)^{α} < ∞},

δ′′_{B} = inf_{α}{∑_{n} ∑_{f∈B_{n}} exp(−αϱ(x, f(z))) < ∞}

where x ∈ △, and ϱ is the non-Euclidean metric on △. Then δ_{B} = δ′_{B} = δ′′_{B}.

Proof. Consider D(z, r) a Euclidean disc of radius r centred at z, which misses P_{c}. Let f be a choice of inverse to B^{m} some iterate of B on △. From the Koebe \frac{1}{4} theorem we have 1 − |f(z)| > \frac{1}{4}|f′(z)|r and the Schwarz–Pick lemma asserts |B^{m′}(x)| ≤ (1 − |B^{m}(x)|^{2})/(1 − |x|^{2}). So for x = f(z),

1 − |f(z)|^{2} ≤ |f′(z)|(1 − |z|^{2}),

and we have

\frac{2}{1 − |z|} ≤ \frac{|f′(z)|}{(1 − |f(z)|)} ≤ \frac{4}{r}.

This shows δ′_{B} = δ_{B}. Now

2^{−s}(1 − |f(x)|)^{s} < exp −sϱ(0, f(x)) < (1 − |f(x)|)^{s}

and varying the basepoint from which the hyperbolic distance is calculated only contributes a constant term showing δ′′_{B} = δ′_{B}.

The same argument shows

Theorem 3.1. Let B be a finite Blaschke product acting on the unit disc, so that the closed disc D = \overline{△(z, r)} ⊂ △_{c}. Let D′ be any pre-image of D, then the hyperbolic area of D′ is bounded with bound only depending on B, D.

Then by analogy with the classical results on Fuchsian groups one easily proves

Lemma 3.3. 0 < δ_{B} ≤ 1.

Proof. The upper estimate is a packing argument [10] in the hyperbolic metric. The lower bound [7] follows easily from the density of repelling points in J_{B}. If B is of degree d, with K = sup_{z∈J_{B}} |B′(z)| then

δ_{B} ≥ \frac{log d}{log K} .

4. A measure on the Julia set

Given a Blaschke product B a t-conformal measure μ for B is a Borel probability measure supported on the Julia set which satisfies the following equation

(4.1.1)  μ(B(A)) = ∫_{A} |B′(z)|^{t} dμ(z)

on any Borel set A on which B is injective.

We follow Patterson’s original construction [11] as described in [10] (pp. 45– 55). Denoting the non-Euclidean distance ϱ(x, y) by (x, y) we form the series

g_{s}(x, y) = ∑_{z∈O^{−}(y)} exp(−s(x, z))

which converges for s > δ_{B} and diverges for s < δ_{B}. It is important that our series diverge at δ_{B} and to this end we introduce a weighting factor h(exp(x, z)) in g_{s}(x, y) [10, Lemma 3.1.1] so that

g_{s}^{∗}(x, y) = ∑_{z∈O^{−}(y)} exp −s(x, z)h(exp(x, z))

diverges for s ≤ δ_{B} and converges for s > δ_{B}. We now construct a set of measures, for s > δ. Let

μ_{x,s} = 1/g_{s}^{∗}(y, y) ∑_{z∈O^{−}(y)} exp −s(x, z)h(exp(x, z))D_{z}

where D_{z} denotes the Dirac point mass at z.

We can control the variation of μ_{x,s} with the basepoint x by the following simple estimate.

Lemma 4.1. Given x, x′ in △ and ξ ∈ ∂△ then

\begin{matrix}exp & (x, w) \\ exp & (x′, w) \end{matrix} ⟶ [\frac{P (x, ξ)}{P (x′, ξ)}]

as w ⟶ ξ. P (x, ξ) is the Poisson kernel (1 − |x|^{2})/(|x − ξ|^{2}).

Follow Nicholls [10, pp. 53–56] to estimate the behaviour of the measures under change of basepoint and the action of B, a series of straightforward estimates. To obtain a measure μ_{x,δ} we note that μ_{x,s}( ¯△) is bounded as s → δ and apply Helley’s theorem to deduce that for s → δ^{+} the measures μ_{x,s} converge weakly to μ_{x,δ} on ¯△. Since the series g_{s}^{∗}( · , ·) diverges at s = δ the measure must be supported on the Julia set. Reading off the behaviour of μ_{x,δ} under the action of B we get

Theorem 4.1. Let B be a Blaschke product with exponent of convergence δ, and suppose μ_{x,δ} is obtained as a limit of μ_{x,s} as s → δ^{+} . Then μ_{x,δ} is a Borel measure supported on the Julia set J_{B}. Further if A is a Borel measurable subset of J_{B} on which B is injective then

μ_{x}(B(A)) = ∫_{z∈A}[\frac{P (x, Bz)}{P (x, z)}]^{δ}|B′(z)|^{δ} dμ_{x,δ}(z).

In particular μ_{0} satisfies

μ_{0}(B(A)) = ∫_{z∈A} |B′(z)|^{δ} dμ_{0}(z)

and so up to a constant multiple is a δ -conformal measure.

We can also control the influence of the change of base point on the measure we have constructed following the argument from Fuchsian groups. Denoting the collection of measures μ_{x,δ} obtained as above by M_{x} we have

Theorem 4.2. Let B be a Blaschke product with exponent of convergence δ. Choose x, x′ ∈ △ and for ν_{x} ∈ M_{x} define a new measure ϕ(ν_{x}) by

ϕ(ν_{x})(E) = ∫_{E}[\frac{P (x′, ξ)}{P (x, ξ)}]^{δ} dν_{x}(ξ).

Then ϕ is a homeomorphism of M_{x} onto M_{x′}. If ν_{x,s_{j}} converges weakly to ν_{x} as s_{j} → δ^{+} then ν_{x′,s_{j}} converges weakly to ϕ(ν_{x}).

5. Conical and parabolic points

In the case of a Fuchsian group an important role is played by the way in which orbits accumulate at the limit set. The orbits control the way in which the Patterson measure accumulates on the limit set. We will make the same division of the Julia set into conical and parabolic points and show that the measure μ_{0} respects this division.

A point z ∈ J_{B} lies in the conical limit set of B if there is a sequence {x_{n}} ⊂ O^{−}(x) such that |x_{n} − z|/(1 − |x_{n}|) is bounded. That is the conical limit set consists of those points which are accumulated by pre-images in a Stolz angle. By Theorem 3.1 the conical limit set is independent of x the point used to define it.

We prove

Theorem 5.1. Let B be a finite Blaschke product with Julia set J_{B}. Then for z ∈ J_{B} either z is a conical limit point or z lies in O^{−}(z_{0}), where z_{0} is a parabolic fixed point of B.

This is the analogue of the theorem of Beardon and Maskit [4] that the limit set of a finitely generated Fuchsian group splits into conical and parabolic points. In particular if B has no parabolic fixed point then every point in the Julia set is conical. We show first that every point in the Julia set which does not lie in O^{−}(z_{0}) is conical, and then show that no member of O^{−}(z_{0}) is conical.

First we observe if x ∈ J_{B} \ O^{−}(z_{0}), then x has a forward limit point which is not z_{0} . If not there is an n_{0} so that for n > n_{0} , B^{n}(x) ∈ △(z_{0}, ε) for some ε > 0. However the dynamics near z_{0} shows this cannot be, since in some neighbourhood of a parabolic fixed point B maps points in J_{B} away from the parabolic point [2].

To show x ∈ J_{B} \ O^{−}(z_{0}) is conical. Choose x_{0} a forward limit point of x, not z_{0} , the parabolic fixed point. There is a disc △(x_{0}, r) ⊂ △_{c}, so all iterates of B have inverses defined and injective on △(x_{0}, r). Let f_{n} be a sequence of such inverses so that f_{n}(x_{0}) → x. The functions f_{n} are univalent and map S^{1} ∩ △(x_{0}, r) to a segment of S^{1} so each function f_{n} is conformally conjugate to a function defined and univalent on the unit disc which maps the real axis to itself. Such a function is typically real. From [8] we have for f an analytic function univalent on the unit disc fixing the origin with sign Im f(x) = sign Im(z) for non-real z then

arg \frac{z}{(1 + z)^{2}} ≤ arg f(z) ≤ arg \frac{z}{(1 − z)^{2}} .

It is easy to show that there is a disc \overline{D} ⊂ H ∩ △ and D ∩ I ≠ 0, f(D) ∩ I ≠ 0 where I is the imaginary axis. By expanding D in the upper-half plane we may arrange that f(D) ∩ I + ε ≠ 0 for all sufficiently small real ε.

Conjugating back to the unit disc and the Blaschke product, we consider the small disc △(x_{0}, r) on which inverses f_{m} of iterates of B are defined, so that f_{m}(x_{0}) accumulate at x. Choose a disc D centred on (0, x_{0}) whose closure is properly contained in △ ∩ △(x_{0}, r). As the inverses f_{m} are conjugate to typically real functions, and the radius (0, f_{m}(y)) intersects f_{m}(D) for all y sufficiently close to x_{0} in S^{1} . We may choose the y = B^{n}(x) for n suitably large. So f_{m}(D) intersects the radius (0, x) for infinitely many m, by Theorem 3.1 f_{m}(D) is of bounded hyperbolic diameter. From which the result follows.

5.1. Parabolic points. We now discuss parabolic points, giving a hyperbolic argument why pre-images of parabolic fixed points cannot be conical and then using some analytic estimates to describe the behaviour at a parabolic point.

Lemma 5.1. Let B be a Blaschke product with z_{0} a parabolic fixed point of B, then z_{0} is not a conical limit point.

Conjugate the Blaschke product so it acts on the upper-half plane with the origin a parabolic fixed point. Classical analysis of parabolic behaviour [2] shows the existence of Π the conformal image of a disc, such that Π ∩ H subtends an angle π at the origin, and B(Π) ⊂ Π. Iterating we have B^{n}(Π) ⊂ Π so if x ∉ Π then no pre-image B^{−n}(x) ∈ Π. If z_{0} is conical then for any x ∈ △ we must have B^{−n}(x) ∈ Π for infinitely many n since Π contains any conical approach to z_{0} , a contradiction for x ∉ Π.

Let B^{q}(z) = z_{0} , z ≠ z_{0} . If z is conical there is a point y so that a sequence of images B^{−m}(y) converge in a cone to z. B^{q} is univalent in some fixed neighbourhood of z and so the ratio

\frac{d(B^{q}(B^{−m}(y)), z_{0})}{d(B^{q}(B^{−m}(y), S^{1})} ≈ \frac{d(B^{−m}(y), z)}{d(B^{−m}(y), S^{1})}.

This is bounded above, just being the condition that z = B^{−q}(z_{0}) is conical, so z_{0} is conical and we have reached a contradiction, proving that the pre-images of a parabolic point are not conical.

Alternatively an induction shows

Lemma 5.2. Let R(z) be an analytic function with Taylor expansion at the origin given by R(z) = z + z^{p+1} + O(z^{2p+1}), and let f denote the inverse to R which fixes the origin, then for z small real and positive

f^{n}(z) = an^{−1/p} + b_{n}(n^{−2/p})

where a = 1/p^{1/p}, and b_{n} is bounded over n and

∞ > C > |f^{n′}(z)|/n^{−(p+1)/p} > c > 0.

This allows us to repeat a classic Fuchsian groups argument [3] which established that groups with a parabolic generator have a definite bound below on their exponent of convergence.

Corollary 5.1. Let B be a parabolic Blaschke product then

δ_{B} ≥ p/(1 + p) ≥ \frac{1}{2}.

Proof. We certainly require

∑_{n>0} |f^{n′}(z)|^{s} < ∞

for s > δ_{B}. Each term is comparable to n^{−s(p+1)/p} forcing s(p + 1) > p for all s > δ_{B}, whence the result.

A stronger argument is required to counterpoint the Fuchsian group argument that δ > \frac{1}{2} for a Fuchsian group with parabolics.

Theorem 5.2. If B is a Blaschke product, with a parabolic fixed point then δ_{B} > \frac{1}{2} .

If B is of the first kind then the previous corollary gives δ_{B} ≥ \frac{2}{3} so we are only concerned about Blaschke products of the second kind. Let f, g be branches of B^{−1} defined in a neighbourhood N of z with f(z) = z and g(N) ∩ N = 0. We consider images of a point y of the form

F_{n_{1}n_{2}...n_{j}}(y) = f^{n_{1}}gf^{n_{2}}g … f^{n_{j}}g(y).

We can now compute a lower bound for δ_{B} since

∑_{n} ∑_{f∈B_{n}} |f′(y)|^{s} ≥ ∑_{j} ∑_{n_{1},...,n_{j}}^{∞} |F′_{n_{1}...n_{j}}(y)|^{s}.

Now we estimate |F′_{n_{1}...n_{j}}(y)| by the chain rule. On some neighbourhood N of z, |g′(x)| > M and for n sufficiently large Lemma 5.2 tells us f^{n′}(g(z)) > cn^{−2} . So

∑_{j} ∑_{n_{1},...,n_{j}}^{∞} |F′_{n_{1}...n_{j}}(y)|^{s} ≥ ∑_{j} ∑_{n_{1}...n_{j}} |M^{j}c^{j}n_{1}^{−2} … n_{j}^{−2}|^{s}.

This is bounded below by

∑_{j}[(Mc)^{s} ∑_{n=m}^{∞} n^{−2s}]^{j}

which diverges if M^{s}c^{s} < ∑_{n=1}^{∞} n^{−2s}, which certainly occurs for some s > \frac{1}{2} .

6. No atoms

Finally we show that the δ -conformal measure we constructed in Section 4 has no atoms. The difficulty is with parabolic points of the second kind, and we treat this following Patterson’s nifty argument in the Fuchsian case [10, §6.5], a different argument to that of [1]. It is clear no conical limit point z can be charged by a conformal measure otherwise |B^{n′}(z)| is uniformly bounded, which is impossible.

Lemma 6.1. Let B be a Blaschke product with J_{B} = S^{1} , then

g_{1}(x, y) = ∑_{n=1}^{∞} ∑_{f∈B_{n}} exp −ϱ(x, f(y))

diverges.

Whence the conformal measure charges no single point.

Proof. Select any point y in the unit disc not fixed by the Blaschke product. If the series in the lemma converges we may form the non-trivial infinite Blaschke product B_{∞} with zeroes at O^{−}(y). This has non-tangential boundary values one almost everywhere, but at all points save the countable pre-images of a parabolic point there are zeroes of the function converging conically, a contradiction.

It is harder to show that for B a Blaschke product of the second kind the critical series diverges. We sketch a proof exhibiting the non-Euclidean behaviour. Conjugate B, a Blaschke product with a parabolic fixed point of the second kind so it acts on the upper-half plane with the parabolic fixed point at infinity. We assume that the series g_{δ}(x, y) converges, and consider how the measure μ_{x,δ} varies as x moves on the imaginary axis. We assume B has a power series expansion at ∞ of the form z(1 − 1/z + O(1/z^{2})). The inverse to this function which fixes ∞ is denoted by f, we may analytically continue it along the positive real axis where it still satisfies the functional relation f(B) = identity. Then we have f^{n} is the inverse to B^{n} fixing ∞ and estimates on the behaviour near parabolic points are provided by [2], or inversion of Lemma 5.2 to give lim_{n→∞} |f^{n}(z) − z|/n = 1 for z real. So the long term behaviour of f^{n} is roughly translation.

To mimic the Fuchsian argument we consider Y the set of points in O^{−}(y) so that {Re z < N : z ∈ O^{−}(y)}, corresponding to a fundamental region for a subgroup of a Fuchsian group generated by a translation. We claim that g_{s}^{∗}(x, y) is boundedly comparable to

Σ_{s,N} = ∑_{z∈Y} ∑_{n=1}^{∞} exp −sϱ(x, f^{n}(z))h(exp ϱ(x, f^{n}(z)))

and so we may estimate the series g_{s}^{∗}(x, y) by the behaviour of Σ_{s,N}. This we will do by controlling the ‘iterates’ of f acting on a bounded part of the plane. Then as we allow x = λi, λ real, to vary we will control g_{s}(x, y) by comparing μ_{x,s}(∞) as λ varies.

We estimate

(6.1.1)  \frac{μ_{λi,s}(∞)}{μ_{i,s}(∞)} = [\frac{P (λi, ∞)}{P (i, ∞)}]^{s} = λ^{s}.

Now we show that if N is chosen appropriately then g_{s}^{∗}(λi, y) ≈ Σ_{s,N}. We have g_{s}^{∗} > c_{s,N}Σ_{s,N} as the latter sum is over a subset of the former up to finite multiplicity. The opposite inequality follows by comparing the mass left at ∞ by the two series, which varies with λ as (6.1.1). As λ increases both series deposit an increasing proportion of their total mass at ∞. So an estimate at λ = 1 gives an estimate of the ratio g_{s}^{∗}/ ∑_{s,N} independent of λ, s, and is non-zero if we choose N sufficiently large.

The estimate we need is the following.

Theorem 6.1. For all ε sufficiently small there are constants A, A′ independent of s, δ < s < δ + 1, so that for λ > 1

g_{s}^{∗}(λi, y) = ∑_{ζ∈O^{−}(y)} h(exp ϱ(λi, ζ)) exp −sϱ(λi, ζ)

≤ Aλ^{ε+1−s} ∑_{{ζ∈O^{−}(y):Re ζ<N}} Im(ζ)^{s}h(exp ϱ(i, ζ))

≤ A′λ^{ε+1−s}g_{s}^{∗}(i, y).

The proof follows that given by Nicholls [10, Theorem 3.5.5] of Patterson’s result, being careful to check that we only introduce bounded distortion on preimages, and the result is as in the Fuchsian case. We finish the proof of the absence of parabolic point masses by computing

g_{s}^{∗}(λi, y) < A′λ^{ε+1−s}g_{s}^{∗}(i, y)

254 Gavin L. Jones

for ε sufficiently small. Taking a limit as s → δ

μ_{λi,δ}( \overline{H} ) < A′λ^{ε+1−δ}μ_{i}( \overline{H} ).
So for some constant C we have μ_{λi}(∞) < Cλ^{ε+1−δ}. But (6.1.1) forces
λ^{δ} < Cλ^{ε+1−δ}

for all ε sufficiently small, and λ > 1. But δ > \frac{1}{2} from Theorem 5.2 and we have reached a contradiction, μ(∞) = 0, and hence by δ -conformality the support of μ has empty intersection with the parabolic points.

References

[1] Aaronson, J., M. Denker and M. Urbanski: Ergodic theory for Markov fibred systems and parabolic rational maps. - Trans. Amer. Math. Soc. 337, 1993, 495–549.

[2] Beardon, A.F.: The Iteration Theory of Rational Maps. - Springer-Verlag, 1991.

[3] Beardon, A.F.: The exponent of convergence of Poincaré series. - Proc. London Math. Soc. 18, 1968, 461–483.

[4] Beardon, A.F., and B. Maskit: Limit points of Kleinian groups and finite sided fundamental polyhedra. - Acta Math. 132, 1974, 1–12.

[5] Denker, M., and M. Urbanski: On Sullivan’s conformal measures. - Nonlinearity 4, 1991, 365–384.

[6] Denker, M., and M. Urbanski: Geometric measures for parabolic rational maps. - Ergodic Theory Dynamical Systems 12, 1992, 53–66.

[7] Garber, V.: On the iteration of rational functions. - Math. Proc. Cambridge Philos. Soc. 84, 1978, 497–505.

[8] Goodman, A.W.: Univalent Functions, Vol. 1. - Mariner, 1983.

[9] Lyubich, M.: The dynamics of rational transforms: the topological picture. - Russian Math. Surveys 41, 1986, 43–117.

[10] Nicholls, P.J.: The Ergodic Theory of Discrete Groups. - London Math. Soc. Lecture Note Ser. 143, Cambridge Univ. Press, 1989.

[11] Patterson, S.J.: The limit set of a Fuchsian group. - Acta Math. 136, 1976, 241–273.

[12] Sullivan, D.: Conformal dynamical systems. - In: Geometric Dynamics, edited by J. Palis, Lecture Notes in Math. 1007, Springer-Verlag, 1981, 725–752.

[13] Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. - Inst. Hautes Études Sci. Publ. Math. 50, 1979, 171–202.

Received 30 August 1994 Received in revised form 11 January 1997