Julia sets are uniformly perfect

A. Eremenko^{∗}

1992

According to Pommerenke [1,2] a compact set E ⊂ C̄ is called uniformly perfect if the doubly connected domains in C̄\E that separate E have bounded moduli. The limit sets of finitely generated non-elementary Kleinian groups as well as Julia sets of hyperbolic rational functions are known to have this property [2]. In [3] Hinkkanen proves that Julia sets of all polynomials are uniformly perfect. Recently he extended his proof to the case of rational functions [4]. In this note we give a simple proof of Hinkkanen’s theorem for arbitrary rational functions.

Let N be an open set in C̄. We assume that each component of N is hyperbolic. A closed curve in N is called trivial if it is freely homotopic to a point in N. A closed curve in N is called peripheric if it is either trivial or homotopic to an arbitrarily small loop around a puncture of N. l_{N}(C) stands for the Poincaré length of a curve C ⊂ N.

Proposition 1 If the Poincaré lengths of all non-trivial closed curves in N are bounded from below by a constant δ > 0 then E = C̄\N is uniformly perfect.

Proof. Let A ⊂ N be a non-degenerate annulus separating E. Denote by C the simple closed curve separating ∂A and having minimal Poincaré length in A. Then C is non-trivial in N. By the generalized Schwarz lemma we have l_{A}(C) ≥ l_{N}(C). So

mod(A) = \frac{π}{l_{A}(C)} ≤ \frac{π}{l_{N}(C)} ≤ \frac{π}{δ} .

∗Supported by NSF grant DMS-9101798

QED.

Remark. The converse is also true. Indeed, assume that there is a simple closed Poincaré geodesic γ ⊂ N having length l. Then by the Collar lemma [5] there exists an annulus A embedded in N, such that γ is non-trivial in A, and the hyperbolic area of A is 2l/ sinh(l/2). Then mod(A) ≥ (l sinh(l/2))^{−1} → ∞, l → 0.

Now we prove that the set of normality N(f) (the complement of the Julia set) of a rational function f, deg f ≥ 2 satisfies the assumption of the Proposition. Let γ be a closed non-trivial curve in N(f). The description of the dynamics of f on the set N(f) due to Fatou and Sullivan [6-8] implies that one of the following statements is true:

a). f^{n}(γ) is trivial in N(f) for some n ∈ N or

b). f^{n}(γ) separates the boundary of Arnold – Herman ring for some n ∈ N ∪ {0}.

In the case b) we are done because: l_{N}(γ) ≥ l_{N}(f^{n}γ), there are finitely many Arnold – Herman rings, and they are all non-degenerate.

In the case a) take n as small as possible and denote by D the component of N(f) containing f^{n}γ. Then f^{n}γ is non-peripheric in D\{critical values of f}. (Otherwise we could deform f^{n}γ to a point in D or to a small loop around a critical value without hitting critical values, so f^{n−1}γ would be trivial in N(f)). Choose three points b_{i} ∈ J = C̄\N(f) and set G = C̄\{b_{1}, b_{2}, b_{3}}. Then f^{n}γ is trivial in G but non-peripheric in G\{critical values of f}, so

l_{N}(γ) ≥ l_{N}(f^{n}γ) ≥ l_{G}(f^{n}γ) ≥ δ,

where δ is the minimal hyperbolic distance between critical values of f with respect to G. QED.

Remark. If f is an entire transcendental function then N(f) can have doubly connected components of arbitrarily large moduli [9].

I thank J. Hubbard for discussions.

References

1. Ch. Pommerenke. Uniformly perfect sets and the Poincaré metric. Arch. Math., 32, 1979, 192-199.

2. Ch. Pommerenke. On uniformly perfect sets and Fuchsian Groups. Analysis, 4, 1984, 299-321.

3. A. Hinkkanen. Julia sets of polynomials are uniformly perfect. Preprint, 1991, 7p.

4. A. Hinkkanen. Julia sets of rational functions are uniformly perfect. Preprint, 1992, 15p.

5. B. Randol. Cylinders in Riemann surfaces. Comment. Math. Helv., 54, 1979, 1-5.

6. D. Sullivan. Quasi conformal homeomorphisms and dynamics I. Ann. Math., 122, 1985, 401-418.

7. A. Beardon. Iteration of Rational Functions: Complex analytic dynamical systems. N. Y., Springer, 1991.

8. A. Eremenko, M. Lyubich. The dynamics of analytic transformations. Leningrad Math. J., 1, 1990, 563-633.

9. I. N. Baker. Multiply connected domains of normality in iteration theory. Math. Z., 81, 1963, 206-214.

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