Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 3–11

NON-ACCESSIBLE CRITICAL POINTS OF CERTAIN RATIONAL FUNCTIONS WITH CREMER POINTS

Lia Petracovici

Western Illinois University, Department of Mathematics 1 University Circle, Macomb, Illinois, 61455, U.S.A.; L-Petracovici@wiu.edu

Abstract. Let R be a rational function with a completely invariant (super)attracting Fatou component. We show that R has a non-accessible critical point in its Julia set, provided that R has a Cremer fixed point with the small cycles property. This extends Kiwi’s result stating that the same is true for polynomials with Cremer fixed points.

1. Introduction

Cremer fixed points of rational functions are irrationally indifferent fixed points in the Julia set. In the case of polynomial maps, the presence of Cremer fixed points forces the Julia set to be not locally connected, while rational maps with Cremer fixed points may have a locally connected Julia set (see [13]). When the Julia set fails to be locally connected, it is of interest to determine which points in the Julia set are accessible from the complement of the Julia set.

Repelling periodic points belong to the Julia set and are always accessible, as they are the landing points of at least one periodic external ray [9]. Since a Cremer fixed point belongs to the Julia set, every neighborhood of it contains infinitely many repelling periodic points, but there is no guarantee that any neighborhood of the Cremer point contains infinitely many periodic orbits, or even one entire orbit. When every neighborhood of a Cremer fixed point contains infinitely many cycles, we say that the Cremer point has the small cycles property, or that is approximated by small cycles. According to Yoccoz [16], Cremer points of quadratic polynomials are always approximated by small cycles. It is not known whether this is true for polynomials of higher degree, or for arbitrary rational functions. According to Perez-Marco [12], rational functions with Cremer periodic points always have some non-accessible points in their Julia sets.

Kiwi [8] has shown that polynomials with a Cremer fixed point having the small cycles property have a non-accessible critical point in their Julia set. We show that the same is true in the case of a rational function with a completely invariant (super)attracting Fatou component. We prove the following results:

2000 Mathematics Subject Classification: Primary 37F10.

Theorem 4.1. Let R be a rational function of degree at least two with a completely invariant (super)attracting Fatou component D. Assume that R has a Cremer fixed point z_{0} . Let C be the connected component of \overline{C} \ D containing the Cremer point z_{0} . Then there exists a polynomial-like map (R; U_{1}, U_{2}) with C ⊂ U_{1} , whose filled Julia set K_{(R;U_{1},U_{2})} coincides with C.

Theorem 4.2. Let R be a rational function of degree at least two with a completely invariant (super)attracting Fatou component. Assume that R has a Cremer fixed point that is approximated by small cycles. Then R has a critical point which is not accessible from the complement of the Julia set.

The paper is organized as follows. In Section 2 we introduce some definitions and collect some results from holomorphic dynamics, which will be used in the sequel. For general background in complex dynamics we refer to [1], [3], [9].

In Section 3 we summarize Kiwi’s results for polynomials [8]. He first proved that polynomials with a Cremer periodic point and with a connected Julia set have a non-accessible critical value, provided that the Cremer fixed point has the small cycles property. For polynomials with disconnected Julia sets, Kiwi’s idea was to extract a polynomial-like mapping which after straightening according to Douady and Hubbard [4] becomes a polynomial with a connected Julia set. Then he applied his first result to obtain a non-accessible critical point [8].

In Section 4 we prove our results for rational functions which have a completely invariant (super)attracting component of the Fatou set. Following Kiwi’s idea, we show that we can extract a polynomial-like mapping which after straightening becomes a polynomial with a connected Julia set, and so it has a non-accessible critical point in the Julia set.

The author is grateful to Professor Aimo Hinkkanen for helpful comments.

2. Cremer fixed points and accessible points

Consider a rational function R of degree at least 2 with a fixed point z_{0} of multiplier λ. The multiplier is equal to R′(z_{0}) when z_{0} is finite and is equal to S′(0) where S(z) = 1/R(1/z) when z_{0} = ∞. By definition, a fixed point is either attracting, repelling or indifferent according as the multiplier satisfies |λ| < 1, |λ| > 1 or |λ| = 1. If λ is a root of unity, we call the corresponding fixed point parabolic or rationally indifferent. For λ = e^{2πiζ}, with ζ real and irrational, we call the fixed point irrationally indifferent. We say that an irrationally indifferent fixed point is a Cremer point if it belongs to the Julia set, and is a Siegel point otherwise.

A fixed point z_{0} is said to have the small cycles property [9] if every neighborhood of z_{0} contains infinitely many periodic orbits. According to Yoccoz [16], Cremer fixed points of quadratic polynomials always have the small cycles property. Yoccoz’s theorem raises the question as to whether every Cremer point of a holomorphic germ has small cycles. The answer was provided by Perez-Marco [10] who showed that for ζ not satisfying a certain diophantine condition, there exists a germ with multiplier λ = e^{2πiζ} that has a Cremer point but no small cycles.

It is not known whether Cremer fixed points of polynomials of arbitrary degree always have small cycles. Also, it is not known whether any rational function can have Cremer points without small cycles. For a detailed discussion of the local dynamics near an irrationally indifferent fixed point see [11].

A point z in the Julia set is called accessible from a Fatou component V if there exists a path γ contained in V that ends at z, i.e.

γ: [0, 1) ⟶ V, and γ(r) → z, r → 1.

When z is not accessible from any Fatou component we say that z is a nonaccessible point. We note that accessibility is completely invariant under the dynamics: z is accessible from V if and only if R(z) is accessible from R(V).

Consider a monic polynomial map P: C ⟶ C of degree d ≥ 2. The complement of the super-attracting basin of infinity, that is the set of all the points z ∈ C with bounded forward orbit under P is called the filled Julia set of P and is denoted by K_{P} .

Assume that K_{P} is connected. From Boetcher’s theorem [1], [3], we know that the complement of the closed unit disk Δ̄ is conformally isomorphic to the complement of the filled Julia set K_{P} . Moreover, there exists a conformal isomorphism:

ϕ: C \ Δ̄ ⟶ C \ K_{P} ,

such that

\begin{aligned}P ∘ ϕ(z) & = ϕ(z^{d}), \\ ϕ(z) & = z + 𝒪(1), z → ∞. \end{aligned}

The question of which points in J_{P} are accessible from C \ K_{P} is closely related to the boundary behavior of ϕ.

The set

R_{t} = ϕ({re^{2πit} : r ∈ (1, ∞)})

is called the external ray with angle t ∈ R. Here the angle t is measured in fractions of a full turn, and not in radians. We say that an external ray R_{t} lands at z, if

lim_{r↘1} ϕ(re^{2πit}) = z.

We see that the landing points of external rays are accessible from C \ K_{P} . The converse is also true: every point accessible from C \ K_{P} is the landing point of some external ray (Lindelöf [14], [15]).

The closure \overline{R_{t}} of an external ray R_{t} is called a closed ray. If R_{t} lands, then the closure \overline{R_{t}} is the union of the external ray and its landing point. Otherwise, if R_{t} fails to land, \overline{R_{t}} is the union of the external ray and a non-trivial connected subset of the Julia set.

Let P^{∘n} denote the n-fold iterate of P. An external ray R_{t} is called periodic if P^{∘n}(R_{t}) = R_{t} for some integer n. In particular, if P (R_{t}) = R_{t}, we say that R_{t} is a fixed ray. Periodic rays always land, and their landing points are parabolic or repelling periodic points. Conversely, parabolic and repelling periodic points are the landing points of at least one periodic ray (see [7], [9]).

3. Kiwi’s results for polynomials

For polynomials with connected Julia sets, we know that every point that is accessible from the complement of the filled Julia set, is the landing point of some external ray (Lindelöf, [14], [15]). For the same case, Kiwi [8] proved a stronger result that states the existence of a non-accessible critical value with the property that, if an external ray accumulates to it, this external ray also accumulates to the Cremer fixed point.

Theorem 3.1 (Kiwi, [8]). Let P be a polynomial with connected Julia set J_{P} and with a Cremer fixed point z_{0} that is approximated by small cycles. Then there exists a critical value v ∈ J_{P} which is not accessible from C \ J_{P} such that: if v ∈ \overline{R}_{t} for some external ray R_{t}, then also z_{0} ∈ \overline{R}_{t}.

When the Julia set is disconnected, Kiwi’s idea is to extract a polynomial-like mapping which after straightening becomes a Cremer polynomial with a connected Julia set. Before we can explain how this is done, we need some definitions and results about polynomial-like mappings (see [4]).

Definition. Let U_{1} ⊂ C and U_{2} ⊂ C be bounded simply connected domains with smooth boundaries such that \overline{U}_{1} ⊂ U_{2} . We say that (f; U_{1}, U_{2}) is a polynomial-like map of degree d if f: \overline{U}_{1} → \overline{U}_{2} is a d-fold branched covering, where d ≥ 2, which is holomorphic in U_{1} .

Definition. The filled Julia set K_{(f;U_{1},U_{2})} of (f; U_{1}, U_{2}) is the set of points in U_{1} for which the forward iterates of f are well-defined:

K_{(f;U_{1},U_{2})} = ⋂ f^{−n}(\overline{U}_{1}).

The Julia set J_{(f;U_{1},U_{2})} is the boundary of K_{(f;U_{1},U_{2})} .

A polynomial-like mapping can be extended to the complex plane in such a way that it is quasiconformally conjugate to a polynomial of the same degree.

Theorem 3.2 (The straightening theorem, Douady–Hubbard, [4]). If (f; U_{1}, U_{2}) is a polynomial-like map of degree d, then there exists a quasiconformal map ϕ: C → C and a polynomial Q of degree d such that ϕ ∘ f = Q ∘ ϕ on \overline{U}_{1} . Moreover, ϕ(K_{(f;U_{1},U_{2})}) = K_{Q}.

For polynomials with disconnected Julia sets, Kiwi proved the following results.

Theorem 3.3 (Kiwi, [8]). Let P be a polynomial of degree at least 2 and let C be a connected component of its filled Julia set K_{P} such that P (C) = C. If C is not a singleton consisting of a repelling fixed point, then there exists a polynomial-like mapping (f; U_{1}, U_{2}) such that C = K_{(f;U_{1},U_{2})} .

Theorem 3.4 (Kiwi, [8]). Let P be a polynomial with a Cremer fixed point z_{0} that is approximated by small cycles. Then there exists a critical point of P in the Julia set which is not accessible from the complement of the Julia set.

4. Our results

We now consider a rational function R of degree two or more with a completely invariant (super)attracting Fatou component. Recall that a set D is completely invariant under the map R means that z belongs to D if and only if R(z) belongs to D.

We extend Kiwi’s Theorem 3.3 as follows.

Theorem 4.1. Let R be a rational function of degree at least two with a completely invariant (super)attracting Fatou component D. Assume that R has a Cremer fixed point z_{0} . Let C be the connected component of \overline{C} \ D containing the Cremer point z_{0} . Then there exists a polynomial-like map (R; U_{1}, U_{2}) with C ⊂ U_{1} such that C = K_{(R;U_{1},U_{2})} .

Proof. Let d be the degree of R and let A = \overline{C} \ D. We will treat the attracting and superattracting components in the same way, and simply call them “attracting”. By replacing R with M^{−1} ∘ R ∘ M for a suitable Möbius transformation M that may depend on the invariant component D, we may assume that the attracting fixed point is located at infinity.

We will work with the generalized Green’s function of the attracting component D, as introduced in [6]. It is known that the Julia set J has positive Hausdorff dimension, and thus it has positive logarithmic capacity ([2], [5]). It follows that ∂D = J also has positive logarithmic capacity, and thus D has a generalized Green’s function g(z) = g(z, ∞) with pole at ∞. The function g is positive and harmonic in D \{∞}, and can be extended to a non-negative subharmonic function in C in the standard way. Hence g is continuous and vanishes in A = \overline{C} \ D. Moreover, all the boundary points of D are regular for the Dirichlet’s problem on D.

Also, we have

g(R(z), ∞) = ∑_{R(w)=∞} g(z, w),

where the sum is taken over all w ∈ \overline{C} with R(w) = ∞ with due count of multiplicity. Note that if R(w) = ∞ then w ∈ D, that there are d terms in the sum, and that w = ∞ occurs at least once in the sum since R(∞) = ∞ (see [6] for the proofs).

Since there are countably many zeros of the gradient of g in D, there exist only countably many positive numbers t > 0 such that the corresponding level sets L_{t} = {z ∈ D : g(z) = t} of g contain zeros of the gradient of g. Also it follows that there exist only countably many positive numbers t such that the image R(L_{t}) = {R(z) : z ∈ D, g(z) = t} contains zeros of the gradient of g. We call t > 0 a regular value for the Green’s function g if neither L_{t} nor R(L_{t}) contains any zero of the gradient of g.

Choose t > 0 such that L_{t} does not contain any zeros of the gradient of g. Then L_{t} consists of finitely many Jordan curves since there is no branching. Indeed, L_{t} is closed, since g is continuous on C, and it is bounded, since g(z, ∞) tends to ∞ as z tends to ∞. Also, L_{t} ∩∂D = ∅, since g(z) → 0, as z → ζ ∈ ∂D.

If γ_{t} is a component of L_{t}, so a Jordan curve, then, recalling that g is subharmonic in C, we know that the maximum of g on the closed Jordan domain enclosed by γ_{t} is attained on γ_{t}, so we have g < t inside γ_{t}. For z ∈ γ_{t}, we have

g(R(z), ∞) = ∑_{R(w)=∞} g(z, w) > g(z, ∞),

since w = ∞ occurs at least once in the sum, and the continuous function

∑_{R(w)=∞} g(z, w) − g(z, ∞)

has a positive minimum on the compact set γ_{t}. Thus, R(L_{t}) lies outside γ_{t}, and we have g > t outside all the closed curves γ_{t}.

Let us denote by V (t) the connected component of {z ∈ C : g(z) < t} containing the Cremer point z_{0} of R. Recall that C denotes the connected component of A = \overline{C} \ D containing the Cremer point z_{0} . We claim that

C = ⋂_{t>0 regular for g} V (t).

Clearly, C is contained in each V (t), since g = 0 on A, so also on C. So, C ⊂ ⋂ V (t). To prove the equality, let us assume the contrary, that there exists a point z in ⋂ V (t) with z ∉ C. Then g(z) = 0 so that z ∈ A = \overline{C} \ D. Let C_{1} be the connected component of A containing z. Since z ∉ C, we know that C and C_{1} are distinct components of A. Then there exists a Jordan curve γ ⊂ D that separates C and C_{1} . Now on γ ⊂ D, the Green’s function g attains a positive minimum δ_{0} , so g ≥ δ_{0} > 0 on γ. Choose t with 0 < t < δ_{0} . Then we can find two distinct components γ_{t} and γ_{t}′ of the level set L_{t}, situated in the two distinct components of C \ γ. So we have C_{1} ∩ V (t) = ∅ (for they are separated by γ), in contradiction with our assumption that z ∈ C_{1} ⋂ V (t). Thus the equality holds.

In particular, if a critical point of R is contained in V (t) for all regular values t of g, then it belongs to C. It follows that there exists a regular value t_{0} of g such that all the critical points of R contained in V (t_{0}) belong to C. Indeed, let c_{1}, c_{2}, …, c_{n} denote the critical points of R. If c_{k} ∉ C for some k, there exists a regular value t_{k} of g such that c_{k} ∉ V (t_{k}). Let t_{0} be a regular value of g with 0 < t_{0} ≤ min{t_{j} : c_{j} ∉ C}. Now V (t_{0}) ⊂ ⋂_{c_{j} ∉C} V (t_{j}), and we conclude that all the critical points of R contained in V (t_{0}) belong to C.

Choose t_{0} as above and such that for any z ∈ D which is a zero of the gradient of g we have R(z) ∉ L_{t_{0}} .

Let γ_{2} be the component of L_{t_{0}} whose interior contains the Cremer point z_{0} , and let U_{2} be the Jordan domain inside γ_{2} (so that C ⊂ U_{2} ). We claim that R(C) = C, and so we may let U_{1} denote the component of R^{−1}(U_{2}) that contains C, and γ_{1} denote its boundary. Then γ_{1} is mapped by R onto γ_{2} and outside γ_{1} , for if g(R(z)) = t_{0} then g(z) < g(R(z)), so g(z) < t_{0} . Thus U_{1} and U_{2} are Jordan domains containing C such that \overline{U}_{1} ⊂ U_{2} .

We still need to verify that R(C) = C. It is easy to see that R(C) ⊂ C. Indeed, first note that R(C) is connected, since C is connected and R is continuous. So R(C) is a connected subset of A, C is the connected component of A containing the Cremer fixed point z_{0} , R(z_{0}) = z_{0} , hence R(C) ⊂ C. Now we show that C ⊂ R(C). Recall that

C = ⋂_{t>0 regular for g} V (t),

so C is a compact connected subset of \overline{C}. Then we know that R^{−1}(C) has at most d components, and each is mapped by R onto C (see [2]). Say R^{−1}(C) = C_{1} ∪ C_{2} ∪ ⋯ ∪ C_{l} so that R(C_{j}) = C for all j with 1 ≤ j ≤ l. We know that R(C) ⊂ C, and so by applying R^{−1} we get

C ⊂ R^{−1}(R(C)) ⊂ R^{−1}(C) = C_{1} ∪ C_{2} ∪ ⋯ ∪ C_{l} ⊂ A.

Since C is connected, we get that C ⊂ C_{k} ⊂ A, for some k. But C is a connected component of A, so we must have C = C_{k}, and hence R(C) = C.

The proper holomorphic map

R: U_{1} ⟶ U_{2}, U_{1} ⟶^{onto} U_{2}, ∂U_{1} ⟶ ∂U_{2}

is a k -fold branched covering. Here k is the degree of R as a covering map of C onto C, all other inverse images of C being excluded.

Note that R: U_{1} ⟶ U_{2} is not a conformal isomorphism. Assuming R is a conformal isomorphism of U_{1} onto U_{2} , the Denjoy–Wolff fixed point theorem ([15]) applied to the holomorphic map S = R^{−1}: U_{2} ⟶ U_{1} shows that there exists a unique Denjoy–Wolff fixed point α ∈ U_{2} , with S(α) = α and |S′(α)| < 1. Then α belongs to \overline{U}_{1} , and since S maps U_{1} into a smaller domain U_{3} with \overline{U}_{3} ⊂ U_{1} , we have α ∈ U_{1} . The Denjoy–Wolff theorem states the uniqueness of the fixed point α in U_{1} . But R and so S has a Cremer fixed point z_{0} in U_{1} , and thus we must have α = z_{0} , a contradiction since z_{0} is a Cremer fixed point of R and α is a repelling fixed point of R. Hence, R is not a conformal isomorphism.

The above remark forces deg(R | U_{1}) ≥ 2, so (R; U_{1}, U_{2}) is a polynomial-like map of degree at least two.

Let

K_{(R;U_{1},U_{2})} = ⋂_{n≥0} R^{−n}(U_{1})

be the filled Julia set of (R; U_{1}, U_{2}).

Note that C ⊂ K_{(R;U_{1},U_{2})} , since R(C) = C. As we have remarked before, the critical points of the polynomial-like map (R; U_{1}, U_{2}) are contained in C, therefore they never escape from U_{1} (c ∈ C, R(C) = C ⇒ R^{n}(c) ∈ C ⊂ U_{1} , for all n ≥ 0). This means that all the critical points of (R; U_{1}, U_{2}) belong to K_{(R;U_{1},U_{2})} . Therefore the set K_{(R;U_{1},U_{2})} is connected (by Proposition 2 in [4]). It follows that

C = K_{(R;U_{1},U_{2})},

since C ⊂ K_{(R;U_{1},U_{2})} ⊂ A and C is a connected component of A.

We now extend Kiwi’s Theorem 3.4 to obtain the following result.

Theorem 4.2. Let R be a rational function of degree at least two with a completely invariant (super)attracting Fatou component. Assume that R has a Cremer fixed point that is approximated by small cycles. Then R has a critical point which is not accessible from the complement of the Julia set.

Proof. Just as in the Theorem 4.1, let C be the connected component of A = \overline{C} \ D that contains the Cremer fixed point z_{0} . It follows from the proof of Theorem 4.1, that R(C) = C, and that we can extract a polynomial-like map (R; U_{1}, U_{2}). After straightening (R; U_{1}, U_{2}) using the Douady–Hubbard theorem, we obtain a polynomial Q and a homeomorphism ϕ of C such that ϕ ∘ R = Q ∘ ϕ on \overline{U}_{1} and K_{Q} = ϕ(C). Then K_{Q} is connected, hence the Julia set J_{Q} = ∂K_{Q} = ϕ(∂C) is connected.

Every neighborhood U ⊂ U_{1} of z_{0} contains infinitely many cycles of R, hence every neighborhood of ϕ(z_{0}) contains infinitely many cycles of Q. Thus, ϕ(z_{0}) is a Cremer fixed point of Q that is approximated by small cycles. By Theorem 3.1 for polynomials with connected Julia sets, we find a critical point c ∈ J_{Q} of Q that is not accessible from C \ J_{Q}. Since c ∈ J_{Q} = ϕ(∂C), it follows that

Non-accessible critical points of rational functions with Cremer points 11

ϕ^{−1}(c) ∈ ∂C ⊂ J_{R}, so ϕ^{−1}(c) is a critical point of R that belongs to ∂C ⊂ J_{R}. Also, paths in C\J_{Q} correspond under ϕ^{−1} to paths in C\∂C. Thus, the critical point ϕ^{−1}(c) of R is not accessible from C \ ∂C ⊃ C \ J_{R}.

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Received 14 January 2003