Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 3–34

ATTRACTING DYNAMICS OF EXPONENTIAL MAPS

Dierk Schleicher

International University Bremen, School of Engineering and Science Postfach 750 561, D-28725 Bremen, Germany; dierk@iu-bremen.de

Abstract. We give a complete classification of hyperbolic components in the space of iterated exponential maps z ↦ λ exp(z), and we describe a preferred parametrization of those components. More precisely, we associate to every hyperbolic component of period n a finite symbolic sequence of length n − 1, we show that every such sequence is realized by a hyperbolic component, and the hyperbolic component specified by any such sequence is unique. This leads to a complete classification of all exponential maps with attracting dynamics, which is a fundamental step in the understanding of exponential parameter space.

1. Introduction

This paper is part of the program to describe the dynamics of exponential maps λ exp and the structure of parameter space, in the spirit of the well-developed body of knowledge about polynomial dynamics. The polynomial theory was pioneered by Douady and Hubbard [DH1] who systematically investigated the Mandelbrot set as the simplest non-trivial example of a holomorphic parameter space. Since then, there has been a lot of further work in this field, much of it based on methods and results developed initially for the Mandelbrot set. Among transcendental entire maps, the exponential family λ exp stands out as the simplest family. It is expected that a good understanding of this space of maps will guide the way for further progress on a study of more classes of entire maps.

For holomorphic spaces of rational maps, such as the space z ↦ z^{2} + c of quadratic polynomials, it is known from the work of Mañe, Sad and Sullivan [MSS] that the set of structurally stable maps is open and dense: a map is structurally stable if it has a neighborhood in which all maps are topologically (and even quasiconformally) conjugate. The complementary locus is called the bifurcation locus within this space of maps; it is closed and nowhere dense. An investigation of a space of rational maps thus starts with a description of connected components in the space of structurally stable maps, called stable components. In most spaces of rational maps (those in which the bifurcation locus is non-empty and not every map has an indifferent orbit), all known stable components consist of hyperbolic maps: there is a uniformly expanding metric in a neighborhood of the Julia set, which is equivalent to the fact that all critical points converge to attracting or superattracting cycles [M2]. It is conjectured that stable components are always hyperbolic; unfortunately, this has not yet been confirmed for any space.

2000 Mathematics Subject Classification: Primary 30D05, 33B10, 37B10, 37C15, 37C20, 37F10, 37F20, 37F45.

One reason why quadratic polynomials are the simplest non-trivial rational maps is because they have only one critical point (except for the fixed point ∞) and of lowest multiplicity; critical points determine the dynamics of rational maps to a large extent. For the space of quadratic polynomials, the bifurcation locus is the boundary of the Mandelbrot set. Douady and Hubbard [DH1] have developed a complete conjectural description of the topology of the Mandelbrot set and its boundary, and they showed that it is a true description if and only if the Mandelbrot set is locally connected. This would imply that every stable component was hyperbolic. They also provided a complete classification of hyperbolic components as part of the complete description of the topology. The importance of the study of the space of quadratic polynomials stems not only from the fact that they are the simplest class of rational maps, but also because renormalization theory [DH2] shows that quadratic polynomials have universal properties; as a consequence, every non-trivial bifurcation locus in spaces of rational maps contains infinitely many homeomorphic copies of the Mandelbrot set [Mc3] (or of Multibrot sets, which are the analogues for maps z^{d} + c with d > 2).

It will be a long way to establish analogous results for transcendental maps, or even to find out which results have analogues and in which sense. The fact that structurally stable maps are dense in many spaces of transcendental entire maps has been established by Eremenko and Lyubich [EL2]; this includes the space of exponential maps. The decisive role of critical points (or critical values) for rational maps is, for transcendental maps, assumed by either critical values or asymptotic values, which are jointly known as singular values. Exponential maps λ exp have only one singular value of the simplest kind: they have no critical values, 0 is the only asymptotic value, and every λ exp is a universal cover C → C \ {0}. This makes them good candidates for prototypes of transcendental maps. Probably for this reason, the space of exponential maps has been studied more than any other space of transcendental maps. An exponential map will be called hyperbolic if it has an attracting orbit (which necessarily attracts the singular orbit, so there can be at most one attracting orbit). A structurally stable component is called hyperbolic if it consists of hyperbolic maps (this is a slight abuse of notation: hyperbolic dynamics in a strict sense would require a uniformly expanding metric in a neighborhood of the Julia set, but the Julia set is never compact for transcendental maps).

The description of the exponential parameter space was begun in the 1980’s by Baker and Rippon [BR], by Eremenko and Lyubich [EL1], [EL2], [EL3], and by Devaney, Goldberg and Hubbard [DGH]. These papers discuss certain fundamental properties of hyperbolic components and of bifurcations (in the case of [EL1], [EL2], [EL3] as an example of a study of more general entire maps), but a description of the global structure of parameter space was in terms of pictures and conjectures. Eremenko and Lyubich conjectured that every structurally stable exponential map is hyperbolic, so the union of the hyperbolic components would be dense in λ-space.

For a hyperbolic exponential map, the Julia set has measure zero [EL1], [EL3] but Hausdorff dimension two [Mc2]. There are some results on the topology, in particular if the attracting orbit has period one [AO]. It seems possible to give a more complete description of the topology for exponential maps with attracting orbits of arbitrary periods.

In this paper, we give a complete description of the space of hyperbolic exponential maps. This was part of Chapter III of the author’s habilitation thesis [S1] (of May, 1999) which developed a description of the exponential parameter space in analogy to Douady and Hubbard’s Orsay Notes [DH1] about the Mandelbrot set. An earlier version of this paper was circulated as [S3].

The bifurcation locus of exponential maps is not locally connected. It would be interesting to have a topological criterion (analogous to local connectivity of the Mandelbrot set) which would imply the validity of the conjecture of Eremenko and Lyubich. It seems possible that this could be done in terms of fibers as discussed in [S2]. There are two more conjectures in [EL2] which have now been established [S1], [RS]; they are explained at the end of Section 7.

The set of parameters λ for which there is a (necessarily unique) attracting periodic orbit is clearly open; connected components where this happens are hyperbolic components. The period of the attracting orbit is constant throughout the component. Our object is to classify hyperbolic components in the λ parameter plane, where λ ranges over C\{0}. It is known [BR], [EL2], [DGH] that all exponential maps with attracting orbits of period 1 are contained in a single hyperbolic component which is bounded in C; it contains a neighborhood of 0. For period 2, there is a unique hyperbolic component which is contained in a left half plane and unbounded to the left; see also Section 2. All other hyperbolic components have period 3 or more, and are unbounded to the right. Every hyperbolic component is simply connected, except that the period 1 component is punctured at 0.

Here is our main result; it is illustrated in Figure 1.

Theorem 1.1 (Classification of hyperbolic components). For every period n ≥ 3, there are countably many hyperbolic components in the space of exponential maps λ exp. Each of them is characterized by a sequence

s_{1}, s_{2}, …, s_{n−1}

(its “intermediate external address”), where s_{1} , s_{2}, …, s_{n−2} ∈ Z with s_{1} = 0, and s_{n−1} ∈ (Z+ \frac{1}{2}). Conversely, every such sequence is realized by a unique hyperbolic component of period n. These hyperbolic components have a natural vertical order in which they stretch out to +∞ along bounded imaginary parts, and this order is the same as the lexicographic order of the corresponding sequences s_{1}, s_{2}, …, s_{n−1} .

Figure 1. The space of parameters λ for exponential maps λ exp, with hyperbolic components indicated in white. Various hyperbolic components are labeled by their intermediate external addresses, or briefly by their periods (in parentheses). The picture has kindly been contributed by Jack Milnor: for every pixel, an approximate test is performed whether or not the corresponding map λ exp has an attracting orbit (with λ at the center of the pixel); in addition, the boundaries of hyperbolic components have been emphasized in order to show their shapes more clearly.

In particular, between any pair of consecutive hyperbolic components of period n, there are infinitely many hyperbolic components of period n + 1, ordered like Z.

The numbers s_{1}, s_{2}, …, s_{n−1} characterizing any hyperbolic component of period n have a dynamic meaning as follows. Let λ be any parameter in the given period n hyperbolic component, and let

U_{1} →^{≈} U_{2} →^{≈} ⋯ →^{≈} U_{n} → U_{1}

be the unique cycle of periodic Fatou components for λ exp, where 0 ∈ U_{1} , where λ ∈ U_{2} , and where U_{n} contains a left half plane. Here λ exp: U_{n} → U_{1} \ {0} is a universal cover, and all other λ exp: U_{k} → U_{k+1} are conformal isomorphisms.

Attracting dynamics of exponential maps 7

Figure 2. The same space as in Figure 1, drawn differently by a special purpose program of Günter Rottenfußer: this program traces out the boundaries of hyperbolic components, which is possible with arbitrary precision for any given hyperbolic component. Unlike for the Mandelbrot set, in the exponential case it is impossible to test whether the singular orbit “escapes to ∞”; instead, in pixel images it is usually tested whether the singular orbit survives some fixed number N of iterations without producing numbers too large to store. This is quite different from the existence of an attracting orbit for the given value of λ, and logically independent. This picture confirms that pixel test pictures like in Figure 1 are approximately correct.

Define the horizontal lines

L(s_{k}) := {z ∈ C : Im(z) = 2πs_{k} − c};

here c = Im(log(λ)), choosing the branch with |c| < π. Then for k = 1, …, n−1, the component U_{k} contains a curve which is asymptotic to L(s_{k}) as Re(z) → +∞ and which maps to L(s_{n−1}) under (λ exp)^{∘(n−1−k)} . Thus s_{k} specifies precisely which branch of (λ exp)^{−1} carries U_{k+1} to U_{k}.

Similarly, in the λ parameter plane, if n > 3 then the points in the hyperbolic component are asymptotic to the line

Im(λ) = 2πs_{2} as Re(λ) → +∞,

while for n = 3 they form a neighborhood of this line near +∞.

Furthermore, if H_{1} and H_{2} are hyperbolic components of any periods greater than 2, then H_{1} lies above H_{2} if and only if its symbol sequence is greater, using lexicographic ordering.

Our results are easily translated into κ = log(λ)-space: then Theorem 1.1 holds for n ≥ 2, and the condition s_{1} = 0 is lifted. However, κ-space is not a true parameter space: every exponential map is represented countably often; adding a constant integer to every intermediate external address in κ-space yields the same map with a different branch of κ.

In Section 2, we review necessary properties about exponential maps and state results from earlier papers. In particular, we introduce dynamic rays. Then, in Section 3, we give a combinatorial coding to every hyperbolic component in terms of “intermediate external addresses”, and we show that each intermediate external address is realized by at least one hyperbolic component. The at least one is strengthened to exactly one in Section 4 using a variant of spider theory. This finishes the classification of hyperbolic components. The second main result (in Section 7) constructs, for every hyperbolic component, a preferred parametrization (which even extends to the boundary). The main difficulty is in breaking the symmetry and fixing an origin of the parametrization, which is accomplished using a “dynamic root” of every periodic Fatou component: this is a boundary point which is fixed under the first return map of the component and which is the landing point of at least two periodic dynamic rays. Existence and uniqueness of dynamic roots is shown in Section 6, while Section 5 provides the necessary combinatorial properties of periodic dynamic rays landing at a common point: whenever three or more periodic rays land at a common point, then the dynamics permutes all these rays transitively, and two rays in this orbit are singled out as “ characteristic rays”. We conclude this paper with a discussion of further results on hyperbolic components, in particular their boundary properties.

Some notation. We write our exponential maps as z ↦ E_{λ}(z) := λe^{z} = exp(z + κ) with λ = exp(κ), where λ ∈ C \ {0} and κ ∈ C; usually we will use the branch |Im(κ)| ≤ π. We will often need F (t) = e^{t} −1, in particular for t ∈ R. Let D := {z ∈ C : |z| < 1} and D^{∗} := D\{0}. We write that a curve or sequence in C converges to +∞ or to −∞ to indicate that the real parts converge to ±∞, while the imaginary parts are bounded.

Acknowledgments. I would like to thank Misha Lyubich and Jack Milnor for many helpful and inspiring discussions, and to the Institute for Mathematical Sciences at Stony Brook for continued support and encouragement. I am also grateful for several helpful comments from the audience at various seminars where these results were presented, in particular in Stony Brook (spring 1999 and spring 2000) and at Boston University (spring 1999). Helpful and constructive suggestions by Lasse Rempe and by an anonymous referee are also gratefully acknowledged.

2. Exponential dynamics

In this section, we will review known properties of exponential dynamics. The map E_{λ} = λ exp has no critical points or critical values, and a unique omitted value 0 also known as singular value. The singular value plays an equally decisive role for exponential dynamics as the critical values do for polynomial dynamics.

Points z with E_{λ}^{∘k}(z) → ∞ as k → ∞ are known as escaping points; they are completely classified [SZ2]: if the singular value itself does not escape, then the escaping points are on disjoint curves called dynamic rays (or hairs) labeled by external addresses \underline{s} = s_{1}s_{2}s_{3} …, which are infinite sequences over Z (there is a well-understood exception if the singular value does escape; that case does not matter for our purposes). The dynamic ray at external address \underline{s} is an injective curve g\underline{s}: ]t\underline{s}, ∞[ → C with Re(g\underline{s}(t)) → +∞ as t → ∞, while Im(g\underline{s}(t)) is bounded. The quantity t\underline{s} ≥ 0 depends on \underline{s} in a well-understood way; we only need bounded sequences \underline{s}, and those have t\underline{s} = 0. We say that a ray g\underline{s} lands at a point w ∈ C if lim_{t↘t\underline{s}} g\underline{s}(t) exists and is equal to w. A ray tail is an unbounded subcurve of a ray: it is a curve g\underline{s}( [τ, ∞[ ) for τ > t\underline{s}.

Every point on a dynamic ray is an escaping point, and every escaping point is on such a ray, or the unique limit point of such a ray. We have the dynamic relation

E_{λ}(g\underline{s}(t)) = g_{σ(\underline{s})}(F (t))

where σ is the shift map on external addresses, dropping the first entry. The meaning of the external address of a ray is the following: the set E_{λ}^{−1}(R^{−}) is a countable union of horizontal lines, spaced at distance 2πiZ, and C\E_{λ}^{−1}(R^{−}) are horizontal strips, labeled by Z so that the strip with label 0 contains the singular value 0 (perhaps on its boundary). Then at least for sufficiently large t > t\underline{s}, the external address \underline{s} of g\underline{s} is the sequence s_{1}s_{2}s_{3} … of strips visited by the orbit of g\underline{s}(t). Not all possible sequences are allowed; the set of allowed sequences is completely understood: it consists of sequences satisfying a certain exponential growth condition [SZ2], and in particular it contains all bounded sequences.

If an exponential map has an attracting periodic point, then the singular value is in a periodic Fatou component which we call the characteristic Fatou component. All periodic orbits, except the unique attracting one, are repelling. We will need a construction and results from [SZ3, Section 4.3]: let n ≥ 2 be the period of the attracting orbit, let U_{1}, U_{2}, …, U_{n} = U_{0} be the cycle of periodic Fatou components, labeled cyclically modulo n so that U_{1} is the characteristic Fatou component, and let a_{1}, a_{2}, …, a_{n} be the attracting periodic orbit labeled so that a_{k} ∈ U_{k} for all k. Let V_{n+1} be a closed neighborhood of a_{1} corresponding to a disk in linearizing coordinates, large enough so as to contain the singular value in its interior. For k = 0, 1, …, n, let

V_{k} := {z ∈ U_{k} : E_{λ}^{∘(n+1−k)}(z) ∈ V_{n+1}}

and V := V_{0} ∪ V_{1} ∪ ⋯ V_{n+1} . Then V_{n} contains a left half plane, and for k ∈ {n−1, n−2, …, 1}, V_{k} contains a neighborhood of a curve towards +∞ with V_{1} ⊃ V_{n+1} , while V_{0} contains neighborhoods of infinitely many such curves towards +∞, spaced equally at integer translates of 2πi. The construction assures that E_{λ}(V) ⊂ V and that all V_{k} are connected and simply connected.

Let R := C \ V_{0} ; it consists of countably many connected components which we will call “regions” R_{u} : let R_{0} be the region containing the singular value and R_{u} := R_{0} +2πiu, for u ∈ Z. Then R = ⋃_{u} R_{u} . Any orbit (z_{k}) within the Julia set then has an associated itinerary u_{1}, u_{2}, u_{3}, … such that z_{k} ∈ R_{u_{k}} for all k. We should emphasize that this itinerary is different from the external address used for example in the construction of dynamic rays: the external address is constructed using inverse images of the negative real axis, which is dynamically not a natural concept. The itineraries as defined here are dynamically natural; compare [SZ3, Sections 4 and 5] for a discussion of the differences. (Note the different fonts for external addresses s_{1}s_{2} … and itineraries u_{1}u_{2} ….)

For exponential dynamics with an attracting periodic orbit, every periodic dynamic ray lands at a repelling periodic point [SZ3, Theorem 3.1]; every repelling periodic point is the landing point of a finite positive number of periodic rays [SZ3, Theorem 5.4]; and a periodic ray lands at a periodic point if and only if ray and point have identical itineraries [SZ3, Proposition 4.5]. In particular, different periodic points have different itineraries, while different periodic rays have the same itinerary if and only if they land together.

The following results are known from [EL2], [EL3], [BR]: in λ-space, there is a unique hyperbolic component H_{1} of period 1 which is a bounded neighborhood of the puncture λ = 0 of parameter space; it comes with a conformal isomorphism D^{∗} → H_{1} , μ ↦ μ exp^{−μ} (so that λ exp has a fixed point with multiplier μ if and only if λ = μ exp^{−μ}). All λ ∈ C^{∗} with |λ| < 1/e have λ ∈ H_{1} . There is a unique period 2 component H_{2} which “almost” occupies a left half plane (in the sense that for every ϑ ∈]π/2, 3π/2[ , there is an R > 0 such that for all r > R, the parameter λ = r exp(iϑ) ∈ H_{2} ). Every hyperbolic component H of period n ≥ 2 is simply connected, and the multiplier map μ: H → D^{∗} is a universal cover.

Lemma 2.1 (Strong attraction only at far parameters). For every period n ≥ 2 and r < 1, there is an R > 0 such that any parameter λ for which there is an attracting orbit of multiplier |μ| ≤ r has |λ| > R.

For every hyperbolic component H of period n ≥ 2 and λ ∈ H, there is a unique homotopy class of curves γ( [0, ∞] ) → H ∪ {∞} with γ(0) = λ and γ(∞) = ∞ such that μ(γ(t)) → 0 as t → ∞ (homotopy with fixed endpoints).

Proof. If a_{1}, a_{2}, …, a_{n} is any periodic orbit of period n under λ exp, then (E_{λ}^{∘n}(a_{m}))′ = ∏_{k} a_{k}. If the orbit is attracting, then there is some |a_{k}| < 1. Hence if |λ| ≤ R, then all |a_{k}| < ξ for some ξ depending on R and n. Now Re(a_{k}) ≥ −ξ implies that |a_{k+1}| ≥ |λ| exp(−ξ) ≥ exp(−ξ)/e (if |λ| < 1/e, then λ exp has an attracting fixed point). Hence we have a lower bound for all |a_{k}| and hence for |μ|.

It follows that any curve γ′: ( [0, ∞[ ) → D^{∗} with lim_{t→∞} γ′(t) = 0 lifts under any branch of the inverse multiplier map to a curve γ: ( [0, ∞[ ) → H with lim_{t→∞} γ(t) = ∞. Conversely, any two curves γ_{1}, γ_{2}: ( [0, ∞[ ) → H which limit at ∞ and with μ(γ_{1,2}(t)) → 0 as t → ∞ project under μ to two curves γ′_{1}, γ′_{2}: [0, ∞[ → D^{∗}, and these are homotopic in D^{∗}; hence γ_{1} and γ_{2} are also homotopic in H (different branches of μ^{−1} can be compensated by loops around 0 in D^{∗} and do not matter).

Remark. This lemma does not show that any two curves γ_{1,2}: ( [0, ∞] ) → H ∪ {∞} with γ_{i}(∞) = ∞ are homotopic if the condition μ(γ_{1,2}(t)) → 0 is dropped: it would even be conceivable that, for some ϑ ∈ R, a branch of μ^{−1}( ]0, e^{iϑ}[ ) (an internal parameter ray, Definition 4.5) tends to ∞ as |μ| → 1. It was conjectured by Eremenko and Lyubich [EL2] that this does not happen. The proof of this is not easy: see [S1, Section V] and [RS].

3. Classification of hyperbolic components

Hyperbolic components of Multibrot sets have the helpful property that they have a unique “center” in which the dynamics is postcritically finite [Mc1], [M1], [ES]. If an exponential map has an attracting orbit, it can never be postsingularly finite (with multiplier 0); the center of hyperbolic components “is at ∞” (in the sense of Lemma 2.1). Fairly enough, it turns out that hyperbolic components of exponential maps have a different feature unknown to the polynomial case: since they stretch out to +∞ like parameter rays, they can be described by a slight generalization of external addresses: we need finite sequences of integers, followed by a half-integer.

Definition 3.1 (Intermediate external address). An intermediate external address of period n ≥ 2 is a finite sequence s_{1}s_{2} … s_{n−2}, s_{n−1} with s_{k} ∈ Z for k ≤ n − 2 and s_{n−1} ∈ (Z + \frac{1}{2}).

The lexicographic order on external addresses (infinite sequences over Z) extends naturally to intermediate external addresses such as \underline{s} = s_{1}s_{2} … s_{n−1} . Intermediate external addresses of period n (which consist of n − 1 numbers) label hyperbolic components of period n.

As usual, we start with a dynamic consideration.

Definition 3.2 (Attracting dynamic ray). Consider an exponential map E_{λ} with an attracting orbit of period n ≥ 2 and let \underline{s} = s_{1}s_{2} … s_{n−1} be an intermediate external address of period n. As always, let a_{1} be the attracting periodic point in the characteristic Fatou component U_{1} . We say E_{λ} has an attracting dynamic ray at external address \underline{s} if there is a curve γ: [0, ∞[ → U_{1} such that the following hold:

– γ(0) = a_{1} ;

– lim_{t→∞} E_{λ}^{∘k}γ(t) = +∞ for k = 0, 1, …, n − 2;

– lim_{t→∞} E_{λ}^{∘(n−1)}γ(t) = −∞;

– every dynamic ray at an external address \underline{s}′ < \underline{s} is below γ;

– every dynamic ray at an external address \underline{s}′ > \underline{s} is above γ.

Remark. Since γ is in a Fatou component, it must be disjoint from every dynamic ray. Both γ and any given dynamic ray tend to +∞ at bounded imaginary parts, so the ray must be above or below γ in the following sense: for real ξ sufficiently large, γ cuts the half plane {z ∈ C : Re(z) > ξ} into two unbounded parts, one above and one below γ, and every dynamic ray must tend to +∞ within one of these two parts.

Lemma 3.3 (Attracting dynamics has external addresses). For every exponential map E_{λ} with an attracting orbit of period n ≥ 2, there is a unique intermediate external address \underline{s} = s_{1}s_{2} … s_{n−1} of period n such that there is an attracting dynamic ray at external address \underline{s}. Every exponential map from the same hyperbolic component has an attracting dynamic ray at the same external address \underline{s}.

Proof. The periodic Fatou component U_{0} = U_{n} contains a left half plane. By simple connectivity, it contains a unique homotopy class of curves connecting the attracting periodic point a_{n} to −∞ eventually within a left half plane, even eventually along R^{−}. This homotopy class of curves can be pulled back n−1 steps to a preferred homotopy class of curves within U_{1} connecting a_{1} to +∞. Choose one such curve γ ⊂ U_{1} . This curve avoids dynamic rays, and it is easy to check that the supremum of external addresses of dynamic rays below γ has well-defined n−1 initial entries in Z: the curve γ and its first n−2 iterates tend to +∞ (with bounded imaginary parts), so the first n − 1 entries in the supremum are just the labels of the strips containing the iterates of γ (with respect to inverse images of R^{−} used in the construction of external addresses). Similarly, the infimum of external addresses of dynamic rays above γ supplies n − 1 well-defined first entries which differ from the lower external addresses only in the last entry and only by one. It is easy to confirm that the external address does not depend on the choice of γ or on the parameter chosen from the hyperbolic component of E_{λ} (the intermediate external address is a discrete object which depends continuously on the parameter and on γ).

We thus have a combinatorial coding for every hyperbolic component (the unique component of period 1 is coded by the empty sequence), and our goal is to show that each coding is realized by exactly one hyperbolic component. This will be done in Theorem 3.5 (existence) and Corollary 4.4 (uniqueness).

First we need a lemma to prove the existence of an attracting orbit.

Lemma 3.4 (Singular orbit in horizontal strip). Suppose that for some parameter λ there is a real number h > 3 such Re(λ) > h and the initial segment z_{1} = 0, z_{2} = λ, …, z_{n} of the singular orbit has the property that |Im(z_{k})| < h for 1 ≤ k ≤ n. Suppose moreover that z_{n} is the first point on the singular orbit with Re(z_{n}) < 0. Then the map E_{λ} has an attracting periodic orbit of exact period n, and the attracting basin contains the left half plane Re(z) ≤ Re(z_{n}) + 1. As Re(λ) → ∞ with fixed height h of the strip, the multiplier tends to 0.

The proof needs a couple of unpleasant calculations, but its idea is very simple: the geometry of the strip containing the singular orbit assures that absolute values of orbit points are dominated by the real parts, and the real parts grow exponentially. Once the orbit reaches a point with negative real part, its absolute value dominates the remaining orbit by far, and its image is extremely close to 0. The contraction coming from the exponential map at this point is far greater than the expansion along the previous orbit, starting at the singular value 0. Therefore, any sufficiently small disk around 0 will map after n steps to a much smaller (almost-) disk close to the origin. In order to map this disk into itself, its size has to be chosen so that it is neither too large (or we would lose control in the estimates) nor too small (or it would not contain the images after n steps). It turns out that things work if we choose the disk so that its image at z_{n} has radius 1.

Proof. The points z_{2}, …, z_{n−1} of the orbit are contained within the strip S := {z ∈ C : |Im(z)| < h} at positive real parts. We show that they all have real parts greater than h. Indeed, this is true for z_{2} = λ by assumption, and for the others it follows by induction using |λ| > h > 1: |z_{k}| = |λ| exp(Re(z_{k−1})) > he^{h}, so Re(z_{k}) > h for k = 2, …, n − 1 and Re(z_{n}) < −h.

Now we show for m ≤ n

(1)  ∏_{k=2}^{m}(|z_{k}| + 1) < (|z_{m}| + 1)^{2}.

Indeed, for m = 1 the empty product on the left equals 1, while Re(z_{1}) > 3 by assumption. For the inductive step, we only need to prove (|z_{m}|+1)^{2} < |z_{m+1}|+1. We will use the inequality (\sqrt{2} x + 1)^{2} < 3 exp(x) for all x ≥ 2 and estimate for Re(z_{m}) > 0 as follows:

(|z_{m}| + 1)^{2} < (\sqrt{2} Re(z_{m}) + 1)^{2}
< 3 exp(Re(z_{m})) < |λ| | exp(z_{m})| < |z_{m+1}| + 1.

Our next claim is about z_{n}, the first point with negative real part:

(2)  e|λ| exp(Re(z_{n})) < (|z_{n}| + 1)^{−2}.

Indeed, we have |z_{n}| = |λ| exp(Re(z_{n−1})) > he^{h} > 3 exp(3) > 60, thus |Re(z_{n})| > he^{h}/\sqrt{2} > 40 and |z_{n}| + 1 < \sqrt{2} |Re(z_{n})|. Using the inequality 2ex^{3} < e^{x} for all x ≥ 8, it follows

e|λ|(|z_{n}| + 1)^{2} < 2e|λ||Re(z_{n})|^{2} ≤ 2e|Re(z_{n})|^{3} < exp(|Re(z_{n})|).

Since the real part of z_{n} is negative, the claim follows.

Now we can start the actual proof of the lemma. Let D_{n} be the open disk of radius 1 around the point z_{n}. Pulling back by the dynamics, we obtain open neighborhoods D_{n−1} around z_{n−1}, …, D_{1} around z_{1} = 0. These pull-backs are contracting at every step: the derivative of E_{λ} at a point z′_{k} ∈ D_{k} is equal to E_{λ}(z′_{k}), and its absolute value is bounded above by |z_{k+1}|+1 (using the inductive fact that dist(z_{k}, ∂D_{k}) ≤ 1). The inverse map is thus contracting with contraction factor at most 1/(|z_{k+1}|+1), and the domain D_{1} contains a disk around the origin with radius at least ϱ = ∏_{k=2}^{n}(|z_{k}| + 1)^{−1} > (|z_{n}| + 1)^{−2} by equation (1) above (since we have a bound on the contraction for every point on the disks, the claim follows without invoking Koebe’s theorems).

On the other hand, all the points in D_{n} are contained in the left half plane Re(z) ≤ Re(z_{n}) + 1. For every point z in this left half plane, |E_{λ}(z)| ≤ |λ| exp(Re(z_{n}) + 1) = e|λ| exp(Re(z_{n})) < ϱ by equation (2) above. It follows that E_{λ}(D_{n}) ⊂ D_{1} and hence E_{λ}^{∘n}(D_{1}) ⊂ D_{1} . This is a proper inclusion, so there is an attracting orbit of period at most n. The period clearly cannot be smaller than n. If within the same strip with imaginary parts bounded by h, Re(λ) becomes large, the size of the image of D_{n} within D_{1} gets much smaller compared to the size of D_{1} , and the multiplier tends to 0.

Now we come to the existence theorem. We restrict to periods n ≥ 3 because the hyperbolic components of periods 1 and 2 are completely classified: the unique component of period 1 in λ-space is coded by the empty external address, and the component of period 2 is coded by the address \underline{s} = 0 of length 1.

Theorem 3.5 (Existence of hyperbolic components). For every n ≥ 3 and every intermediate external address \underline{s} = s_{1}s_{2} … s_{n−1} of period n with s_{1} = 0, there is a hyperbolic component in λ-space in which every exponential map has an attracting dynamic ray at external address \underline{s}.

This hyperbolic component contains an analytic curve tending to +∞ with imaginary parts converging to 2πs_{2} such that along this curve the multipliers of the attracting orbit tend to 0.

Proof. Let s_{n−1}^{±} := s_{n−1} ± \frac{1}{2} ∈ Z and define two periodic external addresses

of period n − 1 via s^{−} := \overline{s_{1}s_{2} … sn−1^{−}} and s^{+} := \overline{s_{1}s_{2} ⋯ sn−1^{+}} . Let A := 1 +

max_{k}{|s_{k}|}.

In [SZ2, Proposition 3.4] (or [SZ1, Theorem 2.3]), the existence of dynamic rays g_{\underline{s}}^{+} , g_{\underline{s}}^{−} was shown: these are curves g_{\underline{s}}^{±}: ]0, ∞[ → C satisfying

(3) g_{\underline{s}}^{±}(t) = t − κ + 2πis_{1} + r_{\underline{s}}^{±}(t) with |r_{\underline{s}}^{±}(t)| < 2e^{−t}(|κ| + 2 + 2πAC′)

for t ≥ 1 + 2 log(|κ| + 3), where C′ < 2.5 is a universal constant. The same statement with the same bound holds also for all σ^{k}(\underline{s}^{±}) (replacing s_{1} by the appropriate entry, of course). In particular, if we let

τ := 1 + 2 log(|κ| + 3 + 2πAC′),

then τ ≥ max{1 + 2 log(|κ| + 3), log 2 + log(|κ| + 2 + 2πAC′)} so that (3) holds and

|g_{\underline{s}}^{±}(t) − (t − κ + 2πis_{1})| = |r_{\underline{s}}^{±}(t)| < e^{−(t−τ)}

for t ≥ τ. After n − 2 iterations, the ray tails g_{\underline{s}}^{±}( [τ, ∞[ ) map to

E_{λ}^{∘(n−2)}(g_{\underline{s}}^{±}( [τ, ∞[ )) = g_{σ^{n−2}(\underline{s}^{±})}( [F^{∘(n−2)}(τ), ∞[ )

with

g_{σ^{n−2}(\underline{s}^{±})}(t) = t − κ + 2πis_{n−1}^{±} + r^{±}(t)

with |r^{±}(t)| < e^{−(t−τ)} ≤ 1. Define a curve

γ′_{κ}: [F^{∘(n−2)}(τ), ∞[ → C via γ′_{κ}(t) = t − κ + 2πis_{n−1};

it has the property that E_{λ}(γ′_{κ}) ⊂ R^{−}. The construction assures that the two ray tails g_{σ^{n−2}(\underline{s}^{±})}( [F^{∘(n−2)}(τ), ∞[ ) are above respectively below γ′_{κ}( [F^{∘(n−2)}(τ), ∞[ ) (asymptotically by iπ), and all three curves are disjoint. Moreover, every dynamic ray g\underline{s}′ with \underline{s}′ < s_{n−1} (that is, s′_{1} < s_{n−1} ) is eventually below γ′, and if \underline{s}′ > s_{n−1} (that is, s′_{1} > s_{n−1} ), then the ray g\underline{s}′ is eventually above γ′.

Pulling back n − 2 times along equal branches of E_{λ}^{−1} on C \ R0^{−}, it follows that there is a curve γ_{κ}: [τ, ∞[ → C between the two rays g_{\underline{s}}^{±}( [τ, ∞[ ) with

(4)  E_{λ}^{∘(n−2)}(γ_{κ}(t)) = γ′_{κ}(F^{∘(n−2)}(t)) = F^{∘(n−2)}(t) − κ + 2πis_{n−1}.

Just like dynamic rays, this curve inherits the bound for t ≥ τ

(5) γ_{κ}(t) = t − κ + 2πis_{1} + r_{κ}(t) with |r_{κ}(t)| < 2e^{−t}(|κ| + 2 + 2πAC′) < e^{−(t−τ)}

(it follows from the construction of γ_{κ} using n − 2 pull-backs that γ_{κ} satisfies asymptotically the same bounds (3) as the two rays g_{\underline{s}}^{±} which surround γ_{k}; this can also be verified in the proof of (3) in [SZ1], [SZ2]).

The curve γ_{κ} clearly satisfies the second and third conditions for attracting dynamic rays; the last two are asymptotically satisfied in the sense that for every \underline{s}′, the ray g\underline{s}′ is above or below γ_{κ} (as needed) for sufficiently large t depending on \underline{s}′. The first condition requires an attracting orbit, which not every exponential map has.

For any R ≥ 0, set κ_{R}^{±} := R + 2πis_{1} ± iπ = R ± iπ and I_{R} := [κR^{−}, κR^{+}]

(a vertical interval of length 2π); then |κ| ≤ R + π for all κ ∈ I_{R}. Set t_{R} := 1+2 log(R+π+3+2πAC′) > τ. For t ≥ t_{R}, the bound (5) implies Im(γ_{κ_{R}^{−}}(t)) < 0 and Im(γ_{κ_{R}^{+}}(t)) > 0. For all κ ∈ I_{R}, we have

Re(γ_{κ}(t_{R})) < t_{R} − R + 1.

Now fix R large enough so that t_{R}−R+1 < 0; this is possible since t_{R} = O(log R). As κ moves from κR^{−} to κR^{+} , by continuity there must be an intermediate value κ^{∗} where γ_{κ^{∗}}(t^{∗}) = 0 for some t^{∗} > t_{R} > τ. The point of this construction is, of course, that κ^{∗} has an attracting periodic orbit of period n with the required properties, at least when R is sufficiently large.

Indeed, with λ^{∗} := exp(κ^{∗}), the first n postsingular points 0, E_{λ^{∗}}(0), …, E_{λ^{∗}}^{∘(n−1)}(0) are in the strip |Im(z)| ≤ 2πA + π + 1: the 2πA comes from the bound |s_{k}| < A; the π is the bound on Im(κ), and the final 1 is the error bound for the rays. The postsingular orbit E_{λ^{∗}}(0) = λ^{∗}, …, E_{λ^{∗}}^{∘(n−2)}(0) has positive real parts, while Re(E_{λ^{∗}}^{∘(n−1)}(0)) ≪ 0. Now if R is large enough, then Lemma 3.4 shows that there is an attracting orbit of exact period n for κ^{∗}, and E_{λ^{∗}}^{∘(n−1)}(γ_{κ^{∗}}( [t^{∗}, ∞[ )) ⊂ U_{n}. Hence γ_{κ^{∗}}( [t^{∗}, ∞[ ) is in the attracting basin with γ_{κ^{∗}}(t^{∗}) = 0, so γ_{κ^{∗}}( [t^{∗}, ∞[ ) ⊂ U_{1} .

Connect γ_{κ^{∗}} to the attracting periodic point a_{1} ∈ U_{1} and call this resulting curve γ: ( [0, ∞[ ) → C. It clearly satisfies the first three conditions for attracting dynamic rays. We argued above that the last two conditions were satisfied at least for large t. But since the curve γ is in the characteristic Fatou component, it is disjoint from all dynamic rays, and it is indeed an attracting dynamic ray at external address \underline{s} = s_{1}s_{2} … s_{n−1} .

By Lemma 3.3, every parameter λ in the hyperbolic component of λ^{∗} has an attracting dynamic ray at external address \underline{s}. This finishes the existence part of the theorem.

To justify the asymptotics, start with large R, hence large t^{∗} > t_{R} = O(log R), and observe that γ_{κ^{∗}}(t^{∗}) = 0 implies, using (5), that κ^{∗} = t^{∗} + 2πis_{1} + r_{κ^{∗}}(t^{∗}) with |r_{κ^{∗}}(t^{∗})| → 0, hence Im(κ^{∗}) → 2πs_{1} = 0. For n = 3, we have λ^{∗} = E_{λ^{∗}}(γ_{κ^{∗}}(t^{∗})) = γ′_{κ^{∗}}(F (t)) = F (t) − κ + 2πis_{2} , hence Im(λ^{∗}) → 2πs_{2} . For n > 3, we argue similarly:

\begin{aligned}λ^{∗} & = E_{λ^{∗}}(γ_{κ^{∗}}(t^{∗})) = g_{σ(\underline{s}^{±})}(F (t^{∗})) + o(1) = F (t^{∗}) − κ^{∗} + 2πis_{2} + o(1) \\ & = F (t^{∗}) − t^{∗} + 2π(s_{2} − s_{1}) + o(1). \end{aligned}

This way, we have shown the existence of a map R ↦ (κ^{∗}, t^{∗}) (for sufficiently large R) with Re(κ^{∗}) = R and γ_{κ^{∗}}(t^{∗}) = 0. Note that E_{λ^{∗}}^{∘(n−1)}(t^{∗}) ∈ R^{−} for all t^{∗} = t^{∗}(R). It is quite easy to check that (∂/∂κ^{∗})(E_{λ^{∗}}^{∘(n−1)}(t^{∗})) → ∞ as t^{∗} → ∞, so the implicit function theorem shows that the graph of R ↦ κ^{∗}(R) is analytic. It follows from Lemma 3.4 that the attracting multiplier tends to 0 as R → ∞.

We know from Lemma 2.1 that every hyperbolic component H has a preferred homotopy class of curves γ: [0, ∞] → H ∪ {∞} with γ(0) ∈ H and γ(∞) = ∞ such that multipliers tend to 0 along this curve. These preferred homotopy classes of curves give a natural vertical order to hyperbolic components, much as the order for dynamic rays: we say that some hyperbolic component is above another hyperbolic component if the corresponding preferred homotopy classes of curves have the appropriate vertical order.

Corollary 3.6 (Relative position of hyperbolic component). The vertical order of hyperbolic components is the same as the lexicographic order of their intermediate external addresses.

Proof. This follows from the previous proof as follows: if \underline{s}′ > \underline{s}′′ are two intermediate external addresses, then there is a periodic external address \underline{s} between them: \underline{s}′ > \underline{s} > \underline{s}′′. In the construction in the proof of Theorem 3.5, the curve γ_{κ} for \underline{s}′ is always above g\underline{s}, while the corresponding curve for \underline{s}′′ is always below g\underline{s}.

Remark. This vertical order can also be expressed in terms of parameter rays [S1], [F]: these are differentiable curves G\underline{s}: (t\underline{s}, ∞) → C with G\underline{s}(t) → +∞ as t → ∞ such that for λ = G\underline{s}(t), the singular value escapes with 0 = g\underline{s}(t); parameter rays are thus disjoint from hyperbolic components. With the notation of the proof of Corollary 3.6, there is a parameter ray G\underline{s}, and the hyperbolic components for \underline{s}′ and \underline{s}′′ are above, respectively below, this ray.

As this paper was being submitted, a manuscript by Devaney, Fagella and Jarque [DFJ] was released which contains the same sufficient condition for the existence of hyperbolic components as in our Theorem 3.5.

4. Uniqueness of the classification

In the following lemma, we construct a curve γ = γ_{−} ∪ {a_{1}} ∪ γ_{+} in the dynamical plane of every exponential map which has an attracting orbit with positive real multiplier. The curve γ_{+} will be used in this section to construct fundamental domains for exponential dynamics, while γ_{−} will be used in Section 7 to construct a “dynamic root”, which helps parametrizing hyperbolic components in a dynamically meaningful fashion.

Lemma 4.1 (Attracting dynamic ray to boundary fixed point). Let E_{λ} be an exponential map which has an attracting periodic orbit with positive real multiplier and with period n ≥ 2. Then there is a proper analytic curve γ: R → U_{1} which contains the orbit of 0 under E_{λ}^{∘n}. This curve is unique up to reparametrization and can be written γ = γ_{−} ∪ {a_{1}} ∪ γ_{+} with two disjoint subcurves γ_{±} which have the following properties:

– γ_{+} is an attracting dynamic ray and contains the orbit of 0 under E_{λ}^{∘n}; it connects a_{1} to +∞;

– γ_{−} starts at a_{1} and lands at some w ∈ ∂U_{1} with E_{λ}^{∘n}(w) = w.

Proof. There is a unique open neighborhood D of a_{1} which corresponds to a round disk in linearizing coordinates around a_{1} and which contains the singular value 0 on its boundary. Let γ_{D} ⊂ D be the curve corresponding to a diameter in linearizing coordinates such that 0 ∈ \overline{γ_{D}}. Then the first return map E_{λ}^{∘n} of U_{1} sends D into itself and γ_{D} into itself. Any proper analytic curve which contains the orbit of 0 under E_{λ}^{∘n} must be an extension of γ_{D} because the orbit of 0 accumulates at a_{1} , so γ is unique if it exists.

The point a_{1} cuts γ_{D} into two radii; let γ′_{+} be the one which ends at 0 and γ′_{−} the other one (then γ_{D} = γ′_{+} ∪ {a_{1}} ∪ γ′_{−}). The set E_{λ}^{∘(−n)}(γ′_{+}) ∩ U_{1} consists of countably many curves; let γ_{+} be the one which extends γ′_{+} : it starts at a_{1} and lands at ∞, running through 0. Since γ′_{+} is differentiable at 0, every branch of E_{λ}^{−1}(γ′_{+}) has bounded imaginary parts, and γ_{+} approaches +∞ along bounded imaginary parts. Then γ_{+} is an attracting dynamic ray: it satisfies the first three conditions of Definition 3.2, and by Lemma 3.3, the intermediate external address of γ_{+} is uniquely determined by the hyperbolic component containing E_{λ}.

There is a unique curve γ_{−} ⊂ U_{1} \ {a_{1}} which extends γ′_{−} and which satisfies E_{λ}^{∘n}(γ_{−}) = γ_{−}: such a curve can be constructed by an infinite sequence of pullbacks, starting at γ′_{−} and always choosing the branch which extends γ′_{−}. Then γ_{−} is analytic and E_{λ}^{∘n}: γ_{−} → γ_{−} is a homeomorphism.

The curve γ_{−} can be parametrized (in many ways) as γ_{−}: R → U_{1} so that E_{λ}^{∘n}(γ_{−}(t)) = γ_{−}(t + 1). We have lim_{t→+∞} γ_{−}(t) = a_{1} . We want to show that lim_{t→−∞} γ_{−}(t) exists in ∂U_{1} . We will use a modification of the known standard proofs for the landing of external dynamic rays of polynomials. Let

U′ := U_{1} \ \overline{⋃_{k≥0} E_{λ}^{∘kn}(0)} and U′′ := E_{λ}^{−(∘n)}(U′) ∩ U_{1}.

Then both domains are open and E_{λ}^{∘n}: U′′ → U′ is a holomorphic covering map, hence a local isometry with respect to the unique normalized hyperbolic metrics of U′′ and U′, and the inclusion U′′ ↪ U′ is a contraction. Therefore, the hyperbolic distance in U′ between any γ_{−}(t − 1) and γ_{−}(t) is less than between γ_{−}(t) and γ_{−}(t+1). By continuity, there is an s > 0 such that the hyperbolic distance in U′ between γ_{−}(t) and E_{λ}^{∘n}(γ_{−}(t)) is at most s, for all t < 0. But since γ_{−}(t) → ∂U_{1} as t → −∞ (points γ_{−}(t) for large negative t need longer and longer to iterate near a_{1} ), and the density of the hyperbolic metric tends to ∞ near ∂U_{1} , it follows that |γ_{−}(t) − E_{λ}^{∘n}(γ_{−}(t))| → 0 as t → −∞. Therefore, any limit point of γ_{−} is either ∞ or a fixed point of E_{λ}^{∘n}. Since the limit set is non-empty and connected, while the set of fixed points is discrete, it follows that γ_{−} lands at a well-defined boundary point of U_{1} which is either ∞ or fixed under E_{λ}^{∘n}. If the landing point is a fixed point of E_{λ}^{∘n}, then the curve of the claim is γ := γ_{−} ∪ {a_{1}} ∪ γ_{+} . All that remains is to show that the curve γ_{−} does not land at ∞.

Suppose that γ_{−}(t) → ∞ as t → −∞. Then also E_{λ}^{∘n}(γ_{−}(t)) = γ_{−}(t − 1) → ∞ as t → −∞. Since γ_{−} ⊂ U_{1} , it follows that E_{λ}^{∘(n−1)}(γ_{−}) ⊂ U_{0} = U_{n} and E_{λ}^{∘(n−1)}(γ_{−}(t)) → ∞ as t → −∞. If we had lim inf Re(E_{λ}^{∘(n−1)}(γ_{−}(t))) < +∞ as t → −∞, then lim inf|E_{λ}^{∘n}(γ_{−}(t))| = lim inf |γ_{−}(t + 1)| < ∞, and the landing point of γ_{−} would be in C. Therefore, Re(E_{λ}^{∘(n−1)}(γ_{−}(t))) → +∞ as t → −∞, and this curve must have bounded imaginary parts because the translates of U_{1} cut {z ∈ U_{n}: Re(z) > 0} into parts with bounded imaginary parts. But then, for t ≪ 0 and Re(γ_{−}(t)) ≫ 0, the entire orbit of γ_{−}(t) must have large real parts, and standard estimates similarly as in Lemma 3.4 show Re(E_{λ}^{∘n}(γ_{−}(t))) ≫ Re(γ_{−}(t)), which is a contradiction: for every t, the point γ_{−}(t) converges to a_{1} under iteration of E_{λ}^{∘n}.

Remark. In fact, it is not difficult to show that γ is a hyperbolic geodesic of U_{1} [S1]: this is easier if the first return dynamics is conjugated to the map M ∘ exp: H^{−} → H^{−}, where H^{−} is the left half plane, exp: H^{−} → D^{∗} is a universal cover and M: D → H^{−} is an appropriate conformal isomorphism.

Using the analytic curve γ from Lemma 4.1, we construct fundamental domains for E_{λ} as follows (provided the attracting multiplier is positive real): there is a subcurve γ′ ⊂ γ which connects the singular value 0 to +∞ (in fact, γ′ = γ_{+} \ (γ′_{+} ∪ {0})); the full inverse image E_{λ}^{−1}(γ′) is a countable collection of analytic curves which connect −∞ to +∞ and which differ by translation by 2πiZ. The connected components of the complement in C of these curves are fundamental domains for E_{λ}; each of them is mapped by E_{λ} conformally onto C \ γ′, and each has bounded imaginary parts because γ′ is differentiable in its endpoint 0.

Theorem 4.2 (Conformal conjugation). Suppose that two exponential maps have attracting orbits of equal period n ≥ 2 with equal positive real multipliers, and both have attracting dynamic rays at the same intermediate external address s_{1}s_{2} … s_{n−1} . Suppose in addition that for both maps, the fundamental domains as constructed above are such that a single fundamental domain contains both the periodic point a_{0} and the singular value. Then both maps are conformally conjugate.

Proof. Step 1. Let E_{λ} and E_{λ′} be two exponential maps satisfying the assumptions of the theorem. Let V_{1} be an open round disk with respect to linearizing coordinates of a_{1} , large enough so as to contain 0 in its interior. For k = 2, 3, …, n+1, let V_{k} ⊂ U_{k} be the domain E_{λ}^{∘(k−1)}(V_{1}), and let V_{0} := E_{λ}^{−1}(V_{1}). Then V_{1} ⊃ V_{n+1} and V_{0} ⊃ V_{n}. Denote the corresponding sets for E_{λ′} as U′_{k} and V′_{k}, where the size of V′_{1} is chosen so that the dynamics on it is conformally conjugate to the dynamics on V_{1} , respecting the singular value.

Step 2. We will construct a quasiconformal homeomorphism φ: C → C from the dynamic plane of E_{λ} to the dynamic plane of E_{λ′} which will eventually turn into a conformal conjugation. Since the multipliers at the attracting fixed points are the same, we can define φ|_{V_{k}}: V_{k} → V′_{k} as conformal isomorphisms for k = 1, …, n which respect the dynamics on V_{k} in the sense that for z ∈ V_{k}

(6)  φ(E_{λ}(z)) = E_{λ′}(φ(z)).

It becomes unique on V_{1} by the requirement that φ(0) = 0, and in view of (6) it is unique on all V_{k}.

Since for k = 1, …, n, the sets \overline{V}_{k} are disjoint closed disks with analytic boundaries, the map φ can be extended from the conformal isomorphism φ: ∪V_{k} → ∪V′_{k} to a quasiconformal homeomorphism φ: C → C. On the multiply connected domain C \ (∪V_{k}), we need to specify the homotopy class of φ. We will do this using the attracting dynamic ray γ_{+} from Lemma 4.1: let γ_{k} := E_{λ}^{∘(k−1)}(γ_{+}) for k = 1, 2, …, n. Every γ_{k} ⊂ U_{k}, and it connects V_{k} to +∞ (for k = 1, 2, …, n−1) or to −∞ (for k = n). Now φ(γ_{k}) connects V′_{k} to ∞ for every k, and we require that φ(γ_{k}) be homotopic to the analogous curve γ′_{k} relative to V′_{1} ∪ ⋯ ∪ V′_{n}. This fixes the homeomorphism φ: C → C uniquely up to homotopy, and it is well known that φ may be chosen so as to be quasiconformal. Note that we do not claim that the extension of φ away from the V_{k} respects the dynamics.

Step 3. Our goal is to promote φ to a conformal conjugation via a sequence of quasiconformal maps φ_{j} with φ_{0} := φ and

(7)  φ_{j} ∘ E_{λ} = E_{λ′} ∘ φ_{j+1}.

We will show that φ_{j} → id uniformly on compact sets, which implies that E_{λ} = E_{λ′}. The construction is inspired by the theory of spiders [HS].

By induction over j, we will assure that φ_{j}|_{V_{k}} = φ_{0}|_{V_{k}} for all j and all k, and in particular φ_{j}(0) = 0 and φ_{j}(λ) = λ′; moreover, φ_{j}(γ_{k}) is homotopic to φ_{0}(γ_{k}) and hence to γ′_{k}, relative to V′_{1} ∪ ⋯ ∪ V′_{n}, for all k and j.

To construct φ_{j+1} from φ_{j}, note that both φ_{j} ∘ E_{λ} and E_{λ′} are universal covers from C to C^{∗}, so there is a homeomorphism φ_{j+1}: C → C which satisfies (7), and it becomes unique by fixing the value at a single point. Since φ_{j} ∘E_{λ}(0) = λ′ = E_{λ′}(0), we can and will require φ_{j+1}(0) = 0 to fix φ_{j+1} everywhere uniquely. Note that (7) implies φ_{j+1}(z + 2πi) = φ_{j+1}(z) + 2πi for all z.

By (7) and (6), we have

(E_{λ′} ∘ φ_{j+1})|_{V_{k}} = (φ_{j} ∘ E_{λ})|_{V_{k}} = (φ_{0} ∘ E_{λ})|_{V_{k}} = (E_{λ′} ∘ φ_{0})|_{V_{k}} ⊂ V′_{k+1}.

To justify φ_{j+1}|_{V_{k}} = φ_{0}|_{V_{k}}, all that remains to show is that φ_{j+1} maps V_{k} onto V′_{k} and not onto another branch of (E_{λ′})^{−1}(V′_{k+1}). It is here that the assumption about the attracting dynamic rays comes in.

Step 3a: the case k = 1, 2, …, n−1. Since (E_{λ′} ∘φ_{j+1})(γ_{k}) = (φ_{j} ∘E_{λ})(γ_{k}) = φ_{j}(γ_{k+1}) is homotopic to γ′_{k+1} = E_{λ′}(γ′_{k}) relative V′_{1} ∪ ⋯ ∪ V′_{n}, it follows that φ_{j+1}(γ_{k}) is homotopic to (a 2πiZ-translate of) γ′_{k} relative E_{λ′}^{−1}(V′_{1} ∪ ⋯ ∪ V′_{n}) ⊃ V′_{1} ∪ ⋯ ∪ V′_{n}. Now φ_{j+1}(0) = 0 implies φ_{j+1}|_{V_{1}} = φ_{0}|_{V_{1}} , and since γ_{1} is attached to V_{1} , it follows that φ_{j+1}(γ_{1}) is homotopic to γ′_{1} .

For k = 2, 3, …, n − 1, the number of 2πiZ-translates of γ_{k} between γ_{k} and γ_{1} is coded in the intermediate external address, and it equals the number of 2πiZtranslates of γ′_{k} between γ′_{k} and γ′_{1} . This quantity must be respected by φ_{j+1} in the sense that the number of translates of γ′_{k} between γ′_{1} and φ_{j+1}(γ_{k}) is the same as between γ_{k} and γ_{1} , so φ_{j+1}(γ_{k}) is homotopic to γ′_{k} and φ_{j+1}|_{V_{k}} = φ_{0}|_{V_{k}}. For k = 2, this implies that φ_{j+1}(λ) = λ′. (For later use in Corollary 4.4, we note that this reasoning is unchanged if a constant integer is added to all s_{k′}: only the differences s_{k′} − s_{1} matter).

Step 3b: the case k = 0. Since V_{0} is invariant under translation by 2πiZ and the curve γ_{n} runs towards −∞, not +∞, a different argument is needed to show that φ_{j+1}|_{V_{0}} = φ_{0}|_{V_{0}} . This is built into the construction: since φ_{j+1}(0) = 0, it follows that φ_{j+1} maps the E_{λ} -fundamental domain containing 0 to the E_{λ′} - fundamental domain containing 0 (up to homotopies of the boundary curve). Since a_{0} and a′_{0} are in the same fundamental domains of E_{λ} respectively of E_{λ′} as 0, it follows φ_{j+1}(a_{0}) = a′_{0} = φ_{0}(a_{0}); now φ_{j+1}|_{V_{0}} = φ_{0}|_{V_{0}} follows from the covering property.

Finally, φ_{0}(γ_{n}) connects a′_{n} to −∞ in the complement of V′_{1} ∪⋯∪V′_{n−1} and their associated curves γ′_{k′}; since there is only one homotopy class of such curves, it follows that φ_{j+1}(γ_{n}) is homotopic to φ_{0}(γ_{n}) rel V′_{1} ∪ ⋯ ∪ V′_{n}.

Step 4. We have a sequence (φ_{j}) of homeomorphisms C → C; it follows from (7) that all are quasiconformal with the same maximal dilatation as φ_{0} . All φ_{j} coincide on ⋃_{k} V_{k}, and all fix the points 0, 2πi, and ∞, and quasiconformal homeomorphisms with these properties form a compact space. On the domain E_{λ}^{∘(−(j−1))}(⋃_{k} V_{k}) with j ≥ 1, the map φ_{j} is conformal and coincides with all φ_{j′} for j′ ≥ j. Now ⋃_{j} E_{λ}^{∘(−(j−1))}(⋃_{k} V_{k}) fills up the entire Fatou set, while the Julia set has measure zero by [EL1], [EL3]. Therefore, the support of the bounded quasiconformal dilatation of the φ_{j} converges to zero, so the φ_{j} converge uniformly on compact subsets of C to an automorphism of C fixing 0, 2πi and ∞, hence to the identity. Finally, φ_{j}(λ) = λ′ implies E_{λ} = E_{λ′}. This is what we claimed.

In order to conclude that every hyperbolic component is uniquely described by its intermediate external address, we first state a lemma with routine proof.

Lemma 4.3 (Same fundamental domain). Every hyperbolic component of exponential maps with period n ≥ 2 contains a map for which the attracting orbit has positive real multiplier, and for which a single fundamental domain contains both the periodic point a_{0} and the singular value.

Proof. Let H be the hyperbolic component and let μ: H → D^{∗} be the multiplier map. Since it is a universal covering, we may find a map E_{λ_{0}} ∈ H with μ(λ_{0}) > 0. There is a curve in H starting at λ_{0} so that the μ-image of this curve describes a circle in D^{∗} around the origin in positive orientation; let λ_{1} ∈ H be the endpoint of the curve.

For E_{λ_{0}} , recall the analytic curve γ_{+} from Lemma 4.1: it is an extension of a diameter in linearizing coordinates around a_{1} . Let γ1_{+} be the subcurve between 0 and E_{λ_{0}}^{∘n}(0). As the parameter λ is deformed from λ_{0} to λ_{1} , the curve γ1_{+} extends to a homotopy γ1_{+}(λ) of curves within U_{1}(λ) connecting 0 to E_{λ}^{∘n}(0) in the complement of the respective singular orbits. In particular, we obtain a curve γ1_{+}(λ_{1}) connecting 0 to E_{λ_{1}}^{∘n}(0) within U_{1}(λ_{1}) in the complement of the singular orbit, and this curve is unique up to homotopy. This is not a closed curve, but the beginning and endpoint are on the same radius with respect to linearizing coordinates of a_{1} , so it makes sense to say that γ1_{+}(λ_{1}) winds once around a_{1} . For the map E_{λ_{1}} , Lemma 4.1 provides another curve which connects 0 to E_{λ_{1}}^{∘n}(0) (part of a linearizing radius), and this has winding number zero. Call this curve

γ̃1_{+}(λ_{1}).

For k = 1, 2, …, n, pull back both curves under E_{λ_{1}}^{∘k}, choosing the branch ending at E_{λ_{1}}^{∘(n−k)}(0) (as in Lemma 4.1): this yields two curves γ0_{+}(λ_{1}) and γ̃0_{+}(λ_{1}) connecting 0 to +∞; their winding numbers around a_{1} differ by 1.

All the countably many E_{λ_{1}} -preimages of γ̃0_{+}(λ_{1}) bound the fundamental domains of E_{λ_{1}} as defined before Theorem 4.2, while the E_{λ_{1}} -preimages of γ0_{+}(λ_{1}) are deformations of the fundamental domain boundaries of E_{λ_{0}} : their difference shows that between E_{λ_{0}} and E_{λ_{1}} , the periodic point a_{0} has jumped one fundamental domain up.

A finite repetition of this process (forward or backward) can bring a_{0} into the fundamental domain which contains 0.

Corollary 4.4 (Uniqueness of classification). The intermediate external addresses associated to hyperbolic components associate a bijection between hyperbolic components and intermediate external addresses with s_{1} = 0 of any given period.

Proof. By Lemma 3.3, every hyperbolic component is associated to a unique intermediate external address s_{1}, s_{2}, …, s_{n−1} ; this defines a “classification map” from hyperbolic components to intermediate external addresses. Every component contains a parameter as described in Lemma 4.3, say with multiplier μ = \frac{1}{2} . By Theorem 4.2 two such exponential maps are identical if their external addresses s_{1}s_{2} … s_{n−1} and s′_{1}s′_{2} … s′_{n−1} satisfy s_{k} − s_{1} = s′_{k} − s′_{1} for all k, so the classification map is injective. Since every address with s_{1} = 0 is realized (Theorem 3.5), only addresses with s_{1} = 0 are realized, and the classification map is a bijection.

Remark. In the proof that the intermediate external address uniquely describes hyperbolic components, we singled out attracting exponential maps for which the same fundamental domain contains the periodic point a_{0} and the singular value 0. This was convenient for the proof, but such maps are dynamically not special enough so that the corresponding locus in parameter space would stand out. We do not believe that these maps have any real significance other than that they are helpful in the proof. There is, however, a dynamically significant locus within every hyperbolic component, called “central internal ray”: this ray has significance both dynamically and in parameter space and allows us to give a preferred parametrization of hyperbolic components. This will be discussed in Section 7, but some preparations are more conveniently done here.

Recall that the multiplier map μ: H → D^{∗} is a universal covering map for every hyperbolic component H of period n ≥ 2, hence there is a conformal isomorphism Φ: H → H with μ = exp ∘Φ, and Φ is unique up to translation by 2πiZ in the range (throughout this paper, H denotes the left half plane). For our preferred parametrization of the component, we have to specify a choice of this integer translation, which is a combinatorial problem.

Definition 4.5 (Internal rays of hyperbolic components). An internal ray at angle ϑ ∈ R/Z of a hyperbolic component H is any branch of μ^{−1}(]0, e^{2πiϑ}[ ), where μ: H → D^{∗} is the multiplier map. In other words, an internal ray is a connected component of the locus in H where arg(μ) is constant.

For every fixed angle ϑ ∈ R/Z, there are countably many parameter rays with the same angle (except for the unique period 1 component in λ space), and on each of them the map |μ| induces a homeomorphism onto ]0, 1[ .

The internal parameter rays at angle 0 are the loci where the multiplier is real and positive, and they can be distinguished dynamically by the following generalization of Lemma 4.3. Note that the conformal isomorphism Φ: H → H induces a vertical order of parameter rays induced by imaginary parts within H; this order does not depend on the ambiguity in the definition of Φ (but since H is a left half plane, for hyperbolic components with unbounded positive real parts this order is “upside down” when parameter rays are ordered with respect to imaginary parts in their approach to +∞).

Lemma 4.6 (Internal parameter rays at angle 0). Every exponential map which has an attracting orbit of period n ≥ 2 with real positive multiplier has an associated integer-valued index Δ′ which specifies how many fundamental domains the periodic point a_{0} is above the singular value (or below, if the index is negative). The index Δ′ is constant for exponential maps on the same internal parameter ray. For every hyperbolic component, the index Δ′ induces an order preserving bijection between Z and parameter rays at angle 0. In particular, for every conformal isomorphism Φ: H → H with μ = exp ∘Φ, there is an integer m = m(H, Φ) depending only on H and Φ such that every λ ∈ H with μ(λ) > 0 has Im(Φ(λ)) = 2π(Δ′ − m).

Proof. Both a_{0} and the singular value are in some fundamental domain, and the latter are ordered like Z; therefore, there is a well-defined integer Δ′ which specifies how many fundamental domains a_{0} is above 0. By continuity, the index is constant along internal parameter rays at angle 0.

Moving one parameter ray up within the hyperbolic component (i.e. moving to a parameter ray on which Im(Φ) is greater by 2π for any given Φ) can be achieved by a deformation as in the proof of Lemma 4.3, and this increases the index Δ′ by one.

5. Characteristic rays and permutations

In this section, we investigate periodic points at which at least two periodic dynamic rays land, and show that the first return map of the periodic points permutes their rays transitively. This property is well known from quadratic polynomials; it depends on the fact that there is a single singularity, not on the degree of the map.

Definition 5.1 (Essential orbit, characteristic point and rays). A periodic orbit will be called essential if at least two dynamic rays land at each of its points. Suppose that a point z on an essential orbit is the landing point of two dynamic rays which separate the singular value from all the other points on the orbit of z; then the point z will be called the characteristic point of its orbit. The characteristic rays of the orbit will be the two dynamic rays landing at the characteristic point which separate the singular value from all the other rays landing at the same orbit.

The following result describes the combinatorics of dynamic rays landing together. The statement is the same as for polynomials, but the usual proof (using “widths of sectors”) does not apply without modification. Still, essential ideas are borrowed from Milnor [M3].

Lemma 5.2 (Permutation of dynamic rays). Every essential periodic orbit has exactly one characteristic point and exactly two characteristic rays at every point. If more than two dynamic rays land at any periodic point, then the first return map of the periodic point permutes these rays cyclically.

Proof. Let z_{1} , z_{2}, …, z_{n} = z_{0} be a periodic orbit of period n, labeled in the order of the dynamics, and let r ≥ 2 be the number of dynamic rays at each of these points. This number is constant along the orbit. The r rays landing at any given point z_{k} cut the complex plane into r open connected components which will be called the “sectors” at z_{k}. These dynamic rays connect z_{k} to +∞, so exactly one of the sectors at z_{k} contains a left half plane.

Consider any sector which does not contain a left half plane. Let m be the position of the first difference in the external addresses of the two rays bounding the sector (with m = 1 if the first entries are different); this will be called the singular index of the sector. For the sector which does contain a left half plane, we let the singular index be m := 0. Clearly, any sector at z_{k} with index m ≥ 1 maps homeomorphically onto a sector at z_{k+1} with index m − 1 (for m = 1, it follows from the fact that the sector must contain a horizontal line segment which stretches infinitely to the right and maps onto an infinite segment of R^{−}). It follows that for each sector, the index equals the number of iterations it takes for this sector to map over a left half plane.

If the index of a sector S at some z_{k} equals 0 so that the sector contains a left half plane, then S will not map forward homeomorphically. If g\underline{s} and g\underline{s}′ are the two dynamic rays bounding S, then E_{λ}(g\underline{s}) = g_{σ(\underline{s})} and E_{λ}(g\underline{s}′) = g_{σ(\underline{s}′)} bound a sector S′ at z_{k+1} containing the singular value; we call S′ the image sector of S. If the image sector also contains a left half plane, then the index remains 0; otherwise, the rays at z_{k+1} separate the singular value from a left half plane, and the index is strictly greater than 0. Each z_{k} has exactly one sector with index 0 (the unique sector containing a left half plane).

Every sector at every point z_{k} is periodic (in the sense just defined, not as a subset of C), and so is the sequence of the indices. The index sequence of each sector must of course contain at least one index 0, and it cannot be the constant sequence: if all the entries of one cycle of sectors were equal to 0, then all the other cycles of sectors could never have index 0, a contradiction.

Among the nr sectors based at the n points z_{1} ⋯ z_{n}, there is at least one with largest index. Let S be such a sector and let z_{1} be the periodic point at which S is based (possibly after cyclic relabeling). If S contains a periodic point z_{k} ≠ z_{1} , then at least one of the sectors at z_{k} has index at least as large as S; after replacing S with such a sector, we may assume that S contains no periodic point z_{k}. Let S_{0} be the unique sector (at z_{0} ) with image sector S. Then S_{0} must contain a left half plane (or the index of S_{0} would exceed that of S), hence S contains the singular value. This makes z_{1} the characteristic point of its orbit, and the rays bounding S are the characteristic rays. This shows the first statement.

Let α_{1} > α_{2} > ⋯ > α_{r} be the indices of the sectors at z_{1} ; no two of them can be equal because otherwise the corresponding sectors would map forward homeomorphically until they contained a left half plane at the same time. Of course, α_{r} = 0 is the sector containing a left half plane.

Consider any cycle C of sectors and let α > 0 be the largest index within its period. Since indices are always decreasing unless they are equal to 0, the index α must occur for a sector containing the singular value but not a left half plane. Let z_{k} be the periodic point at which this sector is based. If z_{k} = z_{1} , then the sector with index α is the sector at z_{1} containing the singular value. Hence all sectors for which the largest index is realized at z_{1} are on the same orbit. This is true even if the sequence of indices contains several maxima and one of them is realized at z_{1} .

If, however, z_{k} ≠ z_{1} , then α ≤ α_{r−1} because the point z_{1} is within the sector at z_{k} with index α, and so are all the sectors at z_{1} with indices α_{1} > ⋯ > α_{r−1} . The cycle C of sectors must map through z_{1} , but the only sector at z_{1} it can map through is the sector containing a left half plane (the maximum α is not assumed at z_{1} ).

It follows that there are at most two cycles of sectors: their representatives at z_{1} must include either the sector containing the singular value or the sector containing a left half plane, or both. Suppose that not all sectors are on the same orbit. Then the sector at z_{1} containing a left half plane is fixed under the first return map of z_{1} and has period n, and all the other r − 1 sectors at z_{1} are on the same orbit, so they are permuted transitively by the first return map of the dynamics and have period (r − 1)n. But all sectors must have equal periods because all dynamic rays have equal periods, and this is possible only if r = 2.

6. Dynamic roots

For an understanding of the dynamics, the most important rays are those which land together. We will now show that such are associated to attracting Fatou components.

We will need the partition constructed in [SZ3, Section 4.3] and reviewed in Section 2. Let u_{1}, u_{2}, …, u_{n−1} be the first n − 1 entries of the itinerary of the singular value (such that the singular value 0 is in the region labeled u_{1} , etc.). By definition of the labels, u_{1} = 0. The nth entry is not defined.

For a dynamic ray g\underline{s} with bounded external address \underline{s}, it may happen that there are two curves within U_{1} which connect the singular value to +∞ such that they separate g\underline{s} from U_{0} . In this case, we will say that the ray g\underline{s} is surrounded by U_{1} , and rays g\underline{s}′ with bounded external addresses \underline{s}′ sufficiently close to \underline{s} will also be surrounded by U_{1} , so this is an open property in the sequence space 𝒮. The Fatou component U_{1} contains infinitely many non-homotopic curves connecting the singular value to +∞, namely pull-backs of curves connecting E_{λ}^{∘(n−1)}(0) to ∞ within the Fatou component U_{n} containing a left half plane. Hence each ray which is not surrounded by U_{1} is either above or below U_{1} in the sense that the ray approaches +∞ above or below all such curves within U_{1} . Similarly, we will say that rays are above or below U_{2}, …, U_{n−1} (but not U_{0} ). Since every U_{i} (for i = 1, 2, …, n − 1) is disjoint from its 2πiZ-translates, and U_{n−1} contains a preimage of an unbounded part of R^{−}, each U_{i} ≠ U_{0} has bounded imaginary parts.

Lemma 6.1 (Periodic external address above U_{1} ). Let U_{1} be the characteristic Fatou component of an exponential map with attracting orbit of period n ≥ 2. Then

\underline{s}^{+} := inf\{\underline{s} ∈ 𝒮 : g\underline{s} is above U_{1} \} and \underline{s}^{−} := sup\{\underline{s} ∈ 𝒮 : g\underline{s} is below U_{1} \}

define two periodic external addresses of period n with equal itineraries.

Proof. It may not be clear a priori that \underline{s}^{+} and \underline{s}^{−} are well-defined sequences in Z^{N} , but their first entries are certainly integers s1^{+} ≥ s1^{−}. If s1^{+} = s1^{−}, then iteration gives

σ(\underline{s}^{+}) = inf\{\underline{s} ∈ 𝒮: g\underline{s} is above U_{2} \} (and similarly for σ(s^{−}))

so that s2^{+} ≥ s2^{−} are well-defined. Repeating this argument shows that

σ^{k}(\underline{s}^{+}) = inf\{\underline{s} ∈ 𝒮 : g\underline{s} is above U_{k+1} \}

where k is the first integer such that sk+1^{+} > sk+1^{−}. Now U_{k+1} surrounds or contains a preimage of an unbounded part of R^{−}, hence a preimage of a left half plane, and the next step is different:

σ^{k+1}(\underline{s}^{+}) = inf\{\underline{s} ∈ 𝒮 : g\underline{s} is below U_{1} , and U_{k+2} does not separate g\underline{s} from U_{1} \}

(the problem is that the lower part of U_{k+1} maps above U_{1} , while the upper part maps below U_{1} , and we are interested only in the image of the upper part; another way of saying this is that σ^{k+1}(\underline{s}^{+}) is the infimum of sequences which are above those parts of U_{k+2} that are below U_{1} ). Note that the infimum still has finite first entry because some part of U_{k+2} is below U_{1} . We can continue to iterate this:

σ^{k+m}(\underline{s}^{+}) = inf\{\underline{s} ∈ 𝒮 :g\underline{s} is below U_{m},
and U_{k+m+1} does not separate g\underline{s} from U_{m}\}

where m is the first integer such that U_{m} surrounds or contains a preimage of an unbounded part of R^{−}, or there is a preimage of an unbounded part of R^{−} below U_{m} which separates U_{m} from U_{k+m+1} . If that happens, we get

σ^{k+m+1}(\underline{s}^{+}) = inf\{\underline{s} ∈ 𝒮 : g\underline{s} is above U_{k+m+2} \}

and we are back to the initial situation. Repeating these arguments for a total of n − 2 iterations, we see that either

(8)  σ^{n−2}(\underline{s}^{+}) = inf\{\underline{s} ∈ 𝒮 : g\underline{s} is above U_{n−1} \},

or there is a k′ ≤ n such that (8) is false, but

(9)

σ^{n−2}(\underline{s}^{+}) = inf\{s ∈ 𝒮 : g\underline{s} is below U_{k′},
and U_{n−1} does not separate g\underline{s} from U_{k′}\}.

In the case of (8), we get

σ^{n−1}(\underline{s}^{+}) = inf\{s ∈ 𝒮 : g\underline{s} is in the same region R_{u_{1}} as U_{1},
and U_{n} = U_{0} does not separate g\underline{s} from U_{1}\}

(because U_{n−1} and the region surrounded by U_{n−1} together map homeomorphically over all of C except a subset of R_{u_{1}} ), and then

σ^{n}(\underline{s}^{+}) = inf\{s ∈ 𝒮 : g\underline{s} is above U_{1} \}.

Similarly, in the case of (9), we get

σ^{n−1}(\underline{s}^{+}) = inf\{s ∈ 𝒮 : g\underline{s} is in the same region R_{u_{k′} +1} as U_{k′+1},
and U_{n} does not separate g\underline{s} from U_{k′+1}\}

and again σ^{n}(\underline{s}^{+}) = inf\{s ∈ 𝒮 : g\underline{s} is above U_{1} \}.

Thus in every case σ^{n}(\underline{s}^{+}) = \underline{s}^{+} , and this is a periodic sequence over Z with period n. The same applies to \underline{s}^{−}, and both have itineraries of period (dividing) n.

The first n − 1 entries in the itinerary of g_{s^{+}} are u_{1}u_{2} ⋯ u_{n−1} , which are the same as in the itinerary of U_{1} or of the singular value. The nth entry in the itinerary of U_{1} is not defined (because the image component U_{n} extends through all fundamental domains), but we saw above that the nth entry in the itinerary of g_{s^{+}} is either u_{1} or u_{k′+1} , and g_{s^{−}} has the same itinerary.

Theorem 6.2 (Two rays at boundary fixed point). Every periodic Fatou component with attracting dynamics of period n ≥ 2 has a unique point on its boundary which is fixed by the first return map of the component and which is the landing point of at least two periodic dynamic rays. The characteristic point of this periodic orbit is on the boundary of the characteristic Fatou component.

Proof. Existence follows from Lemma 6.1: the periodic dynamic rays g_{s^{+}} and g_{s^{−}} have identical itineraries, so by the results of [SZ3] mentioned in Section 2, they land at a common periodic point z, say. In order to prove that z ∈ ∂U_{1} , let l be the hyperbolic distance of z to ∂U_{1} in the hyperbolic domain C\\overline{⋃_{k≥0} E_{λ}^{∘k}(0)}.

Assume that l > 0. The hyperbolic distance between the unique periodic inverse image of z and U_{0} = U_{n} is then less than l. We take n − 1 further pull-back steps along the periodic orbit of z; since the itinerary of z in those steps is the same as that of the singular orbit, the branches for the pull-back of z are those mapping U_{n} to U_{n−1}, U_{n−2}, …, U_{1} , and hyperbolic distances are decreased in every step. After n steps, z is mapped back to itself and its hyperbolic distance to U_{1} has decreased. This contradiction shows that l = 0 and z ∈ ∂U_{1} .

Let z_{1} be the characteristic point of the orbit of z (cf. Definition 5.1 and Lemma 5.2). The two characteristic rays separate the singular value from the orbit of z (except z_{1} itself) and from a left half plane. If z_{1} ≠ z, then the characteristic rays would separate z ∈ ∂U_{1} from a left half plane, and this is a contradiction. This proves the existence claim for the periodic Fatou component U_{1} , and for the others it follows easily.

Now we show uniqueness. The point z_{1} , together with the two characteristic rays landing at it, cut C into two open parts; let V be the one containing the singular value. Suppose that there is another periodic point z′_{1} ∈ ∂U_{1} which is fixed by the first return map of U_{1} and which is the landing point of two periodic dynamic rays. We have z′_{1} ∈ V because V contains \overline{U}_{1} − \{z_{1}\}. The two characteristic dynamic rays landing at z′_{1} are then contained in V as well. But by symmetry between z_{1} and z′_{1} , it also follows that the two characteristic rays landing at z′_{1} bound an open sector V′ which contains z_{1} and all its rays, and this is a contradiction.

Remark. The same statement holds also for parabolic dynamics; the proof requires only the same modifications as in [SZ3, Section 4.3].

Definition 6.3 (Dynamic root). In any exponential dynamics with attracting orbit of period n ≥ 2, the unique point of the characteristic Fatou component which is fixed under the first return map of the component and which is the landing point of at least two dynamic rays (as described in Theorem 6.2) will be called the dynamic root of the characteristic Fatou component.

Lemma 6.4 (Rays at dynamic root). In attracting exponential dynamics, the two characteristic rays of the dynamic root of the characteristic Fatou component separate this Fatou component from all other periodic Fatou components.

Proof. Let U_{1} be the characteristic Fatou component and let z_{1} be its dynamic root. The characteristic dynamic rays at z_{1} separate the singular value and thus U_{1} from all other points on the orbit of z_{1} . Every periodic Fatou component U_{i} has a unique point z_{i} on its boundary (Theorem 6.2). If z_{i} ≠ z_{1} , then U_{i} is separated from the singular value by the characteristic ray pair. Let the periods of the attracting orbit and of z_{1} be n and k, respectively. The number of different periodic Fatou components with z_{1} on its boundary is exactly n/k and the first return map of z_{1} must permute these k components cyclically. Hence the gaps between cyclically adjacent periodic Fatou components at z_{1} are also permuted cyclically, and at least one of them must contain a periodic dynamic ray landing at z_{1} ; hence all gaps do, and all periodic Fatou components at z_{1} are separated by periodic dynamic rays landing at z_{1} . (Conversely, it follows from Lemma 5.2 that all the rays landing at z_{1} are separated by periodic Fatou components provided at least two periodic Fatou components have z_{1} as their common dynamic root.)

7. Parametrization of hyperbolic components

Theorem 7.1 (Parametrization of hyperbolic components). For every hyperbolic component H of period n ≥ 2, there is a unique conformal isomorphism Φ_{H}: H → H with μ = exp ∘Φ_{H} such that if μ(λ) > 0, then Im(Φ_{H}(λ)) = 2πΔ, where the integer Δ specifies how many fundamental domains the periodic point a_{0} ∈ U_{0} is above the dynamic root z_{0} of U_{0} (or below if Δ < 0).

Proof. We already know from Lemma 4.6 the existence of a conformal isomorphism Φ: H → H with μ = exp ∘Φ_{H}, which is unique up to addition of a constant in 2πiZ in the range, so we only need to justify the combinatorial interpretation of imaginary parts if μ(λ) > 0.

Choose any λ ∈ H. Let z_{1} be the dynamic root of the characteristic Fatou component U_{1} and let \underline{s}_{1} < \underline{s}_{2} be the external addresses of its two characteristic rays. Let z_{0} be the dynamic root of U_{0} (with E_{λ}(z_{0}) = z_{1}) and let \underline{s}′_{2} < \underline{s}′_{1} be the least and greatest external addresses (with respect to lexicographic ordering) of two dynamic rays landing at z_{0} . By continuity, these external addresses are the same for every λ ∈ H.

Since the rays g_{\underline{s}_{1}} and g_{\underline{s}_{2}} surround 0, they bound the characteristic sector at z_{1} , which is the image of the sector at z_{0} containing a left half plane. It follows that σ(\underline{s}′_{1}) = \underline{s}_{1} and σ(\underline{s}′_{2}) = \underline{s}_{2} , and there is an m ∈ Z with \underline{s}′_{2} = m\underline{s}_{2} and \underline{s}′_{1} = (m + 1)\underline{s}_{1} (concatenation). By the translation symmetry, it follows that for every m′ ∈ Z, the rays at addresses m′\underline{s}_{2} and (m′ + 1)\underline{s}_{1} land together at z_{0} +2πi(m′ −m). Since \underline{s}_{1} and \underline{s}_{2} are characteristic external addresses, no σ^{k}(\underline{s}_{1}) or σ^{k}(\underline{s}_{2}) is contained in ]\underline{s}_{1}, \underline{s}_{2}[ ; therefore, no σ^{k}(\underline{s}_{1}) or σ^{k}(\underline{s}_{2}) is contained in any ]m′\underline{s}_{1}, m′\underline{s}_{2}[ for m′ ∈ Z. It follows that there is an m_{1} ∈ Z with

m_{1}\underline{s}_{2} < \underline{s}_{1} < \underline{s}_{2} < (m_{1} + 1)\underline{s}_{1};

since the ray pair g_{\underline{s}_{1}} and g_{\underline{s}_{2}} surround the singular value 0, it follows that the ray pair at addresses m_{1}\underline{s}_{2} and (m_{1} + 1)\underline{s}_{1} surrounds 0 as well. We had seen above that the ray pair at addresses m\underline{s}_{2} and (m + 1)\underline{s}_{1} lands at z_{0} .

Now suppose that μ(λ) > 0. Then z_{0} is exactly m−m_{1} fundamental domains above 0. All we used for this reasoning are the external addresses \underline{s}_{1} , \underline{s}_{2} , \underline{s}′_{1} and \underline{s}′_{2} which are the same for any λ ∈ H, so the index m−m_{1} is the same throughout H.

Recall the index Δ′ from Lemma 4.6 which specifies, whenever μ(λ) > 0, how many fundamental domains a_{0} are above 0. Therefore, Δ := Δ′ − (m − m_{1}) specifies how many fundamental domains a_{0} are above z_{0} . By Lemma 4.6, every conformal isomorphism Φ: H → H with μ = exp ∘Φ has an integer m_{Φ} ∈ Z such that for all λ ∈ H with μ(λ) > 0, Im(Φ(λ)) = 2π(Δ′ − m_{Φ}). Setting Φ_{H} := Φ + 2πi(m_{Φ} − (m − m_{1})) yields the desired conformal isomorphism. It is clearly unique.

Having stated this theorem, we should outline why this parametrization is indeed a preferred one, for example over the one from Lemma 4.6 counting fundamental domains between a_{0} and the singular value. The dynamic root clearly has dynamic significance, and the invariant curve γ^{−} from Lemma 4.1 is part of an analytic curve containing the singular orbit. Both might be linked for any particular exponential map: the following lemma states when this happens. It should come as no surprise that the locus of such maps stands out in parameter space; this is discussed at the end of this section.

Lemma 7.2 (Invariant curve lands at dynamic root). For every exponential map which has an attracting orbit of period n ≥ 2 with positive real multiplier, the invariant curve γ_{−} from Lemma 4.1 lands at the dynamic root z_{1} ∈ ∂U_{1} if and only if Δ = 0.

Proof. The curve γ_{−} starts at a_{1} and lands at a point w_{1} ∈ ∂U_{1} with E_{λ}^{∘n}(w_{1}) = w_{1} . Let γ_{0} ⊂ U_{0} be the branch of E_{λ}^{−1}(γ_{−}) starting at a_{0} ; it lands at a point w_{0} ∈ ∂U_{0} with E_{λ}^{∘n}(w_{0}) = w_{0} . Clearly, w_{1} = E_{λ}(w_{0}), so w_{0} is the dynamic root of U_{0} if and only if w_{1} is the dynamic root of U_{1} .

Recall the analytic curve γ_{+} ⊂ U_{1} from Lemma 4.1: it connects a_{1} to +∞ such that γ′_{+} := E_{λ}^{∘n}(γ_{+}) ⊂ γ_{+} , and γ′_{+} contains the singular orbit within U_{1} . The fundamental domains for E_{λ} are bounded by E_{λ}^{−1}(γ_{+} \ γ′_{+}).

Since γ_{0} is disjoint from the boundary of the fundamental domains, it follows that w_{0} is in the same fundamental domain as a_{0} , so w_{0} can be the dynamic root z_{0} ∈ U_{0} only if a_{0} and z_{0} are in the same fundamental domain, i.e. only if Δ = 0.

Conversely, we show that w_{0} is the only boundary point of U_{0} which is fixed by E_{λ}^{∘n} and which is in the same fundamental domain as a_{0} ; this implies that whenever Δ = 0, then γ_{0} lands at z_{0} and γ_{−} lands at z_{1} .

Suppose there is another point w′_{0} ∈ ∂U_{0} with E_{λ}^{∘n}(w′_{0}) = w′_{0} in the same fundamental domain as w_{0} . There is a curve γ′_{0} ⊂ U_{0} with E_{λ}^{∘n}(γ′_{0}) ⊃ γ′_{0} which lands at w′_{0} (take a point b′_{0} in a linearizing neighborhood of w′_{0} and pull it back repeatedly so as to obtain a sequence (b′_{k}) converging to w′_{0} ; these can be connected by a curve γ′_{0} as required). Let (b_{k}) be an analogous sequence of points converging to w_{0} .

Connect b_{0} to b′_{0} by a differentiable curve Γ_{0} ⊂ U_{0} which avoids the fundamental domain boundaries and the curve E_{λ}^{∘(n−1)}(γ_{+}) which contains the singular orbit within U_{0} . Then E_{λ}^{−n}(Γ_{0}) contains a curve Γ_{1} connecting b_{1} to b′_{1} and homotopic to Γ_{0} in U′_{0} := U_{0} \ (E_{λ}^{−1}(γ_{+}) ∪ {a_{0}}) (the fact that the same branch of E_{λ}^{−n} fixes both w_{0} and w′_{0} uses the assumption that both points are within the same fundamental domain and would fail if w′_{0} was an arbitrary fixed point of E_{λ}^{∘n} in ∂U_{0} ). This can be repeated, and the hyperbolic lengths of Γ_{k} within U′_{0} become shorter each time, while b_{k}, b′_{k} → ∂U_{0} , and this implies w_{0} = w′_{0} .

For the following definition, recall the conformal isomorphism Φ_{H}: H → H which maps parameter rays of H to horizontal lines in H.

Definition 7.3 (Height and central internal ray). Given a hyperbolic component H of period n ≥ 2 with preferred conformal isomorphism Φ_{H}: H → H, we define the height of a parameter ray as the number h ∈ R such that Im(Φ_{H}(λ)) = 2πh for λ on this parameter ray.

The central parameter ray of H is the parameter ray at height h = 0.

Remark. The angle of a parameter ray is the projection of h to R/Z. If λ ∈ H has μ(λ) > 0, then clearly h = Δ. Note that for components of period n ≥ 3 in λ-space (and of period n ≥ 2 in κ-space) a ray with larger height is below a ray with smaller height: see the discussion after Definition 4.5. This is unavoidable if among nearby rays one wants rays at larger heights to have larger angles of their multipliers.

Figure 3. Hyperbolic components of periods 3 and 4, with internal parameter rays at integer heights drawn in. In both components, the central parameter rays are highlighted. Their landing points are the roots of the component (at which the components bifurcate from components of period 1 respectively 2).

Remark. We have classified hyperbolic components in terms of intermediate external addresses. A different coding would be in terms of the external addresses of the two characteristic dynamic rays landing at the root of the characteristic Fatou component: clearly, all exponential maps from the same hyperbolic component have the same external addresses at the dynamic rays landing at the root, so there is an algorithm to turn the intermediate external address of the component into the external addresses of the characteristic rays. The converse is much easier: knowing the external addresses of the two characteristic rays, it is not hard to write down the intermediate external address of the attracting dynamic ray and thus of the component. Both algorithms can be found in [RS]. While the coding in terms of intermediate external addresses leads to the easier classification, the characteristic external addresses are more easily related to the structure of the Multibrot sets ℳ_{d} and their limiting configurations as d → ∞.

The boundary of hyperbolic components and bifurcations. While this paper completely describes hyperbolic components of exponential maps, their boundary properties are discussed in [S1, Section V] and [RS]. To complete the picture, we mention some results here.

By Theorem 7.1, every hyperbolic component H (except the period 1 component in λ-space) comes with a preferred conformal isomorphism Φ_{H}: H → H. It extends as a homeomorphism to the closures \overline{Φ}_{H}:\overline{H} → \overline{H}. This result requires a much better understanding of the exponential parameter space. It implies that every hyperbolic component has connected boundary ∂H = \overline{Φ}_{H}^{−1}(iR), which was conjectured by Eremenko and Lyubich [EL2].

Different hyperbolic components may have common boundary points. This happens if and only if one hyperbolic component bifurcates from another, and then both components have a unique boundary point in common. The structure of bifurcations between hyperbolic components in exponential parameter space is cleared up in [RS]. In particular, this proves a third conjecture in [EL2] which states that there are infinitely many “trees” of hyperbolic components such that two components are in the same tree if and only if they can be connected via a finite chain of components so that adjacent components in the chain have common boundary.

Every internal parameter ray of H at height h lands at a well-defined parameter in ∂H with indifferent orbit so that the landing point depends continuously on h. The landing point of the central parameter ray (h = 0) is called the root of H, and it is significant in several ways: the root of H is the only boundary point which may simultaneously be a boundary point of another hyperbolic component H_{0} , and this happens if and only if H bifurcates from H_{0} . Moreover, every boundary point of H is the landing point of one or two external parameter ray (curves consisting of parameters for which the singular orbit converges to ∞ under iteration, see [S1, Chapter II] or [F]). Boundary points of H at positive heights are the landing points of parameter rays which come from +∞ below H, while boundary points at negative heights are landing points of parameter rays above H. The root point of H is the landing point of two parameter rays, one above and one below H [S1, Chapter IV]. This shows once more the significance of our preferred parametrization from Theorem 7.1.

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34 Dierk Schleicher

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Received 30 May 2000