Dynamic rays of bounded-type transcendental self-maps of the punctured plane

We study the escaping set of functions in the class $\mathcal B^*$, that is, holomorphic functions $f:\mathbb C^*\to\mathbb C^*$ for which both zero and infinity are essential singularities, and the set of singular values of $f$ is contained in a compact annulus of $\mathbb C^*$. For functions in the class $\mathcal B^*$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of $\mathbb C^*$ (and hence, in the class $\mathcal B^*$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every essential itinerary $e\in\{0,\infty\}^\mathbb N$, we show that the escaping set of $f$ contains a Cantor bouquet of curves that accumulate to $\{0,\infty\}$ according to $e$ under iteration by $f$.


Introduction
Complex dynamics concerns the iteration of a holomorphic function on a Riemann surface S. Given a point z ∈ S, we consider the sequence given by its iterates f n (z) = (f • n · · · •f )(z) and study the possible behaviours as n tends to infinity. We partition S into the Fatou set, or set of stable points, F (f ) := z ∈ S : (f n ) n∈N is a normal family in some neighbourhood of z and the Julia set J(f ) := S \ F (f ), consisting of chaotic points. If f : S → S is holomorphic andĈ \ S consists of essential singularities, then there are three interesting cases: • S =Ĉ := C ∪ {∞} and f is a rational map; • S = C and f is a transcendental entire function; • S = C * := C \ {0} and both zero and infinity are essential singularities. We study this third class of maps, which we call transcendental self-maps of C * . Such maps are all of the form f (z) = z n exp g(z) + h(1/z) , (1.1) in one complex variable is [Mil06]. See [Ber93] for a survey on the iteration of transcendental entire and meromorphic functions. Although the iteration of transcendental (entire) functions dates back to the time of Fatou [Fat26], Rådström [Råd53] was the first to consider the iteration of holomorphic self-maps of C * . A complete list of references on this topic can be found in [Mar14]. It is our goal in this paper to continue with the program started in [Mar14] of a systematic study of holomorphic self-maps of C * , extending the modern theory of iteration of transcendental entire functions to this setting.
To that end, we recall the definition of the escaping set of an entire function f I(f ) := {z ∈ C : f n (z) → ∞ as n → ∞} whose investigation has provided important insight into the Julia set of entire maps. For polynomials, the escaping set consists of the basin of attraction of infinity and its boundary equals the Julia set. For transcendental entire functions, Eremenko showed that I(f ) ∩ J(f ) = ∅, J(f ) = ∂I(f ) and the components of I(f ) are all unbounded [Ere89]. Similar properties [Mar14, Theorems 1.1, 1.3 and 1.4] hold for transcendental self-maps of C * once the definition is adapted to take both essential singularities into account. More precisely, the escaping set of a transcendental self-map of C * is given by where ω(z, f ) is the classical omega-limit set and the closure is taken inĈ, ω(z, f ) := n∈N {f k (z) : k n}.
As usual, the singular set sing(f −1 ) which denotes the set of the critical values and the finite asymptotic values of f , plays an important role. In the entire setting, the so-called Eremenko-Lyubich class B := {f transcendental entire function : sing(f −1 ) is bounded} consisting of bounded-type functions was introduced in [EL92] (see also [Six14]). Eremenko and Lyubich showed that if f ∈ B, then I(f ) ⊆ J(f ) or, in other words, the Fatou set has no escaping components. Functions in the class B have many other useful properties; see, for example, [RRRS11,MR13,BF15]. In the context of holomorphic maps of C * , the analogous class to consider is that where the singular values stay away from both essential singularities, hence we introduce the class of bounded-type transcendental self-maps of the punctured plane as B * := {f transcendental self-map of C * : sing(f −1 ) is bounded away from 0, ∞} and prove the following result. As shown in [Mar], functions outside the class B * may have escaping It is a natural question to investigate the relationship between entire functions in the class B and self-maps of C * in the class B * . Keen [Kee88] showed that if g and h are polynomials and n ∈ Z, then the function f (z) = z n exp g(z) + h(1/z) has a finite number of singular values and hence belongs to the class B * . The next theorem extends this results to all functions in the class B when n = 0.
Theorem 1.2. Let g and h be entire functions in the class B. Then the function f (z) = exp g(z) + h(1/z) is in the class B * .
As opposed to the situation for entire functions, there is a deep relation between the bounded-type condition for holomorphic self-maps of C * and their order of growth. To be more precise, recall that the order and lower order of an entire function f can be defined, respectively, as We say that f has finite order if both ρ ∞ (f ) < +∞ and ρ 0 (f ) < +∞. Likewise, we can define two quantities associated with the lower order of such functions, λ ∞ (f ) and λ 0 (f ), by replacing lim sup by lim inf in the expression above. An important property of entire functions f ∈ B is that λ(f ) 1/2 [BE95,Lan95] (see also [RS05a,Lemma 3.5]). The next result shows that, surprisingly, the lower order of a function in C * always equals its order and hence it is not relevant in this setting. Moreover, if the order is finite, then it is an integer. Theorem 1.3. Let f be a transcendental self-map of C * . Then λ 0 (f ) = ρ 0 (f ) and λ ∞ (f ) = ρ ∞ (f ). If f has finite order, then f (z) = z n exp P (z) + Q(1/z) where n ∈ Z and P, Q are polynomials, and therefore ρ 0 (f ), ρ ∞ (f ) ∈ Z and f ∈ B * . In particular, λ 0 (f ), λ ∞ (f ) 1.
In [Ere89], Eremenko conjectured that if f is a transcendental entire function, then the components of I(f ) are all unbounded. A stronger version of this conjecture states that every escaping point can be joined to infinity by a curve of points that escape uniformly. Such curves are called ray tails and their maximal extensions are called dynamic rays. Douady and Hubbard [DH85] were the first to introduce dynamic rays to study the dynamics of polynomials, where I(f ) consists of the attracting basin of infinity which is connected. Devaney and Krych [DK84] showed that for maps in the exponential family E λ (z) = λe z , λ ∈ (0, 1/e), the Julia set consists of dynamic rays (that they called hairs). Devaney and Tangerman [DT86] proved that the same holds for certain finite-type functions, that is, functions with finitely many singular values, satisfying additional technical conditions, such as the sine family S λ (z) = λ sin(z), λ ∈ (0, 1). They coined the term Cantor bouquet to describe the Julia set of these functions. They first defined a Cantor N -bouquet, where N ∈ N, to be a subset of J(f ) homeomorphic to the product of a Cantor set and the half-line [0, +∞), and then a Cantor bouquet to be an increasing union of Cantor N -bouquets. However, this is somewhat different to the definition of Cantor bouquet used more recently (and in this paper) in terms of a topological object called a straight brush which is due to Aarts and Oversteegen [AO93] (see Definition 9.1). Rottenfußer, Rückert, Rempe and Schleicher proved in [RRRS11, Theorem 1.2] that the stronger version of Eremenko's conjecture holds for transcendental entire functions of bounded type and finite order or, more generally, a finite composition of such functions: every escaping point can be joined to infinity by a curve of points that escape uniformly. This result was proved independently by Barański [Bar07, Theorem C] for disjoint-type functions, that is, transcendental entire functions for which the Fatou set consists of a completely invariant component which is a basin of attraction. Shortly after, Barański, Jarque and Rempe proved that, actually, the Julia set of the functions considered in [RRRS11] contains a Cantor bouquet [BJR12,Theorem 1.6].
In this article we prove the existence of dynamic rays for transcendental selfmaps of C * by adapting the construction of [RRRS11] to our setting. We use the notation f n |γ → {0, ∞} to mean that, under iteration by f , the points in γ escape to zero, escape to infinity or accumulate to both of them and nowhere else.
Theorem 1.4. Let f be a transcendental self-map of C * of finite order or, more generally, a finite composition of such functions. Then every point z ∈ I(f ) can be connected to either zero or infinity by a curve γ such that f n |γ → {0, ∞} uniformly in the spherical metric.
Note that in the statement of Theorem 1.4 there is no assumption of boundedtype. This is because, as we mentioned above, finite order transcendental self-maps of C * are always in the class B * (see Lemma 4.6).
Given a holomorphic self-map of C * , a lift of f is an entire functionf satisfying exp •f = f • exp. Bergweiler [Ber95] proved that J(f ) = exp −1 J(f ). Seeing this result one might think that every result about entire functions could be extended to self-maps of C * via their lifts. Unfortunately, this is not possible. In particular, a lift of a map of bounded type is never of bounded type, its singular set is contained in a vertical band and so, we cannot apply directly the results from [RRRS11]. In fact, in the opposite direction, Theorem 1.4 allows to construct dynamic rays for certain entire functions that are not in the class B, but project to functions in the class B * satisfying the hypothesis of Theorem 1.4. Corollary 1.5. Let f be an entire transcendental function of finite order for which there exists k ∈ Z so that f (z + 2πi) = f (z) + k2πi for all z ∈ C, or a finite composition of such functions. Then every point z ∈ I(f ) with | Re f n (z)| → +∞ as n → ∞ can be connected to infinity by a curve of points that escape uniformly.
The main tool to prove Theorem 1.4 is the use of logarithmic coordinates, introduced by Eremenko and Lyubich [EL92], and the expansivity of the logarithmic transform near the essential singularities. The orbit of escaping points eventually enters the tracts (unbounded Jordan domains which are mapped to a neighbourhood of zero or infinity) and remains there. We partition each tract into fundamental domains, each with a corresponding symbol, and consider itineraries on them; see Section 5 for the precise definitions. Observe that the previous theorem contains no claim of which dynamic rays actually exist. Our next result shows that, under the hypothesis of Theorem 1.4, there is a unique dynamic ray for every sequence of fundamental domains that contains only finitely many symbols.
Here P (f ) denotes the postsingular set of f which is the closure of the union of all the (forward) iterates of sing(f −1 ). We say that a dynamic ray γ lands if γ \ γ is a single point.
Theorem 1.6. Let f be a transcendental self-map of C * of finite order or, more generally, a finite composition of such functions, and let (D n ) be an admissible sequence of fundamental domains of f containing finitely many symbols. Then the function f has a unique nonempty dynamic ray γ with itinerary (D n ). Furthermore, if (D n ) is periodic and the set P (f ) is bounded in C * , then the dynamic ray γ lands.
Observe that, for example, Theorem 1.6 implies that every fundamental domain contains exactly one fixed dynamic ray.
We associate to each escaping point an essential itinerary e = (e n ) ∈ {0, ∞} N defined by ∞, if |f n (z)| > 1, for all n ∈ N. Consider, for each e ∈ {0, ∞} N , the set of points whose essential itinerary is eventually a shift of e, that is, Each of the sets I e (f ), e ∈ {0, ∞} N , should be regarded as the analogue of the whole of I(f ) for a transcendental entire function f . In [Mar14, Theorem 1.1] it is shown that, for each e ∈ {0, ∞} N , I e (f ) ∩ J(f ) = ∅. We follow the methods of [BJR12] to show that, in fact, under the hypothesis of Theorem 1.4, each set I e (f ) not only contains periodic ray tails (countably many) but a Cantor bouquet.
Theorem 1.7. Let f be a transcendental self-map of C * of finite order or, more generally, a finite composition of such functions. For each e ∈ {0, ∞} N , there exists a Cantor bouquet X e ⊆ I e (f ) and, in particular, the set I e (f ) contains uncountably many ray tails.
Although Theorem 1.4 is stated in terms of functions of finite order, its proof is more general and applies to a class of functions satisfying certain good geometry properties (see Definition 3.13). Rempe, Rippon and Stallard showed that, assuming an extra condition (namely, that the tracts have what they call bounded gulfs), the ray tails constructed in [RRRS11] consist of fast escaping points [RRS10, Theorem 1.2]. It seems likely that similar conditions would imply that the dynamic rays that we construct here are also fast escaping in the sense of [Mar14].
Remark 1.8. Lasse Rempe-Gillen pointed out that Theorem 1.4 may also be proved using random iteration as described in the last paragraph of [RRRS11, Section 5] by taking, for R > 0 sufficiently large, which both have a logarithmic transform in the class B log and then applying the results of [RRRS11] to a non-autonomous sequence of these two functions. However, it seems natural to provide a direct proof.
Structure of the paper. Roughly speaking, the first half of the paper is devoted to describing the basic properties of functions in the class B * and in the second half we investigate the existence of dynamic rays for these functions. In Section 2, we study what is the relation between the classes B and B * ; the proof of Theorem 1.2 is there. In Section 3, we describe the geometry of logarithmic coordinates of functions in the class B * and give the proof of Theorem 1.1. Finite order functions are introduced in Section 4, where we prove Theorem 1.3, and are shown to be examples of functions with good geometry. In Section 5, we define symbolic dynamics, both in terms of essential itineraries (with respect to essential singularities) and external addresses (with respect to tracts). In contrast to what happens in the entire case, in our setting the Bernoulli shift map is a subshift of finite type, where only some sequences are admissible. In Section 6, we show that if an external address s is periodic, then the set J s (F ) consisting of all points with that address contains an unbounded continuum of fast escaping points -this is used later to prove Theorem 1.6 in Section 9. Dynamic rays are introduced in Section 7. Finally the proofs of Theorem 1.4 and Theorem 1.7 are sketched in Section 8 and Section 9, respectively, focusing on the differences with the proofs of [RRRS11, Theorem 1.2] and [BJR12, Theorem 1.6], which concern entire functions.
Notation. In this paper N = {0, 1, 2, . . .}. If z ∈ C * and X ⊆ C * , then dist(z, X) denotes the Euclidean distance from z to X. If z 0 ∈ C and 0 < r < r , we define the sets D(z 0 , r) := {z ∈ C : |z − z 0 | < r}, A(r, r ) := {z ∈ C : r < |z| < r }. We define the half-planes H + := {z ∈ C : Re z > 0}, H − := {z ∈ C : Re z < 0}, and, for r ∈ R, we put H + r := {z ∈ C : Re z > r}, H − r := {z ∈ C : Re z < −r}, and, for r > 0, H ± r := {z ∈ C : | Re z| > r} = H + r ∪ H − r . If X is a set in C * , then the topological operations X and ∂X are taken in C * unless stated otherwise, and we use X to denote the closure of X inĈ. Finally, if X, Y are disjoint sets, we use X Y to denote the union of X and Y .
Acknowledgements. The authors thank Lasse Rempe-Gillen, Phil Rippon and Gwyneth Stallard for useful discussions during the preparation of this paper and Dave Sixsmith for reading the paper carefully and making very helpful comments. We also thank Lasse Rempe-Gillen for kindly providing the picture from the introduction.

Functions in the class B *
Let f be a transcendental entire function or a transcendental self-map of C * . We say that v ∈Ĉ is a critical value of f if v = f (c) with f (c) = 0. We say that a ∈Ĉ is an asymptotic value of f if there is a continuous injective curve γ : (0, +∞) −→Ĉ (the asymptotic path) such that as t → +∞, γ(t) → α, where α is an essential singularity of f Note that in C * by finite asymptotic value we mean that a / ∈ {0, ∞}. For transcendental self-maps of C * , we can decompose AV(f ) as depending on whether a ∈ AV(f ) has an asymptotic path γ to zero or to infinity. The set AV 0 (f ) ∩ AV ∞ (f ) may be nonempty. Finally, we define the singular set We say that f has bounded type if S(f ) is bounded. Similarly, we say that f has finite type if S(f ) is finite.
The next result relates the singular set and the postsingular set of a transcendental self-map f of C * with the corresponding sets of a liftf of f , which is a transcendental entire function satisfying exp •f = f • exp. Its proof is straightforward and we omit it.
Lemma 2.1. Let f be a transcendental self-map of C * and letf be a lift of f . Then S(f ) = exp −1 S(f ) and P (f ) ⊆ exp −1 P (f ) .
Recall that if f is a holomorphic self-map of C * , we define ind(f ) to be the index of f (γ) with respect to the origin, where γ is any positively oriented simple closed curve around the origin. Observe that, in the hypothesis of the previous lemma, if |ind(f )| = 1, then P (f ) = exp −1 P (f ) .
The following lemma is a basic property about the singular values of the composition of two functions.
Lemma 2.2. Let f and g be meromorphic functions in C. Then we have that Proof. By the chain rule, (g • f ) (z) = g f (z) f (z), and thus Observe that the set f −1 CP(g) may be empty, and hence the other inclusion does not hold in general.
Finally, if γ is an asymptotic path of g • f with asymptotic value a, then either f γ(t) ) → b ∈ AV(f ) as t → +∞, where g(b) = a, or f (γ) is an asymptotic path of g and a ∈ AV(g). Therefore AV(g • f ) ⊆ g AV(f ) ∪ AV(g) and the opposite inclusion follows easily.
Let B and B * be the bounded-type classes defined in the introduction. Observe that, by Lemma 2.2, both B and B * are closed under composition. Recall that Theorem 1.2 establishes a way to construct functions in B * from functions in B.
To prove this theorem, we need the following preliminary result. Note that if n 0, then f ∞ and f 0 are transcendental entire functions, while if n < 0, then they are meromorphic functions with a pole at the origin (which is omitted).
Proof of Proposition 2.3. We can express Outside a disk of radius r, the functions f and e h(0) f ∞ are as close as we want provided that r is large enough. Therefore AV ∞ (f ) = e h(0) · AV(f ∞ ). Differentiating f , we get or, equivalently, It follows easily from [EL92, Lemma 1] that if f ∈ B then there is a constant R 0 > 0 such that and hence If n < 0, the function f ∞ is meromorphic but, since the pole at z = 0 is omitted and sing(f −1 ∞ ) is bounded away from the origin, the same proof of Lemma 3.6 can be used to obtain inequality (2.1) in this case as well. Suppose that f ∞ has bounded type, then Since f ∞ is entire, the components of the set {z ∈ C : |f ∞ (z)| > R} are all unbounded and tend to infinity as R → +∞ (see Lemma 3.2). Therefore, since there exists M, N > 0 such that if |f (z)| > R and |z| 1 then and so Hence, CV(f ) cannot contain a sequence of critical values whose critical points are in C \ D that accumulate to infinity, because if f (z) is a critical value, then the quantity zf (z)/f (z) = 0. Similarly, in a neighbourhood of zero, as R → +∞, and thus f has no critical values accumulating to zero whose critical points are in D. Finally, since we are assuming that the functions 1/f ∞ and 1/f 0 have bounded type too, 0 / ∈ sing(f −1 ∞ ) ∪ sing(f −1 0 ) , so the sets Sixsmith [Six14] showed that if f / ∈ B, then η f = 0, where η f is the quantity defined in (2.2), and thus provided and alternative characterisation of functions in the class B. This was later generalised by Rempe-Gillen and Sixsmith in [RS15].
Theorem 1.2 states that if g, h ∈ B, then the function f (z) = exp g(z)+h(1/z) is in class B * . Thus, it can be used to produce examples of functions in the class B * from functions in the class B (see Example 2.6). Recall that Keen proved that if g and h are polynomials and n ∈ Z, then f (z) = z n exp g(z) + h(1/z) is in the class B * as well (see Proposition 4.5 and Lemma 4.6).
Proof of Theorem 1.2. Let f ∞ = exp • g where g ∈ B. By Lemma 2.2, and both CV(f ∞ ) = exp CV(g) and AV(f ∞ ) are bounded in C. On the other hand, Similarly, since h ∈ B the functions f 0 (z) = exp −h(z) and 1/f 0 have bounded type. Therefore f ∞ and f 0 satisfy the hypothesis of Proposition 2.3 and the function f (z) = exp g(z) + h(1/z) is in the class B * .
Remark 2.4. Observe that if n = 0 and f (z) = z n exp g(z) with g ∈ B, even if CV(g) is bounded the set CV(f ) may accumulate to zero (n > 0) or to infinity (n < 0). Thus Theorem 1.2 is optimal.
Remark 2.5. The converse of Theorem 1.2 is not true in general as the critical values of g can be unbounded in a vertical band and the critical values of f ∞ be bounded in an annulus. For example, the Fatou function g(z) = z + 1 + e −1 is not in the class B while the function f (z) = exp g(z) + 1/z) is in the class B * .
Example 2.6. We give a couple of examples of functions in the class B * constructed from functions in the class B using Theorem 1.2.
(i) The function f (z) = exp (sin z + 1)/z is in B * and sing(f −1 ) is an infinite set which accumulates to z = 1. (ii) The function f (z) = exp(exp z + 1/z) is in B * and has a finite asymptotic value a = 1.

Logarithmic coordinates for the class B *
Let a ∈Ĉ and for r > 0 choose U (r) to be a connected component of f −1 D(a, r) such that if r 1 < r 2 then U (r 1 ) ⊆ U (r 2 ). We say that U is a logarithmic singularity over a if f : U (r) → D(a, r) \ {a} is a universal covering for some r > 0 (see [Ive14] for a classification of the singularities of the inverse). Transcendental self-maps of C * have logarithmic singularities over both zero and infinity.
Definition 3.1 (Logarithmic tract). Let f ∈ B * and let A ⊆ C be a topological annulus bounded away from zero and infinity that contains the set S(f ). Denote W = W 0 ∪ W ∞ , where W 0 and W ∞ are the components of C * \ A whose closure in C contains, respectively, zero and infinity.
The following lemma is a well-known classification of the coverings of the punctured disk D * := D(0, 1) \ {0} (see, for example, [Hat02]). If X is a topological space, we say that two coverings p 1 : X 1 → X and p 2 : X 2 → X of X are equivalent if there exists a homeomorphism p 21 : X 2 → X 1 such that Lemma 3.2 (Coverings of D * ). Let U ⊆Ĉ and let f : U → D * be a holomorphic covering. Then either U is biholomorphic to D * and f is equivalent to z d , or U is simply connected and f is the universal covering, hence equivalent to the exponential map.
In particular, the closure of each tract inĈ contains only one of the essential singularities. Now we are going to introduce a logarithmic change of variables.
contain, respectively, a left and a right half-plane. A logarithmic transform of f is a continuous function F : T → H which makes the following diagram commute.
The connected components of T are called tracts of F and can be classified into four types where the lower index indicates if the tracts have zero or infinity in their closure and the upper index indicates if they are mapped to H 0 or H ∞ by F . We define In the entire case, often the expressions 'lift' and 'logarithmic transform' are used indistinctly to refer to F defined on the tracts. In this paper we reserve the word lift for an entire functionf such that exp •f = f • exp.
Remark 3.4. Observe that we can obtain F as the restriction of a liftf of f to the set T . However, since F is only defined on T , we can add a different integer multiple of 2πi to F on each tract T , and hence F is not necessarily the restriction of a transcendental entire functionf . Moreover, there exists a curve δ ⊆ C * \V joining zero to infinity, where V = exp T .
Proof. These properties follow easily from the fact that the exponential map is a holomorphic cover and, in particular, a local homeomorphism. The fact that there exists a curve δ ⊆ C * \ V joining zero to infinity is a straight consequence from (b) and (c) in the case that V consists of finitely many tracts. Otherwise, this follows from Carathéodory's theorem and the fact that V is locally connected (see [BF15, Lemma 2.1]). Hence, we can define a continuous branch of the logarithm on T .
We denote by B * log the class of holomorphic functions F : T → H satisfying properties (a) to (f) in Theorem 3.5, regardless of the fact that they come from a function f ∈ B * or not. The main advantage of working in the class B log defined in [RRRS11] or, in our case, the class B * log , is that functions satisfy the following expansivity property (3.1) which implies that points in I(f ) eventually escape at an exponential rate.
In particular, there exists Sullivan proved that rational maps have no wandering domain [Sul85]. Following this result, Keen [Kee88], Kotus [Kot87] and Makienko [Mak87] proved independently that transcendental self-maps of C * with finitely many singular values have no wandering domains. In [Kot87], Kotus also showed that finite-type maps in C * have no Baker domains. Here we show that bounded-type functions have no escaping Fatou component adapting the proof that Eremenko and Lyubich gave for class B [EL92, Theorem 1].
Proof of Theorem 1.1. Suppose to the contrary that there is Then, by normality, there exists some R > 0 so that Since the sets B n := f n (B 0 ) are escaping, they are eventually contained in the tracts of f as n → ∞ and we can assume without loss of generality that B n ⊆ V for all n 0. Let C 0 := log B 0 and let C n := F n (C 0 ) for all n 0. Then exp(C n ) = B n accumulates to {0, ∞} as n → ∞ and hence | Re C n | → +∞ uniformly as n → ∞. Take any ζ 0 ∈ C 0 and, for all n > 0, let ζ n := F n (ζ 0 ) ∈ C n and set d n := dist(ζ n , ∂C n ). Koebe's 1/4-theorem tells us that As n → ∞, since | Re F (ζ n )| → +∞, by Lemma 3.6 we have |F (ζ n )| → +∞ and hence d n → +∞. But this contradicts property (e) of functions in the class B * log because T does not contain any vertical segment of length 2π.
Property (a) in Theorem 3.5 says that the set H contains the union of two half-planes of the form for some R > 0 and F satisfies the expansivity property (3.1).
Definition 3.7 (Normalisation). We say that a logarithmic transform F : for some R > 0 and the expansivity property (3.1) is satisfied in all H. We denote this class of functions by B * n log . Logarithmic transforms of transcendental entire functions can be normalised so that H is the right half-plane H. In contrast, in the punctured plane, when we say that F is normalised we need to specify the constant R. The next lemma shows that we can always assume that F is in the class B * n log by restricting the function to a smaller set.
where δ is the curve from Theorem 3.5. For n ∈ N, the sets are bounded and hence their images F (X n ) and F (Y n ) have bounded real part. All the sets F (X n ) and F (Y n ), n ∈ N, are vertical translates of F (X 0 ) and F (Y 0 ) and hence F (T 0 ∩ H + ) and F (T ∞ ∩ H − ) have bounded real part. Therefore, there exists R 1 > 0 sufficiently large such that The following lemma is a stronger version of the expansivity property (3.1) for functions in B * n log , and says that escaping orbits eventually separate at an exponential rate. The construction in the proof of [RRRS11, Lemma 3.1] can be adapted easily to this setting.
Next we introduce a subclass of B * log consisting of the functions F for which the image F (T ) covers the whole of T and have nicer properties.
Definition 3.10 (Disjoint type). We say that a function F : consists of a single doubly-connected component U which is the immediate basin of attraction of a point in A. Note that the classification of doubly-connected Fatou components in [BD98, Theorem 4] does not apply because U is not relatively compact in C * . Remark 3.11. Independently of [RRRS11], Barański showed that the Julia set of bounded-type maps in the class B consists of disjoint hairs that are homeomorphic to [0, +∞) (we call them dynamic rays) and that the endpoints of these hairs are the only points in Example 3.12. The function f (z) = exp 0.3(z + 1/z) is in the class B * and has a logarithmic transform of disjoint type (see Figure 3).
Sometimes tracts exhibit nicer properties that make them easier to study. We will see later on that this is the case of finite order functions.
Definition 3.13 (Good geometry properties). Let F ∈ B * log and let T be a tract of F .
(a) We say that T has bounded wiggling if there exist K > 1 and µ > 0 such that for every z 0 ∈ T , every point z on the hyperbolic geodesic of T that connects z 0 to ∞ satisfies In the case K = 1 and µ = 0 we say that T has no wiggling. A function F ∈ B * log has uniformly bounded wiggling if the wiggling of all tracts of F is bounded by the same constants K, µ. (b) We say that T has bounded slope if there exist constants α, β > 0 such that We say that T has zero slope if this limit is zero.
We say F has good geometry if the tracts of F have bounded slope and uniformly bounded wiggling.
(ii) If F, G ∈ B * n log and G has bounded slope, then G • F has bounded slope with the same constants as G.

Order of growth in C *
The order of an entire function is defined to be the infimum of ρ ∈ R ∪ {∞} such that log |f (z)| = O(|z| ρ ) as z → ∞. Equivalently, Polynomials have order zero and exp(z k ), k ∈ N, has order k. There are also transcendental entire functions of order zero and of infinite order. When we deal with holomorphic self-maps of C * , controlling the growth means looking at how |f (z)| tends to zero or infinity when we approach one of the essential singularities z = 0 or z = ∞. Observe that if f is such map, then 1/f is also holomorphic on C * , and For simplicity, from now on we will write M (r) and m(r) when it is clear what the function f is.
A priori, the notion of order of growth in this context splits into the following four quantities: For entire functions, if f has no zeros then ρ(f ) = ρ(1/f ) as a consequence of the fact that you can write the order in terms of the Nevanlinna characteristic function T (R, f ): Note that in a neighbourhood of infinity the term h(1/z) is not relevant and the same happens with g(z) in a neighbourhood of the origin. Then, putting this into our definition of order for C * and using Jensen's formula we obtain that   for r > 0, where A r := {z ∈ C : 1/r < |z| < r}. It follows from the Maximum principle that M (r, f ) and m(r, f ) are respectively the maximum and minimum of |f (z)| in the whole annulus A r (in the same way that, for an entire function, we have M (r) = max z∈D(0,r) |f (z)|). In our notation, Now we will see that, in fact, every holomorphic self-map of C * that has finite order necessarily has to be as in Example 4.2. We will begin by stating a classical result concerning entire functions of finite order due to Pólya [Pól25]. While there is a huge variety of entire functions of finite order, the next result shows that having finite order in C * is a quite restrictive property.
Proposition 4.5. Every transcendental self-map of C * of finite order is of the form f (z) = z n exp(P (z) + Q(1/z)) for some n ∈ Z and P, Q ∈ C[z].
Keen proved the stronger result that every topological conjugacy class of analytic self-maps of C * contains a function of this form [Kee89, Theorem 1], but we give a direct proof of Proposition 4.5 for completeness.
Proof. We know that every transcendental self-map of C * is of the form f (z) = z n exp g(z) + h(1/z) for some n ∈ Z and g, h non-constant entire functions. It is a well-known fact that if P is a polynomial and f is a general entire function, ρ(P · f ) = ρ(f ). Then ρ(e g ) = ρ ∞ (f ) < +∞ and so it follows from Lemma 4.4 that g has to be a polynomial. On the other hand, ρ(e h ) = ρ 0 (f ) < +∞ and so h has to be a polynomial as well.
Keen also showed that, in C * , finite order implies finite type [Kee89, Proposition 2]. This is very different to what happens for the entire case, where we have functions of finite order in the class B that are not in the Speiser class S of finite-type transcendental entire functions. An example of such a function is given by sin(z)/z which has order one and infinitely many critical values in any open interval in R containing the origin. We state Keen's result for future reference.  Proof. Suppose that ρ ∞ (f ) = p and ρ 0 (f ) = q with p, q 1. By Proposition 4.5, f (z) = z n exp P (z) + Q(1/z) , where n ∈ Z and P, Q are, respectively, polynomials of degree p, q. We focus on the tracts whose closure inĈ contains infinity, the case where the closure contains zero is similar. We have where a ∈ C. Let φ = arg(a). For large values of R, the tracts of f defined by |f (z)| > R are contained in the sectors determined by the preimages of the imaginary axis by the map az p , that is the radial lines of angle (kπ + π/2 − φ)/p, k ∈ Z. Tracts that map to a neighbourhood of infinity lie in the sectors containing the radial lines of angle (2kπ − φ)/p, 0 k < p, while tracts that map to a neighbourhood of zero lie in the sectors containing the radial lines of angle ((2k + 1)π − φ)/p, 0 k < p. The preimages of radial lines by the exponential function are horizontal lines and hence the tracts of F are contained in horizontal bands and have zero slope.
Finally, since the boundaries of the tracts tend asymptotically to those horizontal lines, the tracts of F can be chosen to have no wiggling if R is sufficiently large.
It follows from Proposition 4.5 that, in the punctured plane, functions of finite order (as well as entire functions with no zeros) can only have integer orders ρ 0 (f ) and ρ ∞ (f ). There are always exactly 2ρ ∞ (f ) asymptotic paths to infinity corresponding, asymptotically, to the preimages of the positive (asymptotic value infinity) or negative (asymptotic value zero) real line by z d where d = ρ ∞ (f ). Therefore the asymptotic paths alternate as you go around a circle of large radius (see Figure 4). Similarly, in a neighbourhood of zero there are 2ρ 0 (f ) asymptotic paths with the same structure. Each of these asymptotic paths is contained in a logarithmic tract and vice versa.
Another basic property of entire functions in the class B is that they have lower order greater or equal than 1/2 [RS05a, Lemma 3.5]. This is due to the fact that f is bounded on a path to infinity. Remember that the lower order of an entire function is   Then, there is r 0 > 0 and C > 0 such that Proof of Theorem 1.3. We treat separately the cases where f has finite order and infinite order. For simplicity we only consider ρ ∞ (f ) and λ ∞ (f ), the proof for ρ 0 (f ) and λ 0 (f ) is completely analogous.
Let f (z) = z n exp g(z) + h(1/z) with n ∈ Z and g, h non-constant entire functions. Then, using equation (4.1), Suppose that ρ ∞ (f ) = p < +∞. Then, by Proposition 4.5, g is a polynomial and, since ar p , a > 0, is an increasing function for r R for some R > 0, it is clear that λ ∞ (f ) = ρ ∞ (f ). Now suppose that ρ ∞ (f ) = +∞. We use Lemma 4.8 with R = 2r, there is C > 0 and r 0 > 0 such that M (r) < 2 4A(2r) + C for all r > r 0 . Observe that if F ∈ B * log , then the tracts of F in each of the sets T 0 and T ∞ can be ordered with respect to the vertical position around infinity. Therefore it makes sense to speak about a tract being in between two other tracts. This ordering is known as the lexicographic order (see Definition 5.9) and we will come back to it later on.

Symbolic dynamics and combinatorics
Maps in class B * log are defined on a set T , which is a union of tracts, and, therefore, the orbits of some points in T are truncated if F k (z) / ∈ T for some k ∈ N. We denote by J(F ) the set of points that can be iterated infinitely many times by F . As we will show in the following lemma, the reason why J(F ) is called the Julia set of F is that points of J(F ) project to points in J(f ) by the exponential map. However, note that in the case that F ∈ B * log is the logarithmic transform of a function f ∈ B * , there exists an entire functionf that is a lift of f and then J(F ) ⊆ J(f ) = exp −1 J(f ) by a result of Bergweiler [Ber95].
Lemma 5.2. Let f be a transcendental self-map of C * and let F ∈ B * log be a logarithmic transform of f . If F ∈ B * n log , then exp J(F ) ⊆ J(f ) and, if F is of disjoint type, then exp J(F ) = J(f ).
Proof. Suppose to the contrary that z 0 ∈ exp J(F ) ∩ F (f ) = ∅. Then proceeding as in the proof of Theorem 1.1 we get a contradiction between the expansivity of F (3.1) and the fact that T does not contain vertical segments of length 2π. Note that in the normalised case we are using the expansivity with respect to the Euclidean metric, that is, |F (z)| 2 for all z ∈ T (see Lemma 3.6), while in the disjoint-type case we use the expansivity with respect to the hyperbolic metric on H because T is compactly contained in H.
If F is of disjoint type, the inclusion J(f ) ⊆ exp J(F ) follows from the fact that f (C * \ V) ⊆ A and hence F (f ) consists of the immediate basin of attraction of a point in C * \ V and If f is a transcendental self-map of C * , then the escaping set I(f ) consists of all points that accumulate to {0, ∞}. Essential itineraries describe the way points escape and were introduced in [Mar14]. Let us recall the definition here.
Definition 5.3 (Essential itinerary). Let f be a transcendental self-map of C * . We define the essential itinerary of a point z ∈ I(f ) to be the symbol sequence e = (e n ) ∈ {0, ∞} N such that For each e ∈ {0, ∞} N , we denote by I 0,0 e (f ) the set of escaping points whose essential itinerary is exactly e, We say that two essential itineraries e 1 , e 2 ∈ {0, ∞} N are equivalent if σ m (e 1 ) = σ n (e 2 ) for some m, n ∈ N. If e 1 and e 2 are not equivalent, then I e 1 (f ) ∩ I e 2 (f ) = ∅. We now introduce the escaping set for maps in the class B * log , which is a subset of the Julia set of F . Observe that exp I(F ) ⊆ I(f ) and, in fact, every point in I(f ) eventually enters exp I(F ). As with J(F ), if f is a transcendental self-map of C * andf is a lift of f , then I(F ) ⊆ I(f ) but in general these sets are different asf may have points that escape in the imaginary direction and correspond to bounded orbits of f .
For every function F ∈ B * log , we denote by A (respectively A 0 0 , A ∞ 0 , A 0 ∞ , A ∞ ∞ ) the symbolic alphabet consisting of all tracts in T (respectively T 0 0 , T ∞ 0 , T 0 ∞ , T ∞ ∞ , see Definition 3.3). We associate a symbol sequence (T n ) ∈ A N to each point z ∈ J(F ) that describes to which tract the iterate F n (z) belongs for all n ∈ N.
Definition 5.5 (External address of F ). Let F ∈ B * log and let z ∈ J(F ). We define the external address of z, addr F (z), to be the symbol sequence s = (T n ) ∈ A N such that F n (z) ∈ T n for all n ∈ N.
Remark 5.6. The Bernoulli shift map σ : A N → A N mapping the external address (T n ) to (T n+1 ) is a subshift of finite type on the set where, if e 0 , e 1 ∈ {0, ∞}, the set A e 1 e 0 × A N consists of the sequences in A N whose first symbol is in A e 1 e 0 . Observe that the transition graph of σ is where, if e 0 , e 1 2 {0, 1}, the set A e 1 e 0 ⇥ A N consists of the sequences in A N whose first symbol is in A e 1 e 0 . Observe that the transition graph of is N N N N N N N N N N and, in particular, not all sequences in A N are external addresses of points in J(F ).
We now introduce the notion of admissible external address. Only admissible external addresses can be the external address of a point in J(F ). Note that, if we define A 0 := A 0 0 t A 1 0 and A 1 := A 0 1 t A 1 1 . then an external address s = (T n ) 2 ⌃ has essential itinerary e = (e n ) provided that T n 2 A 0 if and only if e n = 0. In terms of essential itineraries, the corresponding transition graph is the complete graph on two vertices, If F 2 B ⇤n log , then z 2 I(F ) has essential itinerary e if and only if addr(z) has essential itinerary e. However, if F is not normalised, these two sequences may be different for a certain number of iterates (see Lemma 7.6).
There is a natural way to order the tracts with respect to the vertical position that they are attached to infinity. Using this, we can endow the set of sequences A e with the lexicographic order. where, if e 0 , e 1 2 {0, 1}, the set A e 1 e 0 ⇥ A N consists of the sequences in A N whose first symbol is in A e 1 e 0 . Observe that the transition graph of is N N N N N N N N N N and, in particular, not all sequences in A N are external addresses of points in J(F ).
We now introduce the notion of admissible external address. Only admissible external addresses can be the external address of a point in J(F ). Note that, if we define A 0 := A 0 0 t A 1 0 and A 1 := A 0 1 t A 1 1 . then an external address s = (T n ) 2 ⌃ has essential itinerary e = (e n ) provided that T n 2 A 0 if and only if e n = 0. In terms of essential itineraries, the corresponding transition graph is the complete graph on two vertices, If F 2 B ⇤n log , then z 2 I(F ) has essential itinerary e if and only if addr(z) has essential itinerary e. However, if F is not normalised, these two sequences may be different for a certain number of iterates (see Lemma 7.6).
There is a natural way to order the tracts with respect to the vertical position that they are attached to infinity. Using this, we can endow the set of sequences A e with the lexicographic order. where, if e 0 , e 1 2 {0, 1}, the set A e 1 e 0 ⇥ A N consists of the sequences in A N whose first symbol is in A e 1 e 0 . Observe that the transition graph of is N N N N N N N N N N and, in particular, not all sequences in A N are external addresses of points in J(F ).
We now introduce the notion of admissible external address. Only admissible external addresses can be the external address of a point in J(F ). Note that, if we define and A 1 := A 0 1 t A 1 1 . then an external address s = (T n ) 2 ⌃ has essential itinerary e = (e n ) provided that T n 2 A 0 if and only if e n = 0. In terms of essential itineraries, the corresponding transition graph is the complete graph on two vertices, If F 2 B ⇤n log , then z 2 I(F ) has essential itinerary e if and only if addr(z) has essential itinerary e. However, if F is not normalised, these two sequences may be different for a certain number of iterates (see Lemma 7.6).
There is a natural way to order the tracts with respect to the vertical position that they are attached to infinity. Using this, we can endow the set of sequences A e with the lexicographic order.  for some e = (e n ) ∈ {0, ∞} N . In this case, we say that the external address s has essential itinerary e. We denote by Σ the set of all admissible external addresses.
Note that, if we define then an external address s = (T n ) ∈ Σ has essential itinerary e = (e n ) provided that T n ∈ A 0 if and only if e n = 0. In terms of essential itineraries, the corresponding transition graph is the complete graph on two vertices, If F ∈ B * n log , then z ∈ I(F ) has essential itinerary e if and only if addr(z) has essential itinerary e. However, if F is not normalised, these two sequences may be different for a certain number of iterates (see Lemma 7.6).
For every admissible external address, we introduce the set of points that have that external address. Note that sometimes we use the term external address to denote a general sequence in Σ, without being necessarily the external address of any point z ∈ J(F ). Therefore, some of the following sets may be empty. There is a natural way to order the tracts with respect to the vertical position that they are attached to infinity. Using this, we can endow the set of sequences Σ e with the lexicographic order.
Definition 5.9 (Lexicographic order). Let F : T → H be in the class B * log . If T, T are components of T ∞ , then we say that T < T if T is in the upper connected component of the intersection of a right half-plane and the complement of T . If T, T are components of T 0 , then we say that T < T if T is in the lower connected component of the intersection of a left half-plane and the complement of T . Finally, if s, s ∈ Σ e for some e ∈ {0, ∞} N , then we say that s < s if there is k ∈ N such that T n = T n for all n < k and T k < T k .
The set Σ e endowed with the lexicographic order is a totally ordered space. Note that, since the map F preserves the orientation, if T 1 < T 2 in T ∞ and T is a component of T 0 , then with the lexicographic ordering we have F −1 T (T 1 ) < F −1 T (T 2 ). Sometimes it will be useful to consider a partition of the tracts into fundamental domains. The following terminology was introduced in [Rem08].
Definition 5.10 (Fundamental domain). Let f ∈ B * and let F : T → H be a logarithmic transform of f that is in the class B * log . Let δ ⊆ C * \ V be the curve joining zero to infinity from Theorem 3.5.
(i) The preimages exp −1 δ define infinitely many fundamental strips S n , n ∈ Z. Every tract of F is contained in a fundamental strip. (ii) For each tract T n of F , the restriction F |Tn : T n → H is a one-to-one covering of either H 0 or H ∞ . Hence, the set F −1 |Tn H \ exp −1 δ has infinitely many components F n,i ⊆ T n , i ∈ Z, that we call fundamental domains of F . (iii) Similarly, the preimages f −1 (δ) divide each tract V n of f into infinitely many sets D n,i = exp F m,i ⊆ V n , i ∈ Z, for some m ∈ Z, that we call fundamental domains of f . Note that sometimes we will refer to a sequence of fundamental domains using only one subindex when we do not need to specify whether two fundamental domains are a subset of the same tract or not.
Since the orbit of every point in J(F ) avoids exp −1 (δ), we can define external addresses in terms of fundamental domains rather than tracts. This is the approach followed, for example, in [BF15]. However, since each fundamental domain covers a fundamental strip, the fundamental domain F n is determined by tract T n where you are and the fundamental strip containing the next tract T n+1 . Thus, considering external addresses of fundamental domains does not add more information to the symbolic dynamics of F .
We can also consider external addresses for functions f ∈ B * rather than for their logarithmic transforms. In this case, specifying the sequence of tracts in V does not capture the whole combinatorics of f ; we define the external addresses of f in terms of fundamental domains. Let A f denote the symbolic alphabet consisting of the fundamental domains of f . Proof. Let (T n ) be a sequence of tracts of F , then the sequence of fundamental domains (D n ) ⊆ V is given by D n = exp F n which, in turn, is determined by T n and T n+1 .
On the other hand, if (D n ) is a sequence of fundamental domains of f , then the tract T 0 ⊇ F 0 , where exp F 0 = D 0 , is given by the choice of the logarithmic transform F , which is unique up to addition of integer multiples of 2πi, and the rest of tracts in the sequence (T n ) satisfy that T n is the only tract in the fundamental strip F (F n−1 ) containing a component of exp −1 (D n ).
We say that a sequence of fundamental domains (D n ) is admissible if it corresponds to an admissible external address s ∈ Σ. In this paper we use external addresses in terms of tracts mostly and restrict the use of fundamental domains to the moments when we need them, in order to keep the notation simpler.

Unbounded continua in J s (F )
A priori, the set J s (F ) may be empty for some external addresses in s ∈ Σ. Rippon and Stallard showed that, for a general transcendental entire function f , the components of the fast escaping set A(f ) ⊆ I(f ), which was previously introduced by Bergweiler and Hinkkanen [BH99], are all unbounded [RS05b]. Using similar ideas, Rempe showed that if f ∈ B (and the same argument follows for class B log ), then every tract T contains an unbounded closed connected set A consisting of points that escape within T [Rem08, Theorem 2.4]. Sometimes we refer to an unbounded closed connected set X ⊆ C as an unbounded continuum; note, however, that such set is not a continuum in C as it is not compact, but X ∪ {∞} is a continuum inĈ (see Lemma 6.2).
Although [Rem08,Theorem 2.4] only concerns points that escape within a tract, if s ∈ A N is a periodic external address, then it follows that J s (F ) contains an unbounded continuum of escaping points. Indeed, if s = T 0 T 1 . . . T p−1 has period p ∈ N and T k , 0 k < p, are tracts of F , then there is a tract T of F p contained in T 0 such that F k (T ) ⊆ T k , 1 k < p, and the result follows from applying [Rem08, Theorem 2.4] to F p in T .
It was remarked in [BJR12, p. 2107] that if s ∈ A N contains only finitely many symbols, then [Rem08, Theorem 2.4] can be adapted to show that J s (F ) = ∅ and hence J s (F ) contains an unbounded continuum; see [BF15, Proposition 2.11] for a detailed proof of this result.
In [Rem07], Rempe showed that this set can be chosen to be forward invariant. Later on, [BRS08, Theorem 1.1] generalised the result of Rempe for transcendental meromorphic functions in C with tracts (not necessarily in the class B).
For transcendental self-maps of C * , we can define the fast escaping set A(f ) using the iterates of the maximum and minimum modulus functions [Mar14, Definition 1.2], and the components of A(f ) are unbounded in C * . We remind that a set X ⊆ C * is unbounded if its closure X inĈ contains zero or infinity. The following lemma is a combination of [Mar14, Theorem 1.1 and Theorem 1.5] and follows from the constructions in their proofs. Remind that I e (f ) ⊆ I e (f ) is the set of escaping points whose essential itinerary is exactly e.
Lemma 6.1. Let f be a transcendental self-map of C * . For each e = (e n ) ∈ {0, ∞} N , there exists an unbounded closed connected set A e ⊆ I e (f ) which consists of fast escaping points and whose closure A e inĈ contains e 0 .
Lemma 6.1 implies that the set J e (F ) contains at least one unbounded component. The goal of this section is to show that, under certain hypothesis, the set J s (F ) contains an unbounded continuum. We begin by stating the Boundary bumping theorem [Nad92, Theorem 5.6] (see also [RRRS11,Theorem A.4]) which implies that if X ⊆Ĉ is a compact connected set containing zero or infinity and E = X ∩ C * , then every component of E is unbounded in C * . Lemma 6.2 (Boundary bumping theorem). Let X be a nonempty compact connected metric space and let E X be nonempty. If C is a connected component of E, then ∂C ∩ ∂E = ∅ (where boundaries are taken relative to X).
First we show that if J K s (F ) = ∅ for sufficiently large K > 0, then the set J s (F ) contains an unbounded continuum. The following lemma is the analogue of [RRRS11, Lemma 3.3] for the class B log . We include the proof for completeness. Proposition 6.3. Let F ∈ B * log , there exists K 1 (F ) 0 such that if K K 1 (F ), for every s ∈ Σ, if z 0 ∈ J K s (F ), then there exists an unbounded closed connected set A ⊆ J s (F ) with dist (z 0 , A) 2π.
Proof. We may assume without loss of generality that F is normalised with H = H ± R for some R > 0. Let K 1 (F ) > 0 be large enough that if K K 1 (F ), then all bounded components of H ∩ T are in the vertical band V K := {z ∈ C : | Re z| < K}. Note that the set V K can only intersect a finite number of tracts in each fundamental strip.
Let Y ⊆ C be an unbounded continuum such that Y \ B(F k (z 0 ), 2π) has exactly one unbounded component. In that case we denote this component by X k (Y ). Let s = (T n ). For all k 1, we have that ∅ = X k (T k ) ⊆ H and hence F −1 |T k−1 maps X k (T k ) into T k−1 . By the expansivity property (3.1),

Thus we can define the sets
and we put A 0 := X 0 (T 0 ). Observe that here we are using the fact that s ∈ Σ because F −1 T k is only defined in one of the two components of H. Let A k denote the closure of A k inĈ which is a continuum. By construction, A k+1 ⊆ A k and dist(z 0 , A k ) π, thus 2π. Finally, by Lemma 6.2, the set A is unbounded in C * .
Next we show that, as in the entire case, if an external address s ∈ Σ has finitely many symbols, then the set J s (F ) contains an unbounded continuum. Note that in contrast to the previous proposition, now we need to show that J s (F ) = ∅. We use the following lemma which is the analogue of [BF15, Proposition 2.6] for the class B * .
Lemma 6.4. Let F ∈ B * log have good geometry and let F be a finite union of fundamental domains of F . Then for any K > 0 sufficiently large, In the following proposition we adapt the proof of [BF15, Proposition 2.11] to our setting. This is based on the ideas of [Rem08, Theorem 2.4] and will be used later to prove Theorem 1.6. Proposition 6.5. Let F ∈ B * log . There exists K 2 (f ) > 0 such that if K K 2 (F ) and s ∈ Σ contains finitely many different symbols, then J K s (F ) contains a continuum whose points have unbounded real part.
Proof. Suppose that s = (T n ) contains N different symbols for tracts T s 1 , . . . , T s N from T and choose, for each 1 j N , N fundamental domains F s j,k ⊆ T s j so that F (F s j,k ) ⊇ T s k . Let F denote the finite collection of fundamental domains {F s j,k }, and assume K 2 = K 2 (F ) > 0 is sufficiently large that Lemma 6.4 holds for F and K > K 2 (F ). Then define (F n ) to be the sequence of fundamental domains from F satisfying that F n ⊆ T n and T n+1 lies in F (F n ).
Let X 0 be the unbounded component of F 0 ∩ H ± K and, for each n > 0, let X n be the unique unbounded component of where F −1 |Fn is the branch of F −1 that maps the fundamental strip F (F n ) ⊆ H in which F n+1 lies to the fundamental domain F n ⊆ T n . Note that since F is entire, F −1 |Fn maps unbounded sets to unbounded sets. Lemma 6.4 tells us that F −1 (∂H ± K ) ∩ F ⊆ C \ H ± K and therefore for each F n ∈ F, Thus, since F n ∩ ∂H ± K = ∅, we have that X n ∩ ∂H ± K = ∅ for all n ∈ N. As before, let X n be the closures of X n inĈ and define X := k∈N X n which is an unbounded continuum. Since all X n intersect ∂H, X \ {0, ∞} has a component X that intersects ∂H ± K and is unbounded by Lemma 6.2. In particular, Proposition 6.5 includes all the periodic external addresses in Σ.
Observe that considering external addresses that consist of fundamental domains instead of tracts we would obtain the result that for all such sequences containing only finitely many different fundamental domains of f there is an unbounded continuum consisting of escaping points with that extended external address.

Dynamic rays
In Theorem 1.1 we showed that bounded-type functions have no escaping Fatou components. Instead, escaping points often lie in curves tending to the essential singularities -called dynamic rays or, sometimes, hairs-such that in every unbounded proper subset -called ray tail -points escape uniformly. We say that a dynamic ray is broken if one of its forward iterates contains a critical point; this concept was introduced in [BF15, Definition 2.2].
Definition 7.1 (Dynamic ray). Let f be a transcendental self-map of C * . A ray tail of f is an injective curve γ : [0, +∞) → I(f ) such that f n (γ(t)) → {0, ∞} as t → +∞ for all n 0 and f n (γ(t)) → {0, ∞} uniformly in t as n → ∞. A dynamic ray of f is a maximal injective curve γ : (0, +∞) → I(f ) such that γ| [t,+∞) is a ray tail for every t > 0. Similarly, we can define ray tails for any logarithmic transform F of f (only defined on the set T ), and dynamic rays for any liftf of f . We shall abuse the notation and use γ for both the ray as a set and its parametrization.
We say that a dynamic ray γ is broken if f n (γ) contains a critical point for n ∈ N. A non-broken ray γ is said to land if γ \ γ consists of a single point or, in other words, if γ(t) has a limit as t → 0.
Example 7.2. We give a couple of straightforward examples of dynamic rays in C * .
(i) The positive real line is a fixed dynamic ray for f (z) = exp(z + 1/z), and points escape to ∞ under iteration. This is an example of a broken ray because the function f has a critical point at z = 1. (ii) If we now consider the function g(z) = exp(−z + 1/z), the positive real line is again forward invariant but z = 1 is a repelling fixed point of g. In this case, the intervals (0, 1) and (1, +∞) form a cycle of 2-periodic non-broken dynamic rays.
Observe that dynamic rays are allowed to land at an essential singularity; that is, the limit of γ(t) as t → 0 and t → +∞ may coincide. The dynamic ray from the following example is non-broken and goes from zero to infinity.
Example 7.3. The positive real line is a fixed and non-broken dynamic ray for the function f (z) = z exp z 2 + exp(−1/z 2 ) (see Figure 6). Since the exponential function is a local homeomorphism, we have the following correspondence between dynamic rays of transcendental self-maps of C * and those of their lifts.
Lemma 7.4. Let f be a transcendental self-map of C * and letf be a lift of f . Then γ is a dynamic ray of f if and only if any connected componentγ of exp −1 γ is a dynamic ray off . Furthermore, γ lands or is broken if and only if γ lands or is broken, respectively.
It is a well-known result for entire functions that if the postsingular set is bounded then all periodic dynamic rays land. This was first proved for the exponential family [SZ03b,Rem06]. Rempe proved a more general version of the result for Riemann surfaces that applies to maps in the classes B and B * [Rem08, Theo-rem B.1]; see also [Den14, Theorem 1.1] for an alternative proof of this result for the class B. The same techniques imply the following result in our setting.
Proposition 7.5. Let f ∈ B * with postsingular set P (f ) bounded away from zero and infinity. Then all periodic dynamic rays of f land, and the landing points are either repelling or parabolic periodic points of f .
Next we show that, since points in ray tails escape uniformly, each dynamic ray is contained in a set I e (f ) for some essential itinerary e ∈ {0, ∞} N .
Lemma 7.6. Let f be a transcendental self-map of C * and let γ be a dynamic ray of f . Then, for every ray tail γ ⊆ γ, there is ∈ N such that all the points in f (γ ) have the same essential itinerary. Hence, there exists an essential itinerary e ∈ {0, ∞} N such that γ ⊆ I e (f ).
Proof. By definition, ray tails escape uniformly and hence, if γ is a ray tail, there is ∈ N such that f n (γ ) ∩ S 1 = ∅ for all n . Then, all points in f (γ ) have the same essential itinerary; that is, γ ⊆ I ,0 e (f ) for some e ∈ {0, ∞} N . Now suppose that γ is a dynamic ray with z 1 ∈ γ ∩ I e 1 (f ) and z 2 ∈ γ ∩ I e 1 (f ). Then there is a ray tail γ ⊇ {z 1 , z 2 } and ∈ N such that all points in f (γ ) have the same essential itinerary. Thus, e 1 ∼ = e 2 and γ ⊆ I e 1 (f ) = I e 2 (f ).
Actually, since all the images of a dynamic ray are unbounded in C * , dynamic rays are asymptotically contained into tracts which are preimages of the neighbourhood W of the set {0, ∞}. Furthermore, each dynamic ray is asymptotically contained in exactly one of the fundamental domains of the function F .
In the following proposition we show that, in order to prove Theorem 1.4, we only require that every escaping point has an iterate that is on a ray tail (see [RRRS11,Proposition 2

.3]).
Proposition 7.7. Let f be a transcendental self-map of C * and let z ∈ I(f ).
Suppose that some iterate f k (z) is on a ray tail γ k of f . Then either z is on a ray tail, or there is some n k such that f n (z) belongs to a ray tail that contains an asymptotic value of f .
Again, it cannot happen that f (w) ∈ {0, ∞} because γ k (T ) would be an asymptotic value, so f (w) = γ k (t 0 ) for some t 0 ∈ [0, ∞). In this case, γ k−1 could be extended, contradicting its maximality. Note that if w was a critical point we would need to choose a branch of the inverse. Thus, w ∈ {0, ∞} and γ k (T ) is an asymptotic value of f (possibly zero or infinity). Then either we have found a ray tail γ k−1 ⊆ f −1 (γ k ) ⊆ I(f ) connecting f (k−1) (z) to one of the essential singularities or γ k contains an asymptotic value. The result follows from applying the above reasoning inductively.
Note that Proposition 7.7 can also be proved by applying its version for entire functions to a liftf of f and then use the correspondence from Lemma 7.4.
We conclude this section by stating a result about escaping points that follows from the expansivity property (3.1) in Lemma 3.6 (see [RRRS11,Lemma 3.2] for the analogue result on entire functions).
Lemma 7.8. Let F : T → H be in the class B * n log with H = H ± R for some R > 0. If z, w ∈ J s (F ) for some external address s and z = w then Observe that this does not imply that neither the point z nor w escape because both points may have an unbounded orbit but with a subsequence where their iterates are bounded. In the next section we will introduce a condition for F (see Definition 8.1) which implies that, in the situation of Lemma 7.8, both points z and w escape, and hence all points in J s (F ) except possibly one must escape.
Lemma 3.9, Lemma 7.8 and Proposition 6.3 correspond respectively to Lemma 3.1, Lemma 3.2 and Theorem 3.3 in [RRRS11, Section 3] and constitute the main tools to prove Theorem 1.4 in the next section.
8. Proof of Theorem 1.4 In this section we adapt the results in [RRRS11, Sections 4 and 5] to our setting. Since the proof Theorem 1.4 follows closely that of [RRRS11, Theorem 1.2], we only sketch it and emphasize the differences between them.
The head-start condition is designed so that every escaping point is mapped eventually to a ray tail and hence we are able to apply Proposition 7.7 and conclude that either the point itself is in a ray tail or some iterate is in a ray tail that contains a singular value.
Definition 8.1 (Head-start condition). Let F : T → H be a function in the class B * log . We first define the head-start condition for tracts, then for external addresses and finally for logarithmic transforms.
• Let T, T be two tracts in T and let ϕ : R + → R + be a (not necessarily strictly) monotonically increasing continuous function with ϕ(x) > x for all x ∈ R + . We say that the pair (T, T ) satisfies the head-start condition for ϕ if, for all z, w ∈ T with F (z), F (w) ∈ T , • We say that an external address s ∈ Σ satisfies the head-start condition for ϕ if all consecutive pairs of tracts (T k , T k+1 ) satisfy the head-start condition for ϕ, and if for all distinct z, w ∈ J s (F ), there is M ∈ N such that • We say that F satisfies a head-start condition if every external address of F satisfies the head-start condition for some ϕ. If the same function ϕ can be chosen for all external addresses, we say that F satisfies the uniform head-start condition for ϕ.
Notice that in the second part we require that the head-start condition cannot be a void condition for any itinerary. Furthermore, if | Re F M (z)| > ϕ(| Re F M (w)|) and the head-start condition is satisfied for that pair of tracts then for all n > M , The head-start condition allows us to order the points in J s (F ) by the growth of the absolute value of their real parts.
Definition 8.2 (Speed ordering). Let s ∈ Σ be an external address satisfying the head-start condition for a function ϕ. For z, w ∈ J s (F ), we say that z w if there exists K ∈ N such that | Re F K (z)| > ϕ(| Re F K (w)|). We extend this order to the closure J s (F ) inĈ by the convention that 0, ∞ z for all z ∈ J s (F ).
Note that although a dynamic ray may contain both zero and infinity in its closure inĈ, ray tails are a subset of T and hence contain either zero or infinity.
The head-start condition implies that the speed ordering is a total order on the then we would get a contradiction because once we are in one of these situations and the head-start condition is satisfied then it is preserved by iteration, that is, for example, if | Re F M 1 (z)| > ϕ(| Re F M 1 (w)|), then | Re F n (z)| > ϕ(| Re F n (w)|) for all n > M 1 . Therefore z w if and only if there exists n 0 ∈ N such that | Re F n (z)| > | Re F n (w)| for all n > n 0 , and hence the speed ordering does not depend on the choice of the function ϕ.
Lemma 8.3. Let s ∈ Σ e , e ∈ {0, ∞} N , be an external address that satisfies the head-start condition for a function ϕ. Then the order topology induced by the speed ordering on J s (F ) coincides with the topology as a subset ofĈ and, in particular, every connected component of J s (F ) is an arc.
Moreover, there exists K > 0 independent of s such that J K s (F ) is either empty or contained in the unique unbounded component of J s (F ), which is an arc to the essential singularity e 0 all of whose points escape except possibly its finite endpoint.
Proof. The first part follows from the fact that the map id : J s (F ) → ( J s (F ), ≺) is an homeomorphism (see [RRRS11,Theorem 4.4]). Indeed, for all a ∈ J s (F ), the sets (a, +∞) ≺ := {z ∈ J s (F ) : a ≺ z}, (−∞, a) ≺ := {z ∈ J s (F ) : z ≺ a}, are open sets in J s (F ) with the subspace topology ofĈ: let k ∈ N be minimal with the property that | Re F k (a)| > ϕ(| Re F k (z)|) then, by continuity, this inequality holds in a neighbourhood of z. Since J s (F ) and the order topology is Hausdorff, the map id −1 is continuous as well. The theorem follows from the order characterisation of the arc (see [RRRS11,Theorem A5]).
For the second part, if K is the constant from Lemma 7.8(ii) and J K s (F ) = ∅, then J K s (F ) has an unbounded component A which is an arc to ∞. Since e 0 is the largest element of J s (F ) in the speed ordering, the set J s (F ) has only one unbounded component. Using the head-start condition, it can be shown that if z, w ∈ J s (F ) and w z then w ∈ I s (F ) (see [RRRS11,Corollary 4.5]). Finally, the fact that J K s (F ) ⊆ A for some K > K follows from the expansivity of F (see [RRRS11,Proposition 4.6]).
Like in the entire case, the following theorem can be deduced from Lemma 8.3 (see [RRRS11,Theorem 4.2]).
Theorem 8.4. Let F ∈ B * log satisfy a head-start condition. Then, for every escaping point z, there exists k ∈ N such that F k (z) is on a ray tail γ. This ray tail is the unique arc in J(F ) connecting F k (z) to either zero or infinity (up to reparametrization).
Observe that Theorem 8.4 together with Proposition 7.7 imply that if f is a transcendental self-map of C * and z ∈ I(f ), then either z is on a ray tail, or there is some n k such that f n (z) belongs to a ray tail that contains an asymptotic value of f .
Previously we have seen that if f has finite order then any logarithmic transforms F of f has good geometry in the sense of Definition 3.13. To complete the proof of Theorem 1.4 we show that functions of good geometry satisfy a head-start condition.
Theorem 8.5. Let F ∈ B * n log be a function with good geometry. Then F satisfies a linear head-start condition.
Proof. Let s ∈ Σ be an external address and suppose that F has bounded slope with constants (α, β). Then the orbits of any two points z, w ∈ J s (F ) eventually separate far enough one from the other. More precisely, if K 1, there exist a constant δ = δ(α, β, K) > 0 such that if |z − w| δ, then | Re F n (z)| > K| Re F n (w)|+|z −w| or | Re F n (w)| > K| Re F n (z)|+|z −w|, for all n 1 (see [RRRS11, Lemma 5.2]). Hence the external address s satisfies the second part of the head-start condition with the linear function ϕ(x) = Kx+δ.
It remains to check that if s = (T n ), for all k ∈ N and for all z, w ∈ T k such that F (z), F (w) ∈ T k+1 , | Re w| > K| Re z| + δ ⇒ | Re F (w)| > K| Re F (z)| + δ.
We skip the technical computations from this proof which are identical to the ones for the entire case, and just observe that this follows from the fact that the tracts of F have uniformly bounded wiggling with constants K and µ for some µ > 0 if and only if the conditions | Re w| > K| Re z| + M | Im F (z) − Im F (w)| α max{| Re F (z)|, | Re F (w)|} + β imply that | Re F (w)| > K| Re F (z)| + M whenever z, w ∈ T , for some M > 0, and hence F satisfies the uniform linear head-start condition with constants K and M for some M > 0 (see [RRRS11,Proposition 5.4]). This implies, for example, that each fundamental domain D of f contains exactly one fixed ray because the constant external address (D n ) with D n = D for all n ∈ N is unique.
In Lemma 6.1, which summarizes some results from [Mar14], we saw that if f is a transcendental self-map of C * and e ∈ {0, ∞} N , then the set I e (f ) contains an unbounded closed connected subset A e . Furthermore, if f ∈ B * and satisfies the hypothesis of Theorem 1.4, then Theorem 1.6 implies that the set I e (f ) contains a ray tail; note that a dynamic ray may intersect the unit circle and hence contain points that are not in I e (f ). Therefore, in this case, since the set {0, ∞} N has uncountably many non-equivalent sequences e and two such sequences give disjoint sets I e (f ), the escaping set I(f ) contains uncountably many rays.
As explained in the introduction, a stronger result is true, namely Theorem 1.7, which states that for every essential itinerary e ∈ {0, ∞} N , the set I e (f ) contains a Cantor bouquet and, in particular, uncountably many hairs. With the goal in mind of proving this theorem, we start by giving a precise definition of a Cantor bouquet (see [AO93, Definition 1.2]). every (x, y) ∈ B, there exist two sequences of hairs attached respectively at β n , γ n ∈ R \ Q such that β n < y < γ n for all n ∈ N, and β n , γ n → y and t βn , t γn → t y as n → ∞. The set [t y , +∞) ×{y} is called the hair attached at y and the point (t y , y) is called its endpoint. A Cantor bouquet is a set X ⊆ C that is ambiently homeomorphic to a straight brush.
First we are going to show that, for each essential itinerary e ∈ {0, ∞} N , the set J(F ) contains an absorbing set X e consisting of hairs so that every point in the set I e (F ) enters X e after finitely many iterations (see [RRRS11,Theorem 4.7]). Recall that, for e ∈ {0, ∞} N , we defined the set J e (F ) := {z ∈ J(F ) : addr F (z) ∈ Σ e } = s∈Σe J s (F ).
It will be helpful to use the following notation: for each e ∈ {0, ∞} N , we define the set of sequences Proposition 9.2. Suppose that F ∈ B * log satisfies a head-start condition. Then, for every e ∈ {0, ∞} N , there exists a closed subset X e ⊆ J + e (F ) with the following properties: (a) F (X e ) ⊆ X e . (b) The connected components of X e are closed arcs to infinity all of whose points except possibly of its endpoint escape. (c) Every point in I e (F ) enters the set X e after finitely many iterations. If F is of disjoint type, then we may choose X e = J + e (F ) and if F is 2πi-periodic, then X e can also be chosen to be 2πi-periodic.
Proof. Let X e be the union of all unbounded components of the set J e (F ), and define the set X e := n∈N X σ n (e) .
Since unbounded components of J(F ) map to unbounded components of J(F ) by F , we have F (X e ) ⊆ X σ(e) and hence X e is forward invariant. By Lemma 6.2, the closure X e inĈ is the connected component of J + e (F ) ∪ {∞} that contains infinity and hence the set X e is closed. By Lemma 8.3, the set X e consists of arcs to infinity all of whose points except possibly of its endpoint escape.
Let K 0 be the constant from Lemma 8.3, independent of s ∈ Σ, so that J K s (F ) is either empty or contained in the unbounded component of J s (F ) which is contained in X e if s ∈ Σ + e . Then (c) follows from the fact that points in I e (F ) enter a set J K σ n (e) (F ) ⊆ X e , n ∈ N, after finitely many iterations. Finally, if F is of disjoint type, then J e (F ) ∪ {∞} = s∈Σe n∈N F −1 |T 0 · · · F −1 |T n−2 F −1 |T n−1 (H en ) · · · ∪ {∞} which is a union of nested intersections of unbounded continua, hence every component of J e (F ) is an unbounded continuum and we can choose X e = J e (F ). If F is a 2πi-periodic function, then the set X e is also 2πi-periodic.
Following [BJR12], the strategy to prove Theorem 1.7 will be, for each essential itinerary e ∈ {0, ∞} N , to compactify the space of admissible external addresses Σ e by adding a circle of addresses at infinity to show the set X e (and hence X e ) contains a Cantor bouquet. This is done by defining intermediate entries of each set T e 1 e 0 , e 0 , e 1 ∈ {0, ∞}, symbols which correspond to entries in between pairs of adjacent tracts as well as to limits of sequences of tracts. We then add intermediate external addresses to the set Σ e , that is, finite sequences of the form s = T 0 T 1 . . . T n−1 S n , where T j ∈ T e j+1 e j , 0 j < n, and S n is an intermediate entry of the set T e n+1 en . We refer to [BJR12, Section 5] for the details.
Finally we sketch the proof of Theorem 1.7. The main idea is to use the existence of a potential function ρ that 'straightens' the brush X e (see [BJR12,Proposition 7.1]).
Proof of Theorem 1.7. Let F ∈ B * log be 2πi-periodic and satisfy a uniform headstart condition and let X e denote the union of the unbounded components of J e (F ) as in Proposition 9.2. For each e ∈ {0, ∞} N , consider the set Z e := {z ∈ X e : ρ F j (z) K for all j 0} ∪S e , where ρ is a 2πi-periodic continuous function that is strictly increasing on the hairs and such that ρ(z n ) → ∞ if and only if | Re z n | → +∞. Then, there exists R > 0 sufficiently large so that J R e (F ) ⊆ Z e ⊆X e and hence Z e is a comb. Then Lemma 9.7 together with the fact that F satisfies a uniform head-start condition imply that Z e is a hairy arc and, by Lemma 9.6, Z e \S e is ambiently homeomorphic to a straight brush. We can choose the set X e from Proposition 9.2 to be 2πi-periodic and so both J e (F ) and exp(J e (F )) contain an absorbing Cantor bouquet. Note that all the points in exp(J e (F )) belong to I 0,0 e (f ) except, possibly, the finite endpoints of the hairs. Finally, if F is of disjoint type, then the closure of J e (F ) inH e is a one-sided hairy arc, and hence both J e (F ) and exp(J e (F )) are Cantor bouquets.