This is a preprint of: “A bound on the number of rationally invisible repelling orbits”, Anna Miriam Benini, Nuria Fagella, Adv. Math., vol. 370, 107214:1–28, 2020. DOI: [10.1016/j.aim.2020.107214] A bound on the number of rationally invisible repelling orbits Anna Miriam Benini ∗ and Núria Fagella † *Dipartimento di Scienze Matematiche Fisiche e Informatiche, Università di Parma, IT † Dep. de Matemàtiques i Informàtica, Universitat de Barcelona, Catalonia, ES Abstract We consider entire transcendental maps with bounded set of singular values such that periodic rays exist and land. For such maps, we prove a refined version of the Fatou- Shishikura inequality which takes into account rationally invisible periodic orbits, that is, repelling cycles which are not landing points of any periodic ray. More precisely, if there are q < ∞ singular orbits, then the sum of the number of attracting, parabolic, Siegel, Cremer or rationally invisible orbits is bounded above by q. In particular, there are at most q rationally invisible repelling periodic orbits. The techniques presented here also apply to the more general setting in which the function is allowed to have infinitely many singular values. 1 Introduction Consider an entire transcendental function f and let S(f ) be its set of singular values S(f ) := {asymptotic values, critical values}. Then f : C \ f −1 (S(f )) → C \ S(f ) is an unbranched covering of infinite degree. The closure of the orbits of all singular values is called the postsingular set and is denoted by [ P(f ) := f n (s). s∈S(f ),n>0 A singular value v is non-recurrent if it does not belong to its ω-limit set ω(v), defined as the set of accumulation points for the orbit {f n (v)}n∈N . Many of the intricate patterns that arise in the dynamics of holomorphic maps are due to the presence of singular values and to the way in which their orbits interact with each other. ∗ This project has received funding from the European Union’s Horizon 2020 research and innovation pro- gramme under the Marie Sklodowska-Curie Grant Agreement No. 703269 COTRADY † Partially supported by the Spanish grant MTM2017-86795-C3-3-P, the Maria de Maeztu Excellence Grant MDM-2014-0445 of the BGSMath and the catalan grant 2017 SGR 1374. 1 For example, it is not difficult to show that if the unique singular value of a quadratic poly- nomial is non-recurrent then the Julia set is locally connected [DH85, Exposé X]. Similarly, the presence of non-repelling periodic orbits is entangled with the behavior of singular orbits. For example, every immediate attracting or parabolic basin needs to contain a singular orbit, and each Cremer point or point in the boundary of a Siegel disk needs to be accumulated by points in the postsingular set [Fat20, Mil06]. As a consequence of this deep relationship, it is possible to give an upper bound for the number of non-repelling cycles in terms of the number of singular values. This is known as the Fatou-Shishikura inequality [Shi87, EL92], and it states that if an entire map (polynomial or transcendental) has q singular values, then Nnon-repelling ≤ q, where Nnon-repelling stands for the number of attracting, parabolic, Cremer and Siegel cycles. The proof of this celebrated result relies on perturbations in parameter space. However, with additional dynamical assumptions on the map (for example, bounded postsingular set), a more combinatorial approach in the dynamical plane also associates each non-repelling cycle to a singular orbit in a precise mathematical way, and in such a way that the latter cannot be associated to any other non-repelling cycle [Kiw00, BF17b]. To be somewhat more precise on this extra assumption, we must talk about rays [DH85, Mil06, RRRS11]. For polynomials, and for many transcendental maps the escaping set, defined as I(f ) := {z ∈ C; f n (z) → ∞}, consists of injective, mutually disjoint curves G : (0, ∞) → I(f ) tending to infinity as t → ∞. These are called external rays for polynomials and dynamic rays (or hairs) for transcendental maps (see Section 2 for a precise definition), although in this paper we will often call them just rays. A ray G is periodic if f n (G) ⊂ G for some n ∈ N, and we say that it lands at a point z0 ∈ C if G(t) → z0 as t → 0. Periodic rays can only land at parabolic or repelling periodic points by the Snail Lemma [Fat20, Mil06]. Polynomial rays foliate the attracting basin of infinity and hence lie in the Fatou set. Their landing is tightly related to the topology of the Julia set. Indeed, the Julia set is locally connected if and only if all rays land, in which case the Julia set can be parametrized by the unit circle. For transcendental maps, the situation is more complex. To start with, it is not always true that the escaping set is formed by rays, although this is the case for a wide class of entire transcendental functions [RRRS11, BRG17]. This class includes the class Brays of functions which are finite compositions of functions of finite order with bounded set of singular values, and for such functions the escaping set lies entirely in the Julia set. In the case of polynomials or maps in Brays , if the postsingular set is bounded all periodic rays land (at repelling or parabolic) periodic points [DH85, Hub93, Mil06, Rem08, Den14]. Conversely, one may ask whether every repelling or parabolic point is the landing point of a ray or, in other words, whether repelling and parabolic points are always accessible from the escaping set. The answer to this question is not always positive and motivates the following definition. Definition 1.1 (Rationally invisible periodic orbit). A repelling periodic orbit of an entire 2 map (polynomial or transcendental) is called rationally invisible if one of the points in the orbit (and hence all of them) is not the landing point of any periodic ray. The non-existence of rationally invisible periodic orbits, whenever it can be proven, has consequences for the study of parameter spaces. In polynomial dynamics, for example, it represents the starting point for Yoccoz puzzle and for much of the machinery which lead to most of the actual rigidity results. In transcendental dynamics, it is related to the non- existence of ghost limbs attached to hyperbolic components. It is therefore of interest to understand the situations under which these special orbits may exist. As it turns out, rationally invisible orbits, despite being repelling, are also tightly related to the orbits of the singular values and more precisely, to unbounded singular orbits. Indeed, if an entire map (polynomial or transcendental in Brays ) has a bounded postsingular set, then every repelling or parabolic periodic point is the landing point of at least one and at most finitely many periodic rays, and hence there are no rationally invisible orbits (see [DH85, Hub93, Mil06] for polynomials, [BF15, BL14, BRG17] for transcendental). In the absence of this boundness restriction, one would like to give an upper bound for the number of rationally invisible periodic orbits in terms of the number of singular values, so as to produce a refinment of the Fatou-Shishikura inequality. This is indeed the case for polynomials [LP96, Corollary 1], [BCL+ 16], and also for transcendental maps, as we show in the main result of this paper (see also Theorem 5.1 for a stronger statement). As usual we say that a singular value escapes along periodic rays if its orbit converges to infinity and eventually belongs to a cycle of periodic rays. Main Theorem. Let f ∈ Brays such that periodic rays land and assume that there are no singular values escaping along periodic rays. Suppose that f has at most q < ∞ singular orbits which do not belong to attracting or parabolic basins. Let Nindifferent denote the number of Cremer cycles and cycles of Siegel disks, and Ninvisible denote the number of rationally invisible orbits. Then we have Nindifferent + Ninvisible ≤ q. In particular, there are at most q rationally invisible repelling periodic orbits. One may wonder about how strong is the assumption that periodic rays land in the transcendental setting. For polynomials, this assumption is equivalent to the assumptions that no critical points escape along periodic external rays and is implied by the standard assumption of the Julia set being connected. It is expected that also in the transcendental case a periodic ray lands unless its forward orbit contains a singular value. This has been proven for functions in the exponential family using parameter space based arguments [Rem06], which seem to be out of reach even for functions with finitely many singular values. The assumption that there are no singular values escaping along periodic rays is evidently weaker that the hypothesis that periodic rays do not intersect the postsingular set. In fact, the latter hypothesis implies landing of periodic rays [Rem08]. The Main Theorem has the following immediate corollary. Corollary 1.2. Let f ∈ Brays such that periodic rays land and assume that there are no singular values escaping along periodic rays. Suppose that f has at most q singular orbits 3 which do not belong to attracting or parabolic cycles. Then there are at most q repelling periodic orbits which are rationally invisible. The only previous known result in the direction of putting a bound on the number of rationally invisible periodic orbits of transcendental maps is due to Rempe-Gillen [Rem06], and states that for any fc (z) = ez + c there is at most one rationally invisible periodic orbit. The proof uses arguments in the parameter space of the exponential family and relies crucially on the existence and structure of wakes in the parameter plane. Instead, the proof that we present in this paper uses the structure of the dynamical plane carved by periodic rays [BF15], and is of a local nature. As a bonus, it also gives more information about the accumulation behavior of the singular orbits (see Theorem 5.1). As a concluding remark, let us note that in exponential dynamics, for parameters for which the postsingular set is bounded there are no rationally invisible repelling periodic orbits, and this fact implies that such parameters cannot belong to ghost limbs attached to hyperbolic components (See Theorem 4, the final conjecture in [Rem06], and the last section in [BL14]). This has also been used for some of the rigidity results in [Ben15]. This type of results increase our current knowledge of parameter spaces. For families of transcendental functions with more than one singular value this knowledge is currently very limited, but there is no doubt that it will undergo an important development in the next decades. We hope that the results and the techniques developed in this paper will be a little brick in the implementation of this large project. The paper is structured as follows. Section 2 contains the background about functions in class Brays and their combinatorics and presents the Separation Theorem [BF15], a key tool for the proof of the main result. It describes also fundamental tails, objects introduced in [BRG17] which can be seen as intermediate steps in the construction of rays, and which despite their intricate combinatorics, have proven to be useful in the proof of several recent results. In Section 3 we give a characterization of landing of periodic rays in terms of some combinatorics of tails. Meanwhile, Section 5 contains the statement and the proof Theorem 5.1, from which the Main Theorem follows, a relation which is made explicit in Section 5. Section 5 contains also a corollary (see Corollary 5.7) stating that, under our assumptions, the union of the dynamical fibers (as in the definition of [RS08]) of a rationally invisible repelling periodic orbit contains either a singular orbit, or infinitely many singular values whose orbits belong to the fiber for more and more iterations. Notation Let C denote the complex plane, D the unit disk. The Euclidean disk of center z and radius r is denoted by Dr (z). By a (univalent) preimage under f n of an open connected set V we mean a connected component U of the set f −n (V ) (such that f n : U → V is univalent). Given a set A and k ∈ N we denote by {A}k the set A × . . . × A where the product is taken k times. 4 Acknowledgments We are thankful to Misha Lyubich for enlightening discussions and to the Centro di Ricerca Matematica Ennio de Giorgi in Pisa for their hospitality, under the Research in Pairs program. 2 Background Tracts, fundamental domains, and dynamic rays Let f be an entire transcendental function with bounded set of singular values and let D be a Euclidean disk containing S(f ) and f (0). The connected components of f −1 (C \ D) are called tracts [EL92] and are unbounded and simply connected. By definition for any tract T we have that f : T → C \ D is an unbranched covering of infinite degree. Let T be the union of all tracts. It is not difficult to find an analytic curve δ ⊂ C \ (D ∪ T ) connecting ∂D to ∞ ([Rot05]; see also [BF15, Lemma 2.1]). Let Ω := C \ (D ∪ δ). The connected components of f −1 (Ω) are called fundamental domains. It is easy to see that only finitely many fundamental domains intersect D and that for any fundamental domain F we have that f : F → Ω is a biholomorphism. We denote by F the collection of all fundamental domains, as well as their union. The structure of the dynamical plane given by tracts and fundamental domains has been useful to construct dynamic rays. The initial idea of finding curves in the escaping set of transcendental entire functions goes back to [Fat26], was later developed in [DT86], [DGH86], [DK84], [BK07], [Bar07], [RRRS11] among others. Definition 2.1 (Dynamic ray). A (dynamic) ray for f is an injective curve G : (0, ∞) → I(f ) such that: (a) lim |f n (G(t))| = ∞ ∀n ≥ 0; t→∞ (b) lim |f n (G(t))| = ∞ uniformly in [t0 , ∞) for all t0 > 0; n→∞ (c) f n (G(t)) is not a critical point for any t > 0 and n ≥ 0; and such that G(0, ∞) is maximal with respect to these properties. If G(0, ∞) is maximal with respect to (a) and (b) but not with respect to (c), then we call the ray broken. Broken rays could therefore be continued if we allowed critical points and their iterated preimages to be part of the ray, as it is the case in the definition in [RRRS11], where branching might occur and several rays might share one same arc. This situation cannot happen in our setting, i.e. rays are pairwise disjoint. A dynamic ray G is periodic if f p (G) = G for some p ≥ 1, and fixed if p = 1. We say that a dynamic ray lands at a point z0 ∈ C if it is not broken and lim G(t) = z0 as t → 0. Observe that dynamic rays are allowed to land at singular values, but that broken rays are not considered to land. Recall that Brays denotes the class of transcendental entire functions which are finite compositions of functions of finite order with bounded set of singular values. In [RRRS11, 5 Theorem 1.2] it is shown that for any f ∈ Brays and for any escaping point z then f n (z) belongs to a dynamic ray for any n large enough. For this paper we need to take into account a combinatorial description of dynamic rays, which is implicitly contained in [RRRS11] and in several of the aforementioned papers but for which we use the explicit setup that has been presented in [BF15]. We say that a dynamic ray G is asymptotically contained in a fundamental domain F if G(t) ∈ F for all t sufficiently large. It is easy to see that this is always the case, as stated in the following lemma. Lemma 2.2 (See e.g. Lemma 2.3 in [BF15]). Let f ∈ Brays . Then every dynamic ray is asymptotically contained in a fundamental domain. Let us consider the symbolic space formed by all infinite sequences of fundamental domains F N = {s = F0 F1 F2 . . .} endowed with the dynamics of the shift map σ : F N → F N , σF0 F1 F2 . . . = F1 F2 F3 . . .. For s = F0 F1 . . . ∈ F N , the set σ −1 s of its preimages is given by all sequences of the form F s := F F0 F1 . . . where F ∈ F. Definition 2.3. We say that a dynamic ray G has address s = F0 F1 . . . ∈ F N and we denote it by Gs if and only if f j (Gs ) is asymptotically contained in Fj for all j. It follows directly from the construction in [RRRS11] that given an address s the ray Gs , if it exists, is unique, and that for rays which are not broken we have that f (Gs ) = Gσs and that {f −1 Gs } = {GF s : F ∈ F}. This implies that a dynamic ray Gs is periodic if and only if s is periodic. We say that Gs has bounded address if s is bounded, i.e. its entries take values over finitely many fundamental domains. The next proposition is [BF15, Proposition 2.11], where it is proven using results and ideas from [DT86] and [RRRS11]. It previously appeared in different formulations in [Rem08], [BK07]. Proposition 2.4. If f ∈ Brays and s ∈ F N is bounded then there exists a unique dynamic ray Gs with address s for f . Remark 2.5. A generalization of rays for functions with not as beautiful a geometry as functions in class Brays can be found in [BRG17]. The unbounded, connected sets which take the place of rays are called dreadlocks. Despite not being curves, dreadlocks have the same combinatorial structure as rays. The results that are presented for rays in this section also hold for dreadlocks. 6 The Separation Theorem Goldberg and Milnor [GM93] proved that for polynomials with connected Julia set, the set of fixed rays together with their landing point separate the set of fixed points which are not landing points of fixed rays; such points include all attracting, Siegel and Cremer parameters. Goldberg-Milnor’s theorem has been generalized to entire transcendental maps in class Brays , under the assumption that periodic rays land [BF15]. In order to state the theorem we need to introduce the notion of basic regions and interior fixed points, following [GM93] and [BF15]. Fix p and assume that periodic dynamic rays land. Let Γ denote the closed graph formed by the rays fixed by f p together with their landing points. The connected components of C \ Γ are called the basic regions for f p . An interior fixed point for f p is a periodic point for f which is fixed by f p and which is not the landing point of any periodic ray which is fixed by f p . Note that attracting, Siegel and Cremer points as well as rationally invisible repelling periodic points are interior periodic points for f p for all p, while parabolic and repelling periodic points may be interior or not depending on p. For example a fixed point which is the landing point of a cycle of periodic rays of period 3 is interior for f but not for f 3 . Theorem 2.6 (Separation Theorem Entire [BF15]). Let f ∈ Brays , p ∈ N and assume that all periodic rays for f which are fixed by f p land. Then there are finitely many basic regions for f p , and each basic region contains exactly one interior fixed point for f p , or exactly one attracting parabolic basin which is invariant under f p . Theorem 2.6 has many corollaries, including that parabolic points are always landing points of periodic dynamic rays (whose period equals the period of the attracting basins), and that hidden components of a Siegel disk are preperiodic to the Siegel disk itself (see [CR16], [BF17a] for an application of this fact to the existence of critical points on the boundary of Siegel Disks). It has recently been used in [BF17b] to associate non-repelling cycles to singular orbits under the hypothesis that periodic rays land. A couple of useful lemmas The following two general lemmas will be used several times in the sequel. The first of them is Lemma 2.1 in [BRG17] Lemma 2.7. Let f : C → C holomorphic, U ⊂ C be a connected set with locally connected boundary. Then for any compact set K ∈ C, only finitely many connected components of f −1 (U ) intersect K. Lemma 2.8 (Forward invariant boundary). Let f be holomorphic, B be a region whose boundary is forward invariant, V be an open subset of C which does not intersect the boundary of B. Then for any connected component U of f −1 (V ) we have that U is either contained in B or in C \ B. Proof. Otherwise, U ∩ ∂B 6= ∅. Since f (∂B) ⊂ ∂B it follows that V ∩ ∂B 6= ∅ contradicting the hypothesis. 7 3 Fundamental tails for a repelling periodic orbit Fundamental tails are relatively new objects introduced in [BRG17] for functions with bounded postsingular set. They already found application in [EFJS18]. Fundamental tails are preim- ages of fundamental domains under finitely many iterates, and hence are nice open sets. Loosely speaking they can be thought of as approximation of rays, which despite being topo- logically curves, do not necessarily have nice geometric properties. In what follows we give a precise definition of tails under weaker assumptions than in the original setting. Let f ∈ Brays whose periodic rays land. Let z0 be a repelling periodic point of minimal period m and let X = {z0 , z1 . . . zm−1 } be its orbit labeled so that f (zi ) = zi+1 with indices taken modulo m. Let p be a multiple of m. Suppose that z0 is an interior fixed point for f p , and consider the basic regions B0 , . . . Bm−1 for f p which contain the elements of X , namely, zi ⊂ Bi for i = 0, . . . , m − 1. indices of the basic regions {Bi } will also be taken modulo m. Let B denote the union of the Bi . Since there are only finitely many basic regions for fp (see Theorem 2.6), the boundary of B contains finitely many pairs of rays which are fixed under f p , together with their landing points. Let D, δ as in Section 2. Let FB be the collection of fundamental domains intersecting B for D, δ. Fix r > 0 such that r > |zi | for all i ∈ X and let Dr ⊃ D be the Euclidean disk of radius r centered at 0 . Let δr ⊂ δ be the unbounded connected component of δ \ Dr . For B,r any fundamental domain F ∈ FB let F be the unique unbounded connected component of B,r F ∩ B ∩ f −1 (C \ (Dr ∪ δr )). This is the same as saying that we are considering F to be the unique unbounded connected component of the fundamental domains obtained by using Dr , δr instead of D, δ, intersected with B. F3 F2 F1 Dr D δ B,r Figure 1: Definition of F when B is a single basic region. The region B is shown together with the disk D used to define fundamental domains. For simplicity only 3 fundamental domains F1 , F2 , F3 are shown. The circle of radius r and its preimages inside F1 , F2 , F3 are in red. Shaded in light blue are the tails of level 1 for the disk DR corresponding to F1 , F2 , F3 . The curve δr is in purple. B,r For any F ∈ FB we have that F does not intersect ∂B, so by Lemma 2.8 for any n we 8 B,r have that any connected component of f −n (F ) is contained in either B or C \ B. Definition 3.1 (Fundamental tails for z0 ). Let z0 , m, The set of tails of level 1, that we B,r B,r denote by T1 = T1 (r), is the set of F where F ∈ FB and F ⊂ B0 . Since Dr ⊃ D, we have that f |τ : τ → C\(Dr ∪δr ) is univalent for any τ ∈ T1 . We define tails of level n by induction. Suppose that we have defined the set Tn of tails of level n and let us define the set Tn+1 of tails of level n + 1. We say that η is a tail of level n + 1 (for z0 , r) if it satisfies the following. • η is a connected component of f −m (τ ) for some τ ∈ Tn ; • η ⊂ B0 and f m : η → τ is univalent; • f i (η) ⊂ Bi for i = 0, . . . , m − 1. It follows that if η ∈ Tn , f m(n−1) : η → τ is univalent, where τ is some element of T1 . The definition above depends on the choice of z0 , p and r. The point z0 , its period m, and the period p of the basic regions are fixed throughout the section, while r may vary. With this definition all tails of all levels are contained in the basic region B0 which contains z0 , and have the following properties. Lemma 3.2. Let τ be a fundamental tail of level n for z0 , r. Then: • τ is asymptotically contained in a unique fundamental domain F0 ∈ FB which intersects B0 , that is, there is a unique fundamental domain F0 ∈ FB which intersect B0 and such that τ ∩ F0 is unbounded. • For j = 1, . . . , m(n − 1), f j (τ ) is asymptotically contained in a fundamental domain Fj which intersects Bj , that is, there is a unique fundamental domain Fj which intersects Bj and such that τ ∩ Fj is unbounded. Proof. The proof follows from the definition of fundamental tails. Lemma 3.2 gives a way to dynamically associate a finite sequence of fundamental domains (called an address) to each tail τ ∈ Tn , similarly to the way in which we associate addresses to dynamic rays. Compare with Definition 3.7 and 3.8 in [BRG17]. Definition 3.3 (Addresses of fundamental tails). Let τ be a fundamental tail of level n and let s = F0 F1 . . . Fm(n−1) be the sequence of fundamental domains given by Lemma 3.2. We say that s is the (finite) address of τ . Observe that s has length `n = m(n − 1) + 1. When it exists, we define τn (s) to be the unique tail of level n and address s. Uniqueness is given by the fact that for each fundamental domain F we have that f : F → C \ (D ∪ δ) is a homeomorphism. At first glance one may expect that all sequences whose elements are fundamental domains intersecting B should be realized. However, some of these fundamental domains are only partially contained in B, and this prevents the existence of some tails. One can characterize precisely the set of addresses which are realized but this is not needed for our purposes. 9 m(n−1) The set of addresses of tails of level n is contained in {FB }`n = FB × FB . Consider infinite sequences in {FB }N . There is a natural projection πn : {FB }N → {FB }`n which maps an infinite sequence s to the finite address consisting of its first `n entries. In this sense, whenever it exists, we can define the tail of level n and address s ∈ {FB }N as the tail of level n and address πn (s). We refer to elements in {FB }N as (infinite) addresses, despite the fact that not all of them are realized as fundamental tails of arbitrarily high levels. The set of admissible addresses is denoted by AB = AB (z0 , p, r) := {s ∈ {FB }N : the tail τn (s) is well defined for all n}. (3.1) τn+1 (s) F0 τn s fm τn (σ m s) Fm f m(n−1) Fm(n−1) Figure 2: Let s = F0 F1 . . . Fn . . . ∈ AB be an infinite address. The fundamental tail τn+1 (s) is contained in τn (s) for points with large modulus. The fundamental tail τn+1 (s) is mapped to τn (σ m s) under f m . Definition 3.4 (Pullback along an address). Let r > 0 and consider the fundamental tails T1 of level 1 for r. Let s = F0 F1 . . . F`n −1 ∈ {FB }`n such that the tail τn (s) exists for some n. Let ζ ∈ B0 . When it exists, we define ζn (s) to be the unique point in f −nm (ζ) ∩ τn (s). Let τn (s) be a tail of level n. The map f m(n−1) : τn (s) → τ1 (σ m(n−1) s) is univalent, hence f mnis a univalent map from τn (s) to C \ (Dr ∪ δr ) (not necessarily surjective). So, if ζ does have a preimage in τn (s), such a preimage is unique. Lemma 3.5. A point z ∈ τn (s) for some n ≥ 1 and some s ∈ {FB }`n if and only if f jm (z) ∈ τn−j (σ mj (s)) ⊂ B0 for all j = 1 . . . n − 1. Proof. This follows from the definition of fundamental tails. We now show S that up to taking r large enough, all possible fundamental tails of all levels exist unless B = i Bi contains singular orbits which follow the itinerary of z0 with respect to the partition into basic regions for f p . 10 Recall that indices of the orbit of z0 as well as indices of the basic regions containing them are taken modulo m, and that f j (z0 ) ∈ Bj for all j ≥ 0. Let SB (f ) be the set of singular values which are contained in B, that is SB (f ) = S(f ) ∩ B. For every s ∈ SB (f ) let i(s) ∈ {0, . . . , m − 1} be such that s ∈ Bi(s) , and let n(s) maximal be such that for all 0 ≤ j ≤ n(s) we have that f j (s) ∈ Bi(s)+j . Let [ [  PB (f ) := f n (s) . (3.2) s∈SB (f ) n≤n(s) Observe that PB (f ) is smaller than P(f ) ∩ B, and that it is forward invariant in the sense that f (PB (f ) ∩ Bi ) ∩ Bi+1 ⊂ PB (f ). (3.3) Proposition 3.6 (Existence of fundamental tails). Let f ∈ Brays such that periodic rays land. Let X = {z0 , . . . , zm−1 } be a repelling periodic orbit of period m and let p be any multiple of m. Suppose that f (zi ) = zi+1 mod m. Let {Bi }i=0...m−1 be the basic regions for f p containing the elements of X . Then at least one of the following is true. Sm−1 (1) There exists a singular value s for f such that s ∈ i=0 Bi , say s ∈ Bi(s) , and such that for all n ≥ 0 we have that f n (s) ∈ Bi(s)+n . (2) There are infinitely many singular values sj for f in at least one of the basic regions Bi , say B0 , and a sequence nj → ∞ as j → ∞ such that for all n ≤ nj we have that f n (sj ) ∈ Bn . (3) The set P(f ) is bounded, and there exists r > 0 such that all tails of all levels are well defined for z0 , p, r. More precisely, this means the following. Let τ ∈ T1 (r), n ≥ 0, and τ̃ a connected component of f −nm (τ ) for which f j (τ̃ ) ⊂ Bj for j ≤ nm. Then f mn : τ̃ → τ is univalent. Proof. We first claim that if neither case (1) nor case (2) occur, the set PB (f ) is bounded. Indeed we have that N := sup n(s) < ∞, s∈SB (f ) hence [ PB (f ) ⊂ f j (SB (f )) j≤N and each of the sets f j (SB (f )) is bounded because SB (f ) is bounded, and hence a finite union thereof is also bounded. If PB (f ) is bounded, let r > 0 be such that the tails of level 1 for Dr do not intersect PB (f ). This can be done by taking Dr ⊃ (D ∪ PB (f ) ∪ f (PB (f )) (notice that f (PB (f )) is not contained in B). 11 We claim that all tails of all levels are defined for such r. Indeed, suppose that this is not the case. Then there exists a minimal k > 0, a tail τk (s) ∈ Tk (r), and a connected component V ∈ f −m (τk (s)) ∩ B0 such that f m : V → τk (s) is not univalent and such that f j (V ) ∈ Bj (by definition of tails for z0 ). Since τk (s) is simply connected, this occurs if and only if there exists j ≤ m such that f j (V ) ⊂ Bj contains a singular value s. By definition of tails, there exists some τ ∈ T1 such that f m(k−1)−j (f j (V )) ⊂ τ ⊂ B0 , hence f m(k−1)−j (s) ∈ τ . Since the orbit of s follows the orbit of f j (V ) (that is, f ` (s) ∈ f ` (f j (V )) ⊂ B(`+j) ) we have that f m(k−1)−j (s) ∈ PB . This contradicts the fact that by choice of r, T1 ∩ PB (f ) = ∅. Let us point out that P(f ) ∩ B may well be unbounded even if PB is not. Given Propostion 3.6, the strategy for proving the Main Theorem will be to show that in case (3), that is, in the absence of trapped singular values, every repelling periodic point is the landing point of a periodic ray. 4 Definition of landing and shrinking lemma In the current section we give an abstract definition of landing and we prove a lemma that will be used in Section 5 (compare with the abstract characterization of landing in [BRG17]). Let z0 be a repelling periodic point of period m, p be a multiple of m, B and Bi be basic regions for f p as in Section 3. In this section we assume that we are in case (3) of Proposition 3.6, that is, there exists r > 0 such that all tails of all levels are well defined for z0 , p, r. In particular, for any τ ∈ T1 , for every n ≥ 0, and for every connected component τ̃ of f −mn (τ ) for which f j (τ̃ ) ⊂ Bj for j ≤ n, we have that f mn : τ̃ → τ is univalent. Let AB as in (3.1) be the set of addresses for which tails of all levels are well defined for z0 , p, r. The following lemma establishes one of the fundamental relations between rays and tails of the same address. Lemma 4.1 (Rays versus tails). Let s = F0 F1 F2 . . . ∈ AB . Then for every z ∈ Gs (t) there exists nz such that the arc connecting z to infinity in Gs is fully contained in τn (s) for all n ≥ nz . Proof. Let Dr be the Euclidean disk of radius r defined in the proof of Proposition 3.6, and consider the curve δr ⊂ δ which starts from the last intersection of δ with Dr . Only for the proof of this lemma, let {Fi } be the fundamental domains obtained by taking preimages of C \ (Dr ∪ δr ) and F be the union of all fundamental domains for f with respect to this choice of Dr . By Lemma 2.2, the dynamic rays f i (Gs ) are asymptotically contained in Fi for all i. Let Gs (t) : (0, ∞) → I(f ) be a continuous parametrization of Gs such that |Gs (t)| → ∞ as t → ∞. Recall that points in Gs ([T, ∞]) escape uniformly to infinity for every T > 0. So for each z = Gs (T ) there exists nz such that for all points Gs (t) with t > T we have f n (Gs (t)) ∈ F for every n ≥ nz . 12 (otherwise, f n+1 (Gs (t)) would belong to the bounded set Dr , or to the curve δr which is mapped to Dr at the next iterate, contradicting the uniform escape to infinity). Hence we have that f n (Gs (T, ∞)) ⊂ Fn for n ≥ nz and, by definition of tails, Gs (T, ∞) ⊂ τn (s) for all n ≥ nz . The following Lemma is a Euclidean version of a classical Lemma which holds for the spherical metric (see for example [Lyu83], Proposition 3. Similar lemmas have been used in [BL14],[BRG17] and in many other papers). For this lemma we do not need the assumption that all tails are well defined as long as we restrict to univalent preimages. Lemma 4.2 (Shrinking Lemma). Let f be holomorphic. Let V 0 ⊂ C be a simply connected open set intersecting the Julia set. Fix a compact set K ⊂ C. For each n consider all connected components Vn,λ 0 of f −n (V 0 ) which intersect K and which are univalent preimages of V 0 under f n , where λ indicates the chosen branch of f −n . Let V b V 0 , and for each n, λ let Vn,λ = f −n (V ) ∩ Vn,λ 0 . Then for any ε > 0 there exists Nε such that 0 diameucl Vn,λ < ε for any n > Nε and for any λ such that Vn,λ ∩ K 6= ∅. The proof is the same as in [Lyu83] for the spherical metric, and the statement for the Euclidean metric follows from the fact that we are only considering preimages intersecting a given compact set K. Lemma 4.3 (Characterization of landing). Let f ∈ Brays such that periodic rays land, and let z0 , m, Bi , B, AB as above. Let Gs ⊂ B0 be a periodic ray of period mq with s ∈ AB and q ≥ 1. Let ζ ∈ B0 \(Dr ∪δr ) for which ζnq (s) is well defined for all n ∈ N as in Definition 3.4. Then Gs lands at z0 ∈ C if and only if ζnq (s) → z0 as n → ∞. Recall that a dynamic ray Gs lands at a point z ∈ C if Gs (t) → z as t → 0. We observe the following. Let z0 , m, r, AB be as in the beginning of this section. Let Gs be a dynamic ray of period mq for some s ∈ AB , I be an arc in Gs connecting a point z ∈ Gs with its image f mq (z). For n ∈ N let In be the unique preimage of I under f mqn which is contained in Gs . Then Gs lands at z0 if and only if dist(In , z0 ) → 0 as n → ∞. Indeed consider a sequence Gs (tn ) → z0 . We have that f kn (Gs (tn )) ∈ I for some kn → ∞ hence Gs (tn ) ∈ Ikn . Moreover if I is chosen such that f j (I) is contained in F for all j, then In ⊂ τnq (s) for all n ≥ 0. The proof is the same as the proof of Lemma 4.1. Proof of Lemma 4.3. If Gs lands at z0 , by Lemma 4.1 for any point ζ = Gs (t) with t large enough we have that ζnq (s) exists, and ζnq converges to z0 by definition of landing. To prove the other direction let ζ ∈ B0 \ (Dr ∪ δr ) such that ζnq (s) is well defined (such a ζ exists because s ∈ AB ) and converges to z0 ∈ C. Let ζ 0 = Gs (t) and let I be the arc in Gs connecting ζ 0 to f mq (ζ 0 ). Up to taking t large enough we can assume that f j (I) ⊂ F for all j. By Lemma 4.1 we have that I ⊂ τj (s) for all j large enough. By assumption, we also have that ζnq (s) ∈ τnq (s) for n large enough. Hence (I ∪ ζnq ) ⊂ τnq (s) for all n large enough. For one such n let V 0 ⊂ τnq (s) be a simply connected set containing both ζnq (s) and I and let 13 V b V 0 containing both ζnq (s) and I. Note that V intersects the Julia set because dynamic rays are subsets of the Julia set. Let K be a compact connected neighborhood of z0 . For j ∈ N let Vj0 (s) be the connected component of the preimage of V 0 under f mqj which is contained in τq(j+n) (s); observe that the inverse branch ψj : V 0 → Vj0 (s) is univalent because r was chosen so that all tails are well defined. Also, Vj0 intersects K for j large enough because ζqn (s) → z0 . By Lemma 4.2, diameucl Vj (s) → 0 as j → ∞. Since we assumed that f j (I) ⊂ F for all j, In ⊂ τnq (s), hence since f nq |τnq (s) is a homeomor- phism, In ⊂ Vj → z0 and Gs lands at z0 . 5 Rationally invisible orbits and singular orbits The goal of this section is to prove that if tails are well defined for a given repelling periodic orbit with respect to a set of basic regions containing it (case (c) in Proposition 3.6), then the orbit is not rationally invisible. The Main Theorem will be derived in Section 6. In the following Theorem, as usual, indices are taken modulo m. Theorem 5.1 (Main theorem for f p ). Let f ∈ Brays such that periodic rays land and assume that there are no singular values escaping along periodic rays. Let X = {z0 , . . . , zm−1 } be a repelling periodic orbit of period m and let p be any multiple of m. Suppose that f (zi ) = zi+1 . Let {Bi }i=0...m−1 be the basic regions for f p containing the elements of X , and B = ∪Bi . Then at least one of the following is true. S (1) There exists a singular value s for f such that s ∈ q−1 i=0 Bi , say s ∈ Bi(s) , and such n that for all n ≥ 0 we have that f (s) ∈ Bn+i(s) . (2) There are infinitely many singular values sj for f in at least one of the basic regions Bi , say s ∈ Bi(s) , and a sequence nj → ∞ as j → ∞ such that for all n ≤ nj we have that f n (s) ∈ Bn+i(s) . (3) Each point in X is the landing point of at least one and at most finitely many periodic dynamic rays, all of which have the same period. By Lemma 8.2 in [BRG17] (compare to [Mil06], Lemma 18.12 for polynomials), if a repelling periodic point is the landing point of a periodic ray then it is the landing point of finitely many periodic rays, all of which have the same period. So it is enough to show that at least one point in X is the landing point of at least one periodic dynamic ray. This implies that the same is true for all elements in X . Indeed, f is a homeomorphism from a neighborhood of zi to a neighborhood of zi+1 , so a dynamic ray Gs lands at zi if and only if f (Gs ) = Gσs lands at zi+1 . Let z0 ∈ X . Recall the definition of fundamental tails for z0 from Section 3. By Proposi- tion 3.6, if neither case (1) nor case (2) occur, then there is r such that all fundamental tails of all level are well defined. Our aim will be to show that under this assumption z0 is the landing point of at least one periodic ray. Recall that `n = m(n − 1) + 1 denotes the length of the address of a tail of level n. 14 Definition 5.2 (Fundamental pieces). Let n ≥ 1. Let s be an infinite address or an address of length at least `n+1 and assume that the tail τn+1 (s) is well defined for some r > 0. Then we define the fundamental piece of level n and address s, which we denote by Pn (s), as Pn (s) := τn+1 (s) \ τn (s). Fundamental pieces are not necessarily connected, nor exist for all levels and addresses. For example, if s is a disjoint-type address (i.e., contains only fundamental domains which do not intersect the disk D) then there are no fundamental pieces of address s for any level. The idea of using fundamental pieces was originally suggested by L. Rempe-Gillen as a possible way to prove the main theorems in [BRG17]. Recall the definition of PB from Section 3. Lemma 5.3 (Properties of fundamental pieces). Let f ∈ Brays , such that periodic rays land. Let z0 , m, B, Bi as usual, and let n ∈ N. Let s be an infinite address or an address of length at least `n and assume that the fundamental tail τn (s) is well defined. Then f m (Pj (s)) = Pj−1 (σ m s) for all j ≤ n − 1 (5.1) and n−1 [ τn (s) ⊂ τN (s) ∪ Pj (s) for all N ≤ n − 1. (5.2) j=N f mn (Pn (s)) = τ1 (σ mn(s) ) ∩ D for all n ∈ N. (5.3) Proof. The first two properties follow from the definition of fundamental pieces and tails (recall that f m : τj (s) → τj−1 (σ m s) is a homeomorphism). We now prove (5.3). Let ζ ∈ Pn (s). We have that f m(n−1) (ζ) ∈ τ2 (σ m(n−1) s) \ τ1 (σ m(n−1) s), and τ1 (σ m(n−1) s) is the preimage of C \ Dr . Hence f m (f m(n−1) (ζ)) = f mn (ζ) ∈ D. Since f m(n−1) (ζ) ∈ τ2 (σ m(n−1) s), f mn (ζ) ∈ τ1 (σ mn s) proving the claim. Recall that SB is the set of singular values which are contained in B. Recall also that for s ∈ SB the integer i(s) ∈ {0, . . . , m − 1} is such that s ∈ Bi(s) , and n(s) is maximal such that for all j ≤ n(s) we have that f j (s) ∈ Bi(s)+j . In other words the orbit of s follows the orbit of X for exactly n(s) iterates with respect to the partition of the plane induced by the regions Bi . Lemma 5.4 (Good neighborhoods of rays). Let f ∈ Brays such that periodic rays land and assume that there are no singular values escaping along periodic rays. Let X = {z0 , . . . , zm−1 } be a repelling periodic orbit and let p be any multiple of m. Suppose that f (zi ) = zi+1 . Let {Bi }i=0...m−1 be the basic regions for f p containing the elements of X , and let B = ∪Bi . Suppose that cases (1) and (2) in Proposition 3.6 do not hold. Let G be a ray in ∂B0 , which is hence fixed under f p . Let {G0 = G, Gj = f j (G)}j=0,...p−1 be the orbit of G under f (here, indices are taken modulo p), and let ψj : Gj → Gj−1 be the unique the inverse branch of f such that ψ := ψ0 ◦ . . . ◦ ψp−1 fixes G. Since Gj are curves we can write them as Gj (t) : R+ → C, with |Gj (t)| → ∞ as t → ∞. Fix ε, Tj > 0. 15 Then there exist neighborhoods Uj of Gj ((ε, Tj )) which contain Gj ((0, Tj )), which do not contain singular values for f , and such that: a. ψj is defined and univalent on Uj ; b. ψj (Uj ) ⊂ Uj−1 ; S S S c. j Uj ∩ s∈SB f n(s)+1 (s) = j Uj ∩ (f (PB ) \ PB ) = ∅. Proof. Notice first that we can take neighborhoods of Gj ((ε, Tj )) which do not intersect S(f ). If not there would be a sequence of singular values in S(f ) converging to a point z ∈ Gj ([ε, Tj ]), which would need to be a singular value because S(f ) is closed. This contradicts the assumption that Gj does not contain singular values. This shows that we can take a neighborhood of Gj ([ε, Tj ]) for every ε. By taking the union over them, we obtain a neighborhood of Gj ((0, Tj ]). Each ψj is defined on compact subsets of Gj containing the landing point, hence in particular, it is defined on G([0, Tj ]). Since Gj contains no singular values for f by assump- tion, lands, and ψj (Gj ) ⊂ Gj−1 , for each j there is a simply connected neighborhood Uj of Gj ([ε, Tj ]) which contains Gj ((0, Tj )), does not intersect the set S(f ) (recall that the latter is closed and that periodic rays contain no singular values), and such that ψj is well defined on Uj and ψj (Uj ) ⊂ Uj−1 , with T0 > T . Let N = maxs∈SB n(s). Since cases (1) and (2) in Proposition 3.6 do not hold, N < ∞. Since G contains no points in singular orbits, SB does not intersect any preimage of G, and in particular it does not intersect the first N preimages of G, that is the infinitely many rays preperiodic to G of preperiod at most N . Since G lands and contains no points in singular orbits, they all land and only finitely many of them intersect any given compact set. Let D c S(f ) be a closed disk and let K be the compact set given by the first N preimages of rays in the boundary of B intersected with D (compare with the proof of Proposition 3.6). Since SB does not intersect K and both are compact sets, we can find a neighborhood W of K which does not intersect SB , and restrict the sets Ui such that their preimages up to level N do not intersect W . The condition that periodic rays do not contain points in singular orbits can be relaxed by assuming that they do not contain forwards iterates of singular values in SB which moreover follow the correct itinerary between the basic regions in B. Lemma 5.5 (Shrinking of fundamental pieces). Let f ∈ Brays such that periodic rays land and assume that there are no singular values escaping along periodic rays. Let z0 , m, B, Bi as in Theorem 5.1. Suppose that case (1) and (2) in Proposition 3.6 do not hold. Let K be a compact set and consider all fundamental pieces Pn (s) for n ∈ N. Then for any ε > 0 there exists Nε such that diameucl Pn (s) < ε for all n ≥ Nε and all s such that Pn (s) ∩ K 6= ∅. Compare with the proof of Lemma 6.3 in [BRG17]. 16 Proof. Since case (1) and (2) in Proposition 3.6 do not hold, PB is bounded, S and so is its image f (PB ) (which is no longer necessarily contained in B). Let Dr c (PB ∪ k≤m f k (PB )) be a disk of radius r centered at 0. Consider the set of tails T1 of level 1 for r, z0 . Notice that PB is forward invariant in the sense of (3.3). It follows that τn (s) ∩ PB = ∅ for all n ∈ N. For each of the finitely many τ ∈ T1 which intersect Dr let γτ be a crosscut of τ such that τ \ γτ is made of two connected components, one of which is bounded and contains all of the connected components of τ ∩ Dr . Call this component ητ . Let [ V= ητ . τ ∈T1 Since finitely many τ ∈ T1 intersect Dr (see Lemma 2.7) we have that V has finitely many connected components. Notice that if two adjacent tails τ, τ̃ both intersect Dr , their bounded components ητ , ητ̃ belong to the same connected component of V. We first claim that any Pn (s) is contained in a connected components of fλ−n (Vi ) for some i and some branch λ. This is implied by showing that f mn (Pn (s)) ⊂ ητ for some τ ∈ T1 . Let ζ ∈ Pn (s) = τn+1 (s) \ τn (s). By (5.3) f mn (ζ) ∈ τ ∩ Dr for some τ ∈ T1 . The fact that τ does not depend on the choice of ζ (we have to check this because Pn (s) is not necessarily a connected set) follows from the fact that for any U connected component of the preimage of ητ under f −mn which is contained in a tail of level n + 1 (which is the case for fundamental pieces) we have that f m : U → ητ is a homeomorphism. So it is enough to show that, for V which is any of the finitely many connected components of V, the diameters of inverse images of V tend to zero uniformly in the family of inverse branches used to define fundamental pieces. Let V be such a component. To fix notation for the inverse branches let us denote by L the set of inverse branches ϕnλ of f mn which map a component ητ ⊂ V inside another tail τn−1 (s), and which are a priori defined only on V . We claim that there is a simply connected neighborhood V 0 of V such that for any ϕnλ ∈ L we have that ϕnλ can be extended (univalently) to V 0 . The claim is obvious for all V b B0 , since by choice of r we can find a simply connected neighborhood V 0 which is contained in B0 \ PB and hence all inverse branches ϕnλ used to define fundamental pieces can be extended. So let us consider V such that ∂V ∩ ∂B0 6= ∅. Let G on ∂B0 which intersects ∂V , and T such that ∂V ∩ G b G(0, T ). In the following we will assume for simplicity that G is unique, but if not, there are finitely many rays and the reasoning has to be done for each of them. Let {G0 = G, Gj = f j (G)}j=0,...p be the orbit of G under f and let ψj : Uj → Uj−1 as in Lemma 5.4. Let V 0 ⊂ (U0 ∪ B0 ) be a simply connected neighborhood of V which does not intersect PB , and let ϕnλ ∈ L. Choose V 0 so that in addition V 0 ∩ B and V 0 ∩ (C \ B) are simply connected. The inverse branch ϕnλ decomposes (uniquely) as ϕnλ = hnm ◦ . . . ◦ h1 where each hi for i = 1 . . . nm is a branch of f −1 that we want to show to be defined on hi−1 ◦ . . . ◦ h1 (V 0 ). Notice that ϕnλ extends to V 0 ∩ B because the latter does not intersect PB . 17 We claim that the inverse branch h1 is well defined and univalent on all of V 0 . Let us denote by X the connected component of f −1 (V ) which contains h1 (V ). Either X b B or X ∩ ∂B 6= ∅. If X b B then the branch h1 is well defined and univalent because by Lemma 5.4 the neighborhoods Uj do not intersect f (SB ) ⊂ f (PB ). Since they also do not intersect f (PB ), the set X does not intersect PB . Since the latter is forward invariant in the sense of (3.3), the branches hi are well defined also for all i = 2 . . . mn, proving the claim. Let us consider the case X ∩ ∂B 6= ∅. Since h1 (V ) ⊂ B, by Lemma 2.8 we have that f (X ∩ B) = V 0 ∩ B f (X ∩ (C \ B)) = V 0 ∩ (C \ B) f (X ∩ ∂B) = V 0 ∩ ∂B It follows that X ∩ ∂B is contained in Gp−1 which is the only preimage of G0 in ∂B. By univalency of f on Gp−1 , h1 extends continuously to G0 ∩ V 0 and coincides with ψ0 on this set. Since h1 extends holomorphically to a neighborhood of V ∩ G0 (since U0 contains no singular values by choice), by the identity principle h1 equals ψ0 and hence h1 extends as a univalent map on all of V 0 . By property b. in Lemma 5.4, h1 (V 0 \ B) ⊂ Up−1 . This last property allows us to repeat the reasoning for h2 and show that it is defined on h1 (V 0 ). Proceeding by induction this gives the claim. By Lemma 4.2, diameucl (ϕnλ (V )) uniformly in λ as n → ∞, provided ϕnλ (V ) ∩ K 6= ∅ for some compact set K. For all addresses s such that Pn (s) ∩ K 6= ∅, the claim of the Lemma follows because Pn (s) ⊂ ϕnλ (V ) for some λ, n, i. Pn (s) P1 (σ m(n−1) s) τn (s) τn+1 (s) τ2 (σ m(n−1) s) τ1 (σ m(n−1) s) τ1 (σ mn (s)) Dr f m(n−1) fm Figure 3: Mapping properties of fundamental pieces (shown in blue). The last result that we need in order to prove the Main Theorem is Iversen’s Theorem [Ive14]. We state it as it is presented in [BE08]. It is a consequence of Gross Star Theorem [Gro18]. Theorem 5.6 (Iversens Theorem). Let f be holomorphic. Let ψ be a holomorphic branch of the inverse of f which is defined in a neighborhood of some point z0 and let γ : [0, 1] → C be 18 a curve with γ(0) = z0 . Then for every ε > 0 there exists a curve γ̃ : [0, 1] → C satisfying γ(0) = z0 and |γ(t) − γ̃(t)| < ε such that ψ has an analytic continuation along γ̃. Proof of Theorem 5.1 Let z0 , zi , m, p, B, Bi be as in the statement. Recall that indices of zi and Bi are taken modulo m. If z0 is not an interior fixed point for f p , there is nothing to show because it is the landing point of a periodic ray of period at most p. Otherwise, in view of Proposition 3.6 we need to show that, if there exists r > 0 such that for any n ≥ 1 all fundamental tails for z0 are well defined, then z0 is the landing point S of a periodic ray. Without loss of generality up to taking a larger r we can assume that i zi ⊂ Dr and that tails are also defined for a slightly smaller r. Let Tn denote the collection of tails of level n for these choices. Let ψ be the inverse branch of f −m fixing z0 . Let U0 ⊂ B0 be a simply connected neighborhood of z0 such that ψ is well defined on U0 , ψ(U0 ) b U0 , and there exists η > 1 such that |(f m )0 (z)| ≥ η for all z ∈ U0 . Note that f i (ψ(U0 )) ⊂ Bi for i ≤ m, and that more generally for ` ∈ N we have f i (ψ ` (U0 )) ⊂ Bi for i ≤ m`. Let Un := ψ n (U0 ), ε = dist(∂U0 , ∂U1 ). By choice of U0 , f mn : Un → U0 is a homeomorphism. Claim 1. Tn ∩ U0 6= ∅ for all n large enough. Proof: Let F be a fundamental domain for f which intersects B0 and choose ζ ∈ F ∩B0 not an exceptional value. Then by Montel’s theorem there exists n large enough so that f nm (U0 ) 3 ζ. Since f is open, there exists ε0 such that the Euclidean disk Dε0 (ζ) ⊂ f nm (U0 ) ∩ B0 ∩ F and contains no exceptional values. Let γ : [0, 1] → B0 be a curve with γ(0) = z0 , γ(1) = ζ. Let ψ n be the inverse branch of f nm fixing z0 . By Iversen’s theorem there exists a curve γ̃ : [0, 1] → B0 such that γ̃(0) = z0 , γ̃(1) ∈ Dε0 (ζ) and such that ψ n has an analytic continuation along γ̃. Hence γ̃(1) ∈ f nm (U0 ). Since ψ n has an analytic continuation along γ̃ we can ensure that the same is true for ψ j for j ≤ n. Since γ̃ ∩ ∂B = ∅ we have that ψ j (γ̃) ∩ ∂B = ∅ for j ≤ n (see Lemma 2.8). Recall that we have f i (ψ ` (U0 )) ⊂ Bi for all ` ∈ N and all i ≤ m. Hence for any n ∈ N we have that f j ψ n (γ̃) ⊂ Bj for j ≤ m. This implies that the point w = ψ n (γ̃(1)) belongs to some tail τn+1 (s) for some s. Since γ̃(1) ∈ f nm (U0 ), and ψ n is a homeomorphism, we also have that w ∈ U0 , proving the fact that Tn ∩ U0 6= ∅ for n large enough. Observe that if TN ∩ U0 6= ∅ for some N , then Tn ∩ U0 6= ∅ for all n ≥ N , because the preimage under ψ of a point ζ which belongs to a tail in Tn intersecting Uk is a point ψz in a tail in Tn+1 intersecting Uk+1 (see also Lemma 3.5). 4 Let N be such that for all n ≥ N we have that Tn ∩ U0 6= ∅ and that diam Pn (s) < ε for all Pn (s) intersecting U0 . Such N exists by Lemma 5.5. For n ≥ N let Sn be the set of finite addresses of length `n (see Definition 3.3) for which the tail τn (s) intersects Un−N . Observe that Sn is finite for every n by Lemma 2.7. 19 Observe that dist(∂Un , z0 ) → 0 as n → ∞ because in U0 the map ψ is contracting by a factor η −1 < 1. So by Lemma 4.3 it is enough to find a periodic address s∗ of period mq and a point ζ ∈ U0 such that ζi (s∗ ) is well defined for all i and such that ζqn−N (s∗ ) ∈ Uqn−N in order to ensure that Gs∗ lands at limn→∞ ζqn−N (s∗ ) = z0 . We now claim that, for some n0 large enough, ψ induces a well defined map Γ from the finite set Sn0 into itself, implying that Γ has a periodic point of some period q. We do this in several steps. Let n ≥ N , s ∈ Sn and let τn (s) be a tail in Tn which intersects Un−N . For every point ζ ∈ τn (s) ∩ Un−N the point ψζ ∈ Un−N +1 is well defined and belongs to some tail of level n + 1 and address s̃(ζ) depending on ζ. Indeed, τn (s) ∩ Un−N may have several connected components, and it is not clear a priori that ψ maps each of these connected components to the same tail of level n + 1. What is clear however is that for each such s̃ = s̃(ζ) the tail τn+1 (s̃) intersects Un−N +1 and that σ m s̃ = s (see Lemma 3.5). Recall that for an address s̃ we denote by πn s̃ its first `n entries. Claim 2. πn s̃ belongs to Sn regardless of the choice of ζ. Proof: For an illustration of the proof of this claim see Figure 4. Let ψζ ∈ Un−N +1 ∩τn+1 (s̃). Recall that τn+1 (s̃) ⊂ τn (s̃) ∪ Pn (s̃) and that it is a connected set. If ψζ ∈ τn (s̃) it follows directly that the first `n entries of s̃ are in Sn . Otherwise ψζ ∈ Pn (s̃) ∩ Un+1−N hence f m(n−N ) ψζ ∈ PN (σ m(n−N ) s̃) ∩ U1 (see Lemma 5.3). By choice of N , diam PN (σ m(n−N ) < ε, so PN (σ m(n−N ) s̃) b U0 and hence its m(n − N )-th pullback Pn (s̃) b Un−N . Since τn+1 (s̃) is connected, τn (s̃) intersects Un−N and hence the first `n entries of s̃ are in Sn . 4 τn+1 (s) τn (s) Un−N UN −n+1 f mn U1 τN +1 (σ m(n−N ) ) U0 Figure 4: Claim 3. There exists n0 > 0 such that for n ≥ n0 , the map σ m : Sn+1 → Sn is injective, hence has a well defined inverse ψn∗ : Sn → Sn+1 on its image σ m (Sn+1 ). 20 Proof: Observe that z0 does not belong to any tail of level N , and hence there exists M > 0 such that TN ∩ UM = ∅. Indeed, otherwise we would have that f mN (z0 ) = z0 ∈ f mN (TN ) ⊂ C \ Dr , contradicting the choice of r. The main point now is to show that there exists n0 such that if n ≥ n0 and s̃ ∈ Sn+1 then Pn (s̃) ⊂ U1 . Notice that a priori it is not even clear whether Pn (s̃) ∩ U1 6= ∅, because the intersection τn+1 (s̃) ∩ Un+1−N may be contained in τn (s̃). Once this is proven, let s ∈ Sn and suppose by contradiction that there exist s̃1 , s̃2 such that ψ(τn (s) ∩ U0 ) intersects both τn+1 (s̃1 ) and τn+1 (s̃2 ). Then we would have that Pn (s̃1 ) and Pn (s̃2 ) are contained in U1 and are mapped homeomorphically to Pn−1 (s) by f m (recall Lemma 5.3), contradicting the fact that f m : U1 → U0 is a homeomorphism and proving Claim 3. So we now proceed to prove that Pn (s̃) ⊂ U1 if s̃ ∈ Sn+1 and n ≥ n0 . Let n ≥ M + N , with M as above such that TN ∩ UM = ∅, and consider s ∈ Sn+1 , that is, τn+1 (s) ∩ Un+1−N 6= ∅. We claim that there exists ñ = ñ(s) ∈ {n − M, . . . , n} maximal such that Pñ (s) ∩ Un+1−N 6= ∅. This is to ensure that τn+1 (s) ∩ Un+1−N intersects a fundamental piece of address s and of sufficiently high level, namely whose level tends to infinity as fast as n. Since TN ∩ Un+1−N = ∅ (because n ≥ M + N ) and since n [ 6 τn+1 (s) ∩ Un+1−N ⊂ (τN (s) ∪ ∅= Pj (s)) ∩ Un+1−N j=N (see Lemma 5.3), there is some ñ ∈ {N, . . . , n} maximal such that Pñ (s) ∩ Un+1−N 6= ∅. If n = M + N this implies that ñ ≥ N = n − M as desired. Now let us proceed by induction on n. Suppose that for all s ∈ Sn there exists ñ = ñ(s) ∈ {n − 1 − M, . . . , n − 1} maximal such that Pñ (s) ∩ Un−N 6= ∅ and let us show that for all s ∈ Sn+1 there exists ñ = ñ(s) ∈ {n − M − 1, . . . , n − 1} maximal such that Pñ+1 (s) ∩ Un−N +1 6= ∅. Then this would imply the claim for ñ+1. If s ∈ Sn+1 we have that σ m s ∈ Sn and hence by the induction hypothesis we have that Pñ (σs) ∩ Un−N 6= ∅ for some ñ = ñ(σ m s) ∈ {n − 1 − M, . . . , n − 1}. Since f m : Pñ+1 (s) → Pñ (σ m s) is univalent we have that Pñ+1 (s) ∩ Un+1−N 6= ∅ and that ñ(σs) + 1 ≥ n − N as required. Now let n1 such that diam Pn (s) < dist(∂U 2 ,∂U1 ) M +1 for n ≥ n1 and s ∈ Sn (this is possible by Lemma 5.5). Let n0 = max{n1 + M, N + 2}. Let n > n0 and let s ∈ Sn+1 . Then by the previous paragraph there is ñ ∈ {n − M, . . . ,S n} such that Pñ (s) ∩ Un−N +1 6= ∅. By definition S Pñ (s) ⊂ τñ+1 (s), and τn+1 (s) ⊂ τñ+1 (s) ∪ nñ+1 Pj (s). Since τn+1 (s) is connected and nñ+1 Pj (s) dist(∂U2 ,∂U1 ) consists of at most M pieces of diameter at most M +1 and τñ+1 (s) ∩ U2 6= ∅ we deduce that τn+1 (s) \ τñ+1 (s) ⊂ U1 . 4 For n ≥ n0 this induces a map ψn∗ : Sn → Sn+1 , where for s ∈ Sn we define ψn∗ (s) as the unique element in Sn+1 such that σ m ψn∗ (s) = s. Recall that FB is the collection of fundamental domains intersecting B and observe that ψn∗ (s) =: α(si )si for some α(si ) ∈ {FB }m , since we add m symbols when we go backwards once. 21 Let Γ := πn0 ◦ ψn∗ 0 : Sn0 → Sn0 . Since Sn0 has finitely many elements, there exists q ∈ N and s0 ∈ Sn0 such that Γq (s0 ) = s0 . Let si = Γi (s) (hence sq = s0 ). By definition of Γ and of α(si ) we have that si+1 = Γ(si ) = πn0 (α(si )si ). Let s∗ := α(sq ) . . . α(s1 ). Notice that s∗ is a periodic address of period qm. Claim 4. For any n ≥ n0 and for any s ∈ Sn , ψn∗ s = α(πn0 s)s. Proof: For n = n0 this is true by definition, so suppose the claim is true for all s ∈ Sn and let us show the claim for all s ∈ Sn+1 . By definition of ψn+1 ∗ , for s ∈ S n+1 we have ∗ ψn+1 s = F0 . . . Fm−1 s ∈ Sn+2 for some F0 , . . . , Fm−1 ∈ FB . By Claim 2 we have that πn+1 (F0 . . . Fm−1 s) ∈ Sn+1 , hence πn+1 (F0 . . . Fm−1 s) = ψn∗ s̃ for some s̃ ∈ Sn . Hence we have that s̃ = πn (s). By the induction hypothesis, F0 . . . Fm−1 = α(s̃) = α(πn0 s̃) = α(πn0 s). 4 Let ζ ∈ τn0 (s∗ ) ∩ Un0 −N . By definition of ψn∗ and by Claim 4 we have that ψ qn (ζ) ∈ τqn+n0 (s̃) where ∗ s̃ = ψnq+n . . . ψn∗ 0 s∗ = α(sq ) . . . α(s1 ) . . . α(sq ) . . . α(s1 ) s∗ = s∗ 0 | {z } repeated n times by Claim 4, so that ψ qn (ζ) ∈ τqn+n0 (s∗ ) and hence ψ qn (ζ) = ζqn (s∗ ). Then Gs∗ lands at z0 by Lemma 4.3 because ζqn (s∗ ) = ψ qn ζ ∈ Uqn−n0 → z0 . We note the following Corollary of Theorem 5.1. Corollary 5.7. Let f ∈ Brays such that periodic rays land and assume that there are no singular values escaping along periodic rays. Let z0 be a rationally invisible repelling periodic point for f . Let {z0 , . . . , zm−1 } be the orbit of z0 and let X be the union of the dynamical fibers of z0 , . . . , zm−1 as defined in [RS08]. Then X contains either a singular orbit, or infinitely many singular values whose orbits belong to the fiber for more and more iterations. 6 Bound on the number of rationally invisible orbits and gen- eralized Fatou-Shishikura inequality This last section is devoted to the proof of the Main Theorem. Let us recall the Main Theorem from [BF17b]. As usual indices are taken modulo m. Theorem 6.1 (Singular orbits trapped in basic regions). Let f be an entire transcendental map in class Brays whose periodic rays land. Let X be a cycle of Siegel disks, attracting basins, parabolic basins or Cremer points of period m and let p be any multiple of m. Let 22 {Bi }i=0...m−1 be the basic regions for f p containing the elements of X . Then, up to relabeling the indices, at least one of the following is true. S (1) There exists a singular value s for f such that s ∈ m−1 i=0 Bi , say s ∈ B0 , and such that f (s) ∈ Bn for all n ∈ N. The orbit of s accumulates either on the non-repelling cycle n or on the boundary of the cycle of Siegel disks. (2) There are infinitely many singular values sj for f in at least one of the basic regions Bi , say B0 , and a sequence nj −→ ∞ such that f n (sj ) ∈ Bn for all n ≤ nj . The j→∞ orbits {f n (sj )}j∈N,n≤nj accumulate either on the non-repelling interior cycle, or on the boundary of the associated Siegel disk. The first case always occurs if X is attracting or parabolic or if f has only finitely many singular values. Moreover, in case (1), the orbit of s does not accumulate on any other interior periodic cycle or on any point on the boundary of a Siegel disk ∆ ∈ / X (provided the point is not on a periodic ray or a periodic point). We are now ready to prove the Main Theorem. We remark that Theorems 5.1 and Theorem 6.1 are stronger than the Main Theorem in that they do not require finiteness of the number of singular orbits which do not belong to attracting or parabolic cycles. Proof of Main Theorem. Suppose by contradiction that there are at least q + 1 cycles of Siegel disks, Cremer points, or rationally invisible repelling periodic orbits (possibly infinitely many). Let p be the product of their periods. Each element in each of the q + 1 cycles is fixed by f p hence belongs to a different basic region for f p by Theorem 2.6. In particular, there are q + 1 disjoint collections of basic regions whose interior periodic orbit is either a cycle of Cremer point, a cycle of centers of Siegel disks, or a rationally invisible repelling periodic orbit. 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