ON FUNDAMENTAL LOOPS AND THE FAST ESCAPING SET D.J.SIXSMITH 3 1 0 Abstract. The fast escaping set, A(f), of a transcendental entire function 2 f has begun to play a key role in transcendental dynamics. In many cases A(f) has the structure of a spider’s web, which contains a sequence of fun- n damental loops. We investigate the structure of these fundamental loops for a J functions with a multiply connected Fatou component, and show that there existtranscendentalentirefunctionsforwhichsomefundamentalloopsarean- 2 alytic curves andapproximately circles,whileothers aregeometricallyhighly 1 distorted. We do this by introducing a real-valued function which measures therateofescapeofpointsinA(f),andshowthatthisfunctionhasanumber ] S ofinterestingproperties. D . h at 1. Introduction m Suppose that f : C C is a transcendental entire function. The Fatou set [ F(f) is defined as the s→et of points z C such that (fn)n∈N is a normal family 1 in a neighbourhood of z. The Julia set∈J(f) is the complement in C of F(f). An v introduction to the properties of these sets was given in [5]. 6 For a general transcendental entire function the escaping set 7 6 I(f)= z :fn(z) as n { →∞ →∞} 2 was studied first in [11]. This paper concerns a subset of the escaping set, called . 1 the fast escaping set A(f). This was introduced in [7], and can be defined [21] by 0 3 (1.1) A(f)= z :there exists ℓ N such that fn+ℓ(z) Mn(R,f), for n N . 1 { ∈ | |≥ ∈ } : Here,themaximummodulusfunctionM(r,f)=max|z|=r f(z),forr 0,Mn(r,f) v | | ≥ denotes repeated iteration of M(r,f) with respect to the variable r, and R > 0 is i X such that Mn(R,f) as n . For simplicity, we only write down this → ∞ → ∞ r restrictiononR in formalstatements ofresults– elsewherethis shouldbe assumed a to be true. We write M(r) when it is clear which function is being referred to. In [21] several results on A(f) were proved by considering the closed sets A (f)= z : fn(z) Mn(R), n N , R { | |≥ ∈ } where R > 0 is such that Mn(R) as n . Rippon and Stallard [21] used →∞ → ∞ propertiesofA (f)andA(f)todevelopnewresultsrelatingtoEremenko’sconjec- R ture [11] that I(f) contains no bounded components. In addition they introduced the conceptofa spider’s web. AsetE is aspider’s webifE is connectedandthere exists a sequence of bounded simply connected domains (Gn)n∈N such that ∂G E, G G , for n N, and G = C. n n n+1 n ⊂ ⊂ ∈ n∈N [ A transcendental entire function for which A (f) is a spider’s web has very R strong dynamical properties – for example, it is shown in [21] that A(f) and I(f) 1 2 D.J.SIXSMITH arealsospiders’webs,Eremenko’sconjectureholdsandallcomponentsoftheFatou setarebounded. Therearemanylargeclassesoftranscendentalentirefunctionsfor whichit is knownthat A (f) is a spider’sweb; see [21], [17] and[22] for examples. R To understand the structure of A (f) spiders’ webs, Rippon and Stallard [21] R introduced fundamental holes and fundamental loops. When A (f) is a spider’s R web,wedefinethefundamentalholeH asthecomponentofA (f)c thatcontains R R the origin, and the fundamental loop L by L =∂H . Since A (f) is closed, we R R R R have that L A (f). R R ⊂ Ournotationherediffersslightlyfromthatin[21]. ForR>0fixed,Ripponand Stallard define sets Am(f)= z : fn(z) Mn+m(R), n N, n+m 0 , for m 0, R { | |≥ ∈ ≥ } ≥ anddefinethe sequenceoffundamentalholesto bethe componentsofAm(f)c that R contain the origin. Denoting this sequence by (H′ ) , we observe that these m m≥0 notations are related by the equation H =H′ , for m 0. Mm(R) m ≥ Itwasshownin[21,Theorem1.9(a)]thatA (f)isaspider’swebwheneverf is R a transcendental entire function with a multiply connected Fatou component. We now give the first results on the properties of fundamental loops in this case. The first of these gives information on the location of some fundamental loops. We say thatasetU surroundsasetV ifandonlyifV iscontainedinaboundedcomponent of C U. We also write dist(z,U)=inf z w. w∈U \ | − | Theorem 1.1. Suppose that f is a transcendental entire function. Then there exists R′ = R′(f) > 0 such that the following holds. If U is a multiply connected Fatou component of f, such that U surrounds the origin and dist(0,U)>R′, then there exist 0 0 such that the following holds. If U is a multiply connected Fatou component of f, such that U surrounds the origin, dist(0,U)>R′, and h is as defined as in (1.2), then (a) if L U is a fundamental loop, then h(z) is constant on L and so L is an R R R ⊂ analytic Jordan curve; (b) if Γ is a level set of h, then Γ has a component γ which surrounds the origin and there is a fundamental loop L such that L γ. R R ⊂ It follows from these results that the fundamental loops of a transcendental entire function can have very varied geometrical properties. For example, consider the transcendental entire function f given in [8, Example 3]. This has a multiply connected Fatou component U with the property that max log z :z ∂ U out n lim { | | ∈ } = . n→∞ min log z :z ∂outUn ∞ { | | ∈ } ByTheorem1.1,thereisafundamentalloopoff whichcoincideswith∂ U ,and out n so is far from circular for large values of n. However, there are also fundamental loopsoff whichlieinsideU ,foreachn N. ByTheorem1.3(a)theseareanalytic n ∈ Jordan curves, and by [8, Theorem 7.1] can be approximately circular. A key tool in the proofs of these theorems is a function R , defined in (4.1) A below, which for a point z is the largest R such that z A (f). In general this R ∈ functioncanonlybedefinedinasubsetofA(f). InSection6weshowthat,subject to a certain normalisation,this definition can in fact be extended in a natural way to the whole complex plane. We show that, in this case, there is an alternative characterisation of A(f). We also show that the function R has a number of in- A teresting properties. The structure of this paper is as follows. First, in Section 2, we state a number of results required in the proof of our main theorems. With the exception of Lemma 2.6, these are all known results. In Section 3 we prove a new result, which states that if a transcendental entire function has a certain property with respect to a nested sequence of bounded simply connected domains, then there is a fixed point which has a certain ‘attracting’ property. This may be of independent inter- est. In Section 4 we show that the function R can be defined in certain multiply A connected Fatou components, and prove several preparatory lemmas. In Section 5 weproveTheorems1.1,1.2and1.3. Finally,inSection6westateandproveseveral results regarding the case when R can be defined in the whole complex plane. A 2. Background material We use the following notation for an annulus and a disc A(r ,r )= z :r < z 0 0 0 such that, for all 0R , 0 ≥ and (2.4) M(rc′,fn) M(r,fn)c′, for r >R1/c′. ≤ 0 We denote the inverse function of M, when this is defined, by M−1. For simplicity we write M−n, for n N, to denote n repeated iterations of M−1. Observe ∈ that M−1(r) is defined for r [f(0), ) and is strictly increasing. Moreover, ∈ | | ∞ by (2.1) and [16, Theorem 7.2.2],logM−1(es) is a concave and increasing function of s. Also, if R is the constant from Lemma 2.1, then it follows from (2.3) that 0 (2.5) M−1(rc) M−1(r)c, for r >max M(R ), f(0) , c>1. 0 ≤ { | |} We next require a number of results concerning multiply connected Fatou components. Our first is the following well-known result of Baker [1, Theorem 3.1], which we often use without comment. Lemma 2.2. Suppose that f is a transcendental entire function and that U is a multiply connected Fatou component of f. Then each U is bounded and multiply n connected, U surrounds U for large n, and U as n . n+1 n n →∞ →∞ In particular,if U is a multiply connectedFatou component, then U is bounded and so fn : U U is a proper map, for n N; see [15, Corollary 1]. Hence n U =fn(U), for→n N. ∈ n ∈ We need a number of results from[8]. Suppose that f is a transcendentalentire function with a multiply connected Fatou component U = U , and let z U be 0 0 ∈ fixed. Itfollowsfrom[8,Theorem1.2]thatthereexistsα>0suchthat,forlargen, themaximumannuluscentredattheorigin,containedinU andcontainingfn(z ) n 0 is of the form (2.6) B =A(ran,rbn), where r = fn(z ), 0⊂0 is such that Mn(R) n+ℓ as∈n . It follows≥from (2.2) that w∈e can choose N N sufficiently large t→hat∞ →∞ ∈ ∞ (2.11) β >c. k k=N Y Now let z U. We can further assume that N is sufficiently large that ∈ fn(z)c >R , for n N, 0 | | ≥ where R is the constant from Lemma 2.1. Now, by Lemma 2.3, 0 C A ran+2πδn,rbn−2πδn , n ⊂ n n for large values of n. Hence, we ca(cid:0)n assume, by Lemm(cid:1)a 2.4 and Lemma 2.5, that N is sufficiently large that (2.12) log fn+1(z) β logM(fn(z)), for n N. n | |≥ | | ≥ Hence, by (2.12) and (2.4), (2.13) fn+1(z) M(fn(z))βn M(fn(z)βn), for n N. | |≥ | | ≥ | | ≥ By repeated application of (2.13) and (2.4), and by (2.11), we have that fn+m(z) Mm(fn(z)Qmk=−01βn+k) Mm(fn(z)c), for n N, m N, | |≥ | | ≥ | | ≥ ∈ as required. (cid:3) We also need the following [21, Theorem 2.3]. 6 D.J.SIXSMITH Lemma 2.7. Let f be a transcendental entire function and let η >1. There exists R′ =R′(f)>0 such that if r >R′, then there exists 0 0 0 z′ A(r,ηr) A(f) ∈ ∩ with fn(z′) >Mn(r,f), for n N, | | ∈ and hence z′ A (f) and M(ηr,fn)>Mn(r,f), for n N. r ∈ ∈ Finally, in Section 6 we need the following well-known result [20, Lemma 2.1]. Lemma2.8. Letf beatranscendentalentirefunction, let K beacompact set with K E(f) = and let ∆ be an open neighbourhood of z J(f). Then there exists N ∩ N such t∅hat fn(∆) K, for n N. ∈ ∈ ⊃ ≥ HereO−(z)= w :fn(w)=z, for some n N andE(f)= z :O−(z) is finite . { ∈ } { } The set E(f) contains at most one point. 3. A map on a nested sequence of domains In this section we prove a result about the existence and properties of a fixed point for certain transcendental entire functions. This may be of independent in- terest. For a hyperbolic domain V, we write [w,z] for the hyperbolic distance V between w and z in V. The main result of this section is as follows. Theorem3.1. Supposethatf isatranscendentalentirefunction,andthat(G ) n n≥0 is a sequence of bounded simply connected domains such that (3.1) G G and f(∂G )=∂G , for n=0,1,2, . n n+1 n n+1 ⊂ ··· Then there exists α G , a fixed point of f, such that, if K G is compact, then 0 0 ∈ ⊂ [α,fn(z)] 0 as n , uniformly for z K. Gn → →∞ ∈ To proveTheorem3.1we requirethe followinglemma. Define D= z : z <1 . { | | } Lemma 3.1. Suppose that (B ) is a sequence of analytic functions from D to n n≥0 D. Suppose also that there exist α D and λ (0,1) such that ∈ ∈ (3.2) B (α)=α and B′(α) λ, for n=0,1,2, . n | n |≤ ··· Then, if K′ is a compact subset of D, B B (z) α as n , uniformly for z K′. n 0 ◦···◦ → →∞ ∈ Proof. ByconjugatingwithaMo¨biusmapifnecessary,wemayassumethatα=0. A result of Beardon and Carne [3, p.217] states that if g :D D is an analytic → function with g(0)=0, then z + g′(0) g(z) z | | | | , for z D, | |≤| | 1+ g′(0)z ∈ (cid:18) | |(cid:19) in which case r+ g′(0) M(r,g) r | | , for r (0,1). ≤ 1+ g′(0)r ∈ (cid:18) | | (cid:19) ON FUNDAMENTAL LOOPS AND THE FAST ESCAPING SET 7 Now, (r+x)/(1+rx) is an increasing function of x, for r (0,1). Hence, by ∈ (3.2), r+λ (3.3) M(r,B ) r =µ(r)0suchthatthefollowingholds. SupposethatU isamultiplyconnected Fatou component of f, which surrounds the origin and satisfies dist(0, U) > R′. Define G as the complementary component of U which contains the origin, for n n n=0,1,2, . Then ··· (a) G G , for n=0,1,2, ; n n+1 ⊂ ··· (b) f(∂G )=∂G , for n=0,1,2, ; n n+1 ··· (c) for all z U there exists R=R(z) such that z A (f). R ∈ ∈ ON FUNDAMENTAL LOOPS AND THE FAST ESCAPING SET 9 Proof. Firstwe note [21, Theorem1.9(a)] that A (f) is a spider’s web. Hence, by R Lemma 4.1, f has a fixed point α. Let V be a multiply connected Fatou component of f. By Lemma 2.2 there is an N N such that fN(V) surrounds both the origin and α, and also fn+1(V) ∈ surrounds fn(V) for n N. ≥ Choose R > 0 such that Mn(R) as n . Then [21, Theorem 1.2(a)] → ∞ → ∞ there is an L N such that fL(V) A (f). Set M = max L,N , and let R ∈ ⊂ { } R′ = max z : z fM(V) . {| | ∈ } Suppose that U is any multiply connected Fatou component such that U sur- roundstheoriginandsatisfiesdist(0,U)>R′. ThenU certainlysurroundsfM(V). Since U surrounds α, then U = f(U) surrounds α by the argument principle. 1 Moreover,U cannot meet either fM+1(V) or U, since ∂U J(f). Hence, by the 1 1 ⊂ maximumprinciple,U surroundsbothfM+1(V)andU. Inductively,U surrounds 1 k both fM+k(V) and U , for k N. Parts (a) and (c) of the lemma follow from k−1 ∈ this fact and the choice of M. Finally we establish part (b). Choose n = 0,1,2, . Since f(G ) is open and n ··· connected,anditsboundaryisinJ(f),itcannotmeettheboundaryofG . Now, n+1 α G , and so f(G ) G = . Hence ∂f(G ) must lie in G , and so n n n+1 n n+1 ∈ ∩ 6 ∅ f(∂G ) G . n n+1 ⊂ Moreover, f is a proper map on the Fatou component U , and so n f(∂G ) f(∂U )=∂U . n n n+1 ⊂ Thus ∂f(G )=∂G , as required. (cid:3) n n+1 Suppose that U is a multiply connected Fatou component which surrounds the origin, and that dist(0,U)>R′, where R′ is the constant from Lemma 4.2. Then, by Lemma 4.2(c) and by the continuity of M, we may define (4.1) R (z)=max R:z A (f) , for z U. A R { ∈ } ∈ The function R has some strong continuity properties, and shares level sets with A the function h. Lemma 4.3. Suppose that f and U are as in Theorem 1.1, and that R is as in A (4.1). Then R is upper semicontinuous in U and continuous in U. Moreover, if h A is as in (1.2), then there exists a continuous strictly increasing function φ:R R → such that (4.2) R (z)=φ(h(z)), for z U. A ∈ Proof. We first prove that R is upper semicontinuous in U. Suppose that z U A ∈ and that ǫ > 0. By the definition of R , we have that z / A (f). Hence A ∈ RA(z)+ǫ there is anN N such that fN(z) 0 such that fN(z′) 0 such that the following holds. If ∆ U is a neigh- ⊂ bourhood of z, then there is a z′ ∆ such that R (z) ǫ > R (z′), in which A A ∈ − case z′ ∈/ ARA(z)−ǫ(f). There exists, therefore, a sequence (zk)k∈N of points of U, distinct from but tending to z, and a sequence (nk)k∈N of integers such that fnk(z ) 1,forsufficientlylargevaluesofk. Notethatwecanassumethatn k →∞ as k . →∞ We mayassumethatǫ issufficiently smallthatMn(R (z) ǫ) asn . A − →∞ →∞ Hence we can choose N large enough that MN(R (z) ǫ)> R , where R is the A 0 0 − constant from Lemma 2.1. Set logMN(R (z)) r =MN(R (z) ǫ) and c= A >1. A − logr It follows by repeated application of (2.3) that we have logMm(rc) clogMm(r), for m N. ≥ ∈ Hence logMm+N(R (z)) logMm(rc) A = c>1, for m N. logMm+N(R (z) ǫ) logMm(r) ≥ ∈ A − This establishes our claim regarding the left-hand side of (4.3). To establish our claim regarding the right-hand side of (4.3) we use some tech- niques from[19], though we give the full details for completeness. For a hyperbolic domain V, we let ρ denote the density of the hyperbolic metric in V. V Choose any w ,w J(f) with w =w , and put G=C w ,w . Note that 1 2 1 2 1 2 ∈ 6 \{ } [z,zk]U ≥[fnk(z),fnk(zk)]fnk(U) ≥[fnk(z),fnk(zk)]G = ρG(z)|dz|, for k ∈N, ZΓk where Γk is a hyperbolic geodesic in G joining fnk(z) to fnk(zk). By, for example, [13, Theorem 9.14], there exist R>2 and C >0 such that C ρ (z) , for z R. G ≥ z log z | |≥ | | | | Choose K sufficiently large such that fnk(z) > fnk(z ) >2R, for k K. k | | | | ≥ We then have |fnk(z)| dr log fnk(z) (4.4) [z,z ] ρ (z) dz C =Clog | | . k U ≥ZΓk G | |≥ Z|fnk(zk)| rlogr (cid:18)log|fnk(zk)|(cid:19)

On fundamental loops and the fast escaping set

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