For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree $d\geq 2$ is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle $d$-tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of "docile" functions: a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This can be seen as an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljevi\'c for more restrictive classes of entire functions

Alhamed, Mashael

Rempe, Lasse

Sixsmith, Dave

University of Liverpool Repository

ESCAPING SETS ARE NOT SIGMA-COMPACTLASSE REMPEAbstract. Let f be a transcendental entire function. The escaping set I(f) consistsof those points that tend to infinity under iteration of f . We show that I(f) is notσ-compact, resolving a question of Rippon from 2009.1. IntroductionLet f : C→ C be a transcendental entire function. The setI(f) ..= {z ∈ C : limn→∞fn(z) =∞}is called the escaping set of f , wherefn = f ◦ · · · ◦ f︸ ︷︷ ︸n timesdenotes the n-th iterate of f . The escaping set was first studied by Eremenko [Ere89]and has been the subject of intensive research in recent years; see e.g. [Obe09, RRRS11,Ber18, RS19] and their references. The topological study of I(f) turns out to be sur-prisingly intricate. For example Eremenko [Ere89, p. 343] asked whether, for everytranscendental entire function, every connected component of I(f) is unbounded. Thisquestion has become known as Eremenko’s conjecture, and is one of the most famousopen problems in transcendental dynamics. Strengthened versions of the conjecture havesince been proved false in general; compare [RRRS11, Theorem 1.1] and [R16, Theo-rem 1.6]. On the other hand, Rippon and Stallard [RS11b] have shown that I(f)∪{∞}is always connected, so any counterexample to Eremenko’s original conjecture wouldhave rather subtle topological properties.The fast escaping set is a certain subset of the escaping set, first introduced by Berg-weiler and Hinkkanen [BH99]. This set has also received considerable attention recently;in part because it appears to be more tractable than the full escaping set I(f). In par-ticular, the analogue of Eremenko’s conjecture holds for A(f) [RS05]. One differencebetween the two objects lies in the topological complexity of their definitions. Indeed,A(f) can be written as a countable union of closed sets [RS12, Formula (1.3)]; so it isan Fσ set. On the other hand, by definitionI(f) = {z ∈ C : ∀M ≥ 0∃N ≥ 0: |fn(z)| ≥M for n ≥ N}=∞⋂M=0∞⋃N=0∞⋂n=Nf−n(C \D(0,M)),Date: March 19, 2021.2010 Mathematics Subject Classification. Primary 30D05, Secondary 37F10, 54D45.12 LASSE REMPEwhere D(0,M) denotes the disc of radius M around 0. So I(f) is Fσδ: a countableintersection of countable unions of closed sets. This raises the following question, posedby Rippon in 2009 [Obe09, Problem 8, p. 2960].1.1. Question. Is there a transcendental entire function f for which I(f) itself is Fσ?Since any closed subset of C is a countable union of compact sets, it is equivalent toask whether I(f) is ever σ-compact (a countable union of compact sets); compare also[Lip20b]. It is well-known that I(f) cannot be a Gδ (a countable intersection of opensets); see Lemma 2.4 below.As far as we are aware, there is no function f for which Question 1.1 has been pre-viously resolved; the goal of this paper is to give a negative answer in general. In fact,we prove a stronger result, as follows. Let UO(f) consist of all z ∈ C whose orbit{fn(z) : n ≥ 0} is unbounded, and let BU(f) = UO(f) \ I(f) denote the “bungee set”;see [OS16]. Recall that the Julia set J(f) is the set of non-normality of the iterates of f .1.2. Theorem. Let f be a transcendental entire function. Then every σ-compact subsetof UO(f) omits some point of I(f) ∩ J(f) and some point of BU(f) ∩ J(f).In particular, the sets I(f), UO(f), BU(f) and their intersections with J(f) are notσ-compact.Acknowledgements. I thank David Lipham and Phil Rippon for interesting discus-sions. I also thank David Mart́ı-Pete, James Waterman and the referee for helpfulcomments that improved the presentation of the paper.2. Proof of the TheoremWe require a result on the existence of arbitrarily slowly escaping points, which followsfrom work of Rippon and Stallard [RS15]; see also [RS11a, Theorem 1].2.1. Theorem. Let f be a transcendental entire function, and let R0 ≥ 0. Then thereexists M0 > R0 with the following property. If (am)∞m=0 is a sequence with am →∞ andam ≥M0 for all m, then there are ζ ∈ J(f) ∩ I(f) and ω ∈ J(f) ∩ BU(f) withR0 ≤ |fm(ζ)| ≤ am and R0 ≤ |fm(ω)| ≤ am for m ≥ 0.Proof. The result is an easy consequence of [RS15], which studies “annular itineraries”of entire functions. Fix some R > 0 such that M(r) > r for r ≥ R, where M(r) denotesthe maximum modulus of f on the disc D(0, r). Then the annular itinerary of a pointz is the sequence (sm)∞m=0 such that sm = 0 if |fm(z)| < R, andM sm−1(R) ≤ |fm(z)| < M sm(R)otherwise. (Here Mk is the k-th iterate of r 7→M(r).)Theorem 1.2 of [RS15] implies that for suitable R ≥ R0, for an entry sm = n ≥ 0in an annular itinerary, there is a subset Xn ⊂ {0, . . . , n + 1} of allowable next entriessm+1, and that, for infinitely many n, all entries but at most one are allowable. Moreprecisely,(a) any sequence (sm)∞m=0 with sm+1 ∈ Xsm for all m is the annular itinerary of somez ∈ J(f);(b) n+ 1 ∈ Xn for all n;ESCAPING SETS ARE NOT SIGMA-COMPACT 3(c) there is an increasing sequence (nj)∞j=1 such that #Xnj ≥ n+ 1.(With the notation of [RS15], Xnj = {0, . . . , nj+1}\Ij, where #Ij ≤ 1, and Xn = {n+1}when n 6= nj for all j.) We may suppose that the sequence (nj) is chosen such thatn1 ≥ 2.Set M0 ..= Mn1(R) and let (am)∞m=0 be a sequence as in the statement of the theo-rem. We may assume that (am) is non-decreasing. Similarly as in the proof of [RS15,Corollary 1.3 (d)], we can construct an annular itinerary (sm)∞m=0 satisfying (a) suchthat sm →∞ and(2.1) M sm(R) ≤ am for m ≥ 0.Indeed, for each j, either nj ∈ Xnj or nj − 1 ∈ Xnj ; let us denote this element of Xnj byñj. Then any sequence of the form(sm)∞m=0 = n1, ñ1, n1, ñ1, . . . , n1︸ ︷︷ ︸length N1, n1 + 1, n1 + 2, . . . , n2, ñ2, . . . , n2︸ ︷︷ ︸length N2, n2 + 1, n2 + 2, . . .satisfies (a). If Nj is chosen sufficiently large that(2.2) am ≥Mnj+1(R) for m ≥ Nj,then (2.1) holds for m ≥ N1. It also holds for m ≤ N1, since sm ≤ n1 for these values ofm, and am ≥M0 = Mn1(R) by assumption.Let ζ ∈ J(f) have annular itinerary (sm)∞m=0; then ζ ∈ I(f) andR0 ≤ R ≤M sm−1(R) ≤ |fm(ζ)| < M sm(R) ≤ am for all m.Hence ζ has the desired properties. To obtain ω, we instead use an unbounded butnon-escaping sequence of the form(s̃m)∞m=0 = B1, n1 + 1, n1 + 2, . . . , n2, n̂2, B2, n1 + 1, n1 + 2, . . . , n3, n̂3, B3, . . . ,where n̂j ∈ Xnj is either n1 or n1 − 1, and Bj is a block alternating between entries n1and ñ1. To ensure that the sequence satisfies (a), Bj should end in n1; it should beginwith n1 if n̂j = n1 − 1, and with ñ1 if n̂j = n1. The length Nj of the block Bj is againchosen to satisfy (2.2). Proof of Theorem 1.2. If D ⊂ C is closed and z ∈ C, we setnD(z) ..= min{n ≥ 0: fn(z) /∈ D} ≤ ∞,with the convention that nD(z) =∞ if no such n exists. Observe that nD depends uppersemicontinuously on z, as the infimum of upper semicontinuous functions. Indeed,nD(z) = min{χn(z) : n ≥ 0}, where χn(z) ={∞ if z ∈ f−n(D)n if z /∈ f−n(D),and each χn is upper semicontinuous since f−n(D) is closed. Note also that z ∈ UO(f)if and only if nD(z) <∞ for every compact D ⊂ C.Now let R0 ≥ 0 be arbitrary, and choose M0 as in Theorem 2.1. Let X ⊂ UO(f)be σ-compact; say X =⋃∞j=0Kj where each Kj is compact. Define Mj..= M0 + j andDj ..= D(0,Mj). Then nDj(z) <∞ for every z ∈ Kj.4 LASSE REMPESince Kj is compact and nDj is upper semicontinuous,nj ..= maxz∈KjnDj(z)exists for every j. Set N0 ..= n0 and Nj+1 ..= max{nj+1, Nj + 1}. For n ≥ 0, definej(n) ..= min{j : Nj ≥ n} andan ..= Mj(n) = min{Mj : Nj ≥ n}.Then an ≥ M0 for all n and limn→∞ an = ∞. So by Theorem 2.1, there are ζ ∈I(f) ∩ J(f) and ω ∈ BU(f) ∩ J(f) such that |fn(z)| ≤ an for z ∈ {ζ, ω} and all n ≥ 0.Let j ≥ 0. Then, for n ≤ nj ≤ Nj, we have |fn(z)| ≤ an ≤Mj, and hence fn(z) ∈ Dj.So nDj(z) > nj, and z /∈ Kj by choice of nj. Thus z /∈ X =⋃∞j=0Kj, as claimed. By the expanding property of the Julia set, we also obtain the following answer to aquestion of Lipham (personal communication).2.2. Corollary. Let f be a transcendental entire function and let Y be one of the setsI(f) ∩ J(f), UO(f) ∩ J(f) and BU(f) ∩ J(f). Then Y is nowhere σ-compact. That is,if X ⊂ Y is σ-compact, then X is nowhere dense in Y .Proof. We prove a slightly stronger statement. Let Y be as in the statement and letX ⊂ UO(f) ∩ J(f) be σ-compact. Then X ∩ Y is nowhere dense in Y ; i.e., if U ⊂ Cwith U ∩ Y 6= ∅, then X omits some point of U ∩ Y .Let R0 and M0 be as in Theorem 2.1, and set L0 ..= J(f)∩D(0,M0) \D(0, R0). If R0is chosen sufficiently large, then L0 contains no Fatou exceptional point of f . (Recallthat a Fatou exceptional point is a point with finite backward orbit, and that f has atmost one finite exceptional point [Ber93, p. 156].) Since U ∩ J(f) 6= ∅, it follows thatthere is some n such that fn(U) ⊃ L0.The proof of Theorem 1.2 shows that no σ-compact subset of UO(f) ∩ J(f) containsL0 ∩ Y . (Note that we may take a0 = M0 in the proof to ensure that |ζ|, |ω| ≤ M0.)Moreover, fn(X) is σ-compact as the image of a σ-compact set under a continuousfunction. So fn(X) omits some points of L0 ∩ Y . Since Y is backward-invariant, Xomits some points of U ∩ Y , as claimed. We remark that the proof of Theorem 1.2 establishes the following very general prin-ciple:2.3. Proposition (Abstract version of the main theorem). Let f : U → V be continuous,where U, V are non-empty topological spaces and U is σ-compact. Let UO(f) denote theset of z ∈ U such that fn(z) is defined and in U for all n ≥ 0, but the orbit {fn(z)} isnot contained in any compact subset of U .Suppose that X ⊂ UO(f) is σ-compact and ∆ ⊂ U is compact. Then there is asequence (∆n)∞n=0 of compact subsets ∆n ⊂ U with ∆ ⊂ ∆0 ⊂ ∆1 ⊂ . . . and⋃∞n=0 ∆n =U such that X contains no point ζ ∈ UO(f) with fn(ζ) ∈ ∆n for all n ≥ 0.Remark. Note that we are not assuming any relation between the spaces U and V .However, the statement is vacuous when UO(f) = ∅, and in particular when U ∩V 6= ∅.Sketch of proof. Let (Dj)∞j=0 be an increasing sequence of compact subsets Dj ⊂ U with⋃∞j=0Dj = U . Define j(n) as in the proof of Theorem 1.2; then the sets ∆n..= Dj(n)have the desired property. ESCAPING SETS ARE NOT SIGMA-COMPACT 5It follows that the set of escaping points is not σ-compact in any setting where ananalogue of Theorem 2.1 holds. This includes:(a) transcendental meromorphic functions C→ Ĉ [RS11a];(b) transcendental self-maps of the punctured plane [MP18, Theorem 1.2];(c) quasiregular self-maps f : Rd → Rd of transcendental type [Nic16];(d) continuous functions ϕ : [0,∞) → [0,∞) with ϕ(t) 6→ ∞ as t → ∞, and suchthat I(ϕ) 6= ∅ [ORS19, Theorem 2.2].A more general setting than both (a) and (b) is provided by the Ahlfors islands mapsof Epstein; see e.g. [RR12]. These are maps f : W → X, where X is a compact one-dimensional manifold, W ⊂ X is open, and f satisfies certain transcendence conditionsnear ∂W . We may define the escaping set I(f) as the set of points z ∈ W with fn(z) ∈ Wfor all n and dist(fn(z), ∂W ) → 0 as n → ∞. It is plausible that an analogue ofTheorem 2.1 holds for Ahlfors islands maps with W 6= X, using a similar proof asin [RS11a]. This would mean, by Proposition 2.3, that I(f) is not σ-compact for suchfunctions.For completeness, we conclude by giving the simple proof that I(f) is never a Gδ set;compare also [Lip20a, Corollary 3.2].2.4. Lemma. Let f be a transcendental entire function. Then I(f) and I(f)∩J(f) arenot Gδ sets.Proof. Every closed subset of C (or any metric space) is Gδ; so J(f) is Gδ. The inter-section of two Gδ sets is again Gδ, so it is enough to prove the claim for I(f) ∩ J(f).By [Ere89, Theorem 2], I(f) ∩ J(f) is nonempty, and hence dense in J(f) by Montel’stheorem. By Baire’s theorem, any two dense Gδ subsets of J(f) must intersect. Henceit is enough to observe that BU(f) ∩ J(f) contains a dense Gδ by Montel’s theorem,namely the set of points whose orbits are dense in J(f). (See [BD00, Lemma 1].) Lipham has pointed out the following reformulation of Corollary 2.2.2.5. Corollary. Any Gδ set A ⊂ Y ..= J(f) \ I(f) is nowhere dense in Y .In particular, Y is Gδσ but not strongly σ-complete; that is, it cannot be written as acountable union of relatively closed Gδ subsets.Proof. Suppose, by contradiction, that D(ζ, ε) ∩ Y ⊂ A for some ζ ∈ J(f) and ε > 0.The complement of a Gδ set is Fσ, so B ..= J(f) ∩D(ζ, ε) \ A is an Fσ set withD(ζ, ε) ∩ I(f) ∩ J(f) ⊂ B ⊂ I(f) ∩ J(f),which contradicts Corollary 2.2.The set Y is Gδσ by definition. As mentioned in the proof of Lemma 2.4, Y containsa Gδ subset U that is dense in J(f). On the other hand, if (Ak)∞k=0 is a sequence ofrelatively closed Gδ subsets of Y , then Ak is closed and nowhere dense in J(f). Hence∞⋃k=0Ak ⊂∞⋃k=0Ak 6⊃ U ⊂ Y,by Baire’s theorem, and Y is indeed strongly σ-complete. 6 LASSE REMPEReferences[BD00] I. N. Baker and P. Domı́nguez, Residual Julia sets, J. Anal. 8 (2000), 121–137.[Ber93] Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29(1993), no. 2, 151–188.[Ber18] Walter Bergweiler, Lebesgue measure of Julia sets and escaping sets of certain entire func-tions, Fund. Math. 242 (2018), no. 3, 281–301.[BH99] Walter Bergweiler and Aimo Hinkkanen, On semiconjugation of entire functions, Math. Proc.Cambridge Philos. Soc. 126 (1999), no. 3, 565–574.[Ere89] A. È. Eremenko, On the iteration of entire functions, Dynamical systems and ergodic theory(Warsaw, 1986), Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 339–345.[Lip20a] David S. Lipham, A note on the topology of escaping endpoints, Ergodic Theory and Dy-namical Systems (2020, First View), 1–4.[Lip20b] David S. Lipham, The topological dimension of radial Julia sets, Preprint arXiv:2002.00853,2020.[MP18] David Mart́ı-Pete, The escaping set of transcendental self-maps of the punctured plane, Er-godic Theory Dynam. Systems 38 (2018), no. 2, 739–760.[Nic16] Daniel A. Nicks, Slow escaping points of quasiregular mappings, Math. Z. 284 (2016), no. 3-4,1053–1071.[Obe09] Mini-Workshop: The Escaping Set in Transcendental Dynamics, Report on the mini-workshop held December 6–12, 2009, organized by Walter Bergweiler and Gwyneth Stallard.Oberwolfach Reports. Vol. 6 (2009), no. 4, pp. 2927–2963.[ORS19] John W. Osborne, Philip J. Rippon, and Gwyneth M. Stallard, The iterated minimummodulus and conjectures of Baker and Eremenko, J. Anal. Math. 139 (2019), no. 2, 521–558.[OS16] John W. Osborne and David J. Sixsmith, On the set where the iterates of an entire functionare neither escaping nor bounded, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 2, 561–578.[R16] Lasse Rempe, Arc-like continua, Julia sets of entire functions, and Eremenko’s conjecture,Preprint arXiv:1610.06278, 2016.[RR12] Lasse Rempe and Philip J. Rippon, Exotic Baker and wandering domains for Ahlfors islandsmaps, J. Anal. Math. 117 (2012), 297–319.[RRRS11] Günter Rottenfußer, Johannes Rückert, Lasse Rempe, and Dierk Schleicher, Dynamic raysof bounded-type entire functions, Ann. of Math. (2) 173 (2011), no. 1, 77–125.[RS05] P. J. Rippon and G. M. Stallard, On questions of Fatou and Eremenko, Proc. Amer. Math.Soc. 133 (2005), no. 4, 1119–1126.[RS11a] , Slow escaping points of meromorphic functions, Trans. Amer. Math. Soc. 363 (2011),no. 8, 4171–4201.[RS11b] , Boundaries of escaping Fatou components, Proc. Amer. Math. Soc. 139 (2011),no. 8, 2807–2820.[RS12] , Fast escaping points of entire functions, Proc. Lond. Math. Soc. 105 (2012), no. 4,787–820.[RS15] , Annular itineraries for entire functions, Trans. Amer. Math. Soc. 367 (2015), no. 1,377–399.[RS19] , Eremenko points and the structure of the escaping set, Trans. Amer. Math. Soc. 372(2019), no. 5, 3083–3111.Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK,https://orcid.org/0000-0001-8032-8580Email address: l.rempe@liverpool.ac.uk

Geometrically finite transcendental entire functions

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Geometrically finite transcendental entire functions

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