Dynamical sets whose union with infinity is connected
DDYNAMICAL SETS WHOSE UNION WITH INFINITY ISCONNECTED
DAVID J. SIXSMITH
Abstract.
Suppose that f is a transcendental entire function. In 2014, Rippon andStallard showed that the union of the escaping set with infinity is always connected.In this paper we consider the related question of whether the union with infinity ofthe bounded orbit set, or the bungee set, can also be connected. We give sufficientconditions for these sets to be connected, and an example a transcendental entirefunction for which all three sets are simultaneously connected. This function lies, infact, in the Speiser class.It is known that for many transcendental entire functions the escaping set has atopological structure known as a spider’s web. We use our results to give a large classof functions in the Eremenko-Lyubich class for which the escaping set is not a spider’sweb. Finally we give a novel topological criterion for certain sets to be a spider’s web. Introduction
Let f be an entire function. When studying complex dynamics it is usual to partitionthe complex plane into two sets; the Julia set J ( f ), which contains those points in aneighbourhood of which the iterates of f are chaotic, and its complement the Fatou set F ( f ) = C \ J ( f ). The Fatou set is open, and its connected components are called Fatoucomponents . For more information on complex dynamics, including precise definitionsand properties of these sets, we refer to [Ber93].Recently, several authors have worked with an alternative partition. This divides theplane into three sets determined by the nature of the orbits of points; the orbit of apoint z ∈ C is the sequence ( f n ( z )) n ≥ of its iterates under f . We define these sets asfollows. Firstly, the escaping set is given by I ( f ) := { z ∈ C : f n ( z ) → ∞ as n → ∞} . The escaping set for a general transcendental entire function f was first studied byEremenko [Ere89]. He showed that I ( f ) ∩ J ( f ) (cid:54) = ∅ , and that all components of I ( f ) areunbounded. He also conjectured that the same is true of all components of I ( f ). Thisconjecture, which is still open, has since been the focus of much research in complexdynamics.Secondly, the bounded orbit set is defined by BO ( f ) := { z ∈ C : there exists K > | f n ( z ) | < K, for n ≥ } . When f is a polynomial, BO ( f ) is known as the filled Julia set , and has been muchstudied. The set BO ( f ) for a transcendental entire function f was studied in [Ber12]and [Osb13]. Mathematics Subject Classification.
Primary 37F10; Secondary 30C65, 30D05. a r X i v : . [ m a t h . D S ] D ec DAVID J. SIXSMITH
Finally, the bungee set is defined simply as BU ( f ) := C \ ( I ( f ) ∪ BO ( f )). It is easy tosee that if P is a polynomial, then there is a punctured neighbourhood of infinity thatlies in I ( P ), and so BU ( P ) is empty. However, if f is transcendental, then BU ( f ) isnon-empty. In fact, see [OS16, Theorem 5.1], the Hausdorff dimension of BU ( f ) ∩ J ( f )is greater than zero. The properties of BU ( f ) were studied in [Laz17] and in [OS16].From here onwards we assume that f is transcendental. It is now known that the set (cid:98) I ( f ) := I ( f ) ∪{∞} is a connected subset of the Riemann sphere; see [RS11, Theorem 4.1].Moreover, see [ORS17, Theorem 1.1], the same property holds for I ( f ) ∪ BU ( f ) ∪ {∞} .Our principle interest in this paper is to ask if there are conditions that ensure that oneor both of the sets (cid:100) BO ( f ) := BO ( f ) ∪ {∞} and (cid:100) BU ( f ) := BU ( f ) ∪ {∞} can also be connected. In fact we have the following result. Theorem 1.1.
There is a transcendental entire function f such that each of the sets (cid:98) I ( f ) , (cid:100) BO ( f ) and (cid:100) BU ( f ) is connected. Remark.
The function f in Theorem 1.1 has only two singular values (points at whichit is not possible to define some inverse branch). It follows that f is in the Speiser class S , which consists of those transcendental entire functions for which the set of singularvalues is finite.Since (cid:98) I ( f ) is always connected, in order to prove Theorem 1.1 we give sufficient condi-tions for (cid:100) BO ( f ) or (cid:100) BU ( f ) to be connected. We then show that there is a transcendentalentire function with the necessary properties. The first result is as follows. Theorem 1.2.
Suppose that f is a transcendental entire function. If f has an unboundedFatou component in BO ( f ) (resp. BU ( f ) ), then (cid:100) BO ( f ) (resp. (cid:100) BU ( f ) ) is connected. Remark.
The usefulness of the second part of this theorem is limited by the fact thatthere are relatively few examples of transcendental entire functions with Fatou com-ponents in BU ( f ). Examples of a transcendental entire function with such a Fatoucomponent were given in [Bis15] and [EL87]. The only examples of Fatou componentsin BU ( f ) which are also known to be unbounded were given in [Laz17] and [OS16].The second result requires the notion of a finite logarithmic asymptotic value, whichwe define as follows. We use the notation B ( a, r ) := { z ∈ C : | z − a | < r } , for a ∈ C and r > . Definition 1.
Suppose that f is a transcendental entire function. A value α ∈ C isa finite logarithmic asymptotic value of f if there exist r > and a component U of f − ( B ( α, r )) , such that the restriction f : U → B ( α, r ) \ { α } is a universal covering. Our sufficient condition for the connectedness of (cid:100) BU ( f ) is as follows. Theorem 1.3.
Suppose that f is a transcendental entire function. If f has a finitelogarithmic asymptotic value α ∈ J ( f ) , then (cid:100) BU ( f ) is connected. Remark.
In [OS16] the authors asked if there is a transcendental entire function f suchthat BU ( f ) is connected. Although this question is still open, these results give at leasta partial answer to this question. YNAMICAL SETS WHOSE UNION WITH INFINITY IS CONNECTED 3
Recent study of I ( f ) has shown that this set often has a topological structure knownas a spider’s web. The following definition of a spider’s web was first given in [RS12]. Definition 2.
A connected set E ⊂ C is a spider’s web if there exists a sequence ofbounded simply connected domains, ( G n ) n ∈ N , such that ∂G n ⊂ E, G n ⊂ G n +1 , for n ∈ N , and (cid:91) n ∈ N G n = C . Clearly if I ( f ) is a spider’s web, then neither (cid:100) BO ( f ) nor (cid:100) BU ( f ) can be connected. Infact, in some sense, the converse is also true; if (cid:100) BO ( f ) ∪ (cid:100) BU ( f ) is disconnected, then I ( f ) is a spider’s web. See Corollary 4.2 below.There are now many examples of transcendental entire functions f such that I ( f ) isa spider’s web; see, for example, [Evd16], [RS12] and [Six11]. However, none of theseexamples are in the much studied Eremenko-Lyubich class B ; this class consists of thosetranscendental entire functions for which the set of singular values is bounded. Thetechniques used to prove our earlier results can be used to show that there is a largesubclass of class B for which the escaping set is not a spider’s web. Theorem 1.4.
Suppose that f ∈ B is a transcendental entire function. If f has a finitelogarithmic asymptotic value, then I ( f ) is not a spider’s web. In fact we conjecture the following.
Conjecture. If f ∈ B is a transcendental entire function, then I ( f ) is not a spider’sweb. Our final result is the following, which gives a simple topological characterisation ofan I ( f ) spider’s web, and also an A ( f ) spider’s web, for a transcendental entire function f . Here A ( f ) is the so-called fast escaping set , which was introduced in [BH99], and canbe defined, see [RS12], by;(1.1) A ( f ) := { z ∈ C : there exists (cid:96) ∈ N such that | f n + (cid:96) ( z ) | ≥ M n ( R, f ) , for n ∈ N } . Here the maximum modulus function is defined by M ( r, f ) := max | z | = r | f ( z ) | , for r ≥ . We write M n ( r, f ) to denote repeated iteration of M ( r, f ) with respect to the variable r . In (1.1), we assume that R > M n ( R, f ) → ∞ as n → ∞ .Finally, we say that a set E ⊂ C separates a point z ∈ C from infinity if there is abounded open set U such that z ∈ U and ∂U ⊂ E . Theorem 1.5.
Suppose that f is a transcendental entire function. Then I ( f ) (resp. A ( f ) ) is a spider’s web if and only if it separates some point of J ( f ) from infinity. If f has no multiply connected Fatou components, then J ( f ) is a spider’s web if and only ifit separates some point of J ( f ) from infinity. Remark.
It is known that if f is a transcendental entire function, then I ( f ) containsan unbounded component; see [RS12, Theorem 1.1]. This implies that neither BO ( f )nor BU ( f ) can be a spider’s web. DAVID J. SIXSMITH
Structure of the paper.
The structure of this paper is as follows. First, in Section 2we gather some preliminary results. Next, in Section 3 we prove Theorem 1.2 andTheorem 1.3, and then use these results to prove Theorem 1.1. Finally, in Section 4 weprove Theorem 1.4 and Theorem 1.5.2.
Preliminary results
We use the following, which is known as the “blowing-up” property of the Julia set;see, for example, [Ber93, Lemma 2.2]. Here an exceptional point is a point with finitebackward orbit; there is at most one such point.
Lemma 2.1.
Suppose that f is a transcendental entire function, and V is an open setthat meets J ( f ) . If K is a compact set that does not contain an exceptional point, thenthere exists n ∈ N such that f n ( V ) ⊃ K , for n ≥ n . We also require a result on wandering domains. If f is a transcendental entire function,and U is a Fatou component of f , then we say that U is preperiodic if there exist n, m ∈ N with n (cid:54) = m and f n ( U ) = f m ( U ). If this is not the case, then we say that U is wandering .We use the following [OS16, Theorem 1.5] which, roughly speaking, says that most pointson the boundary of a wandering domain have the same behaviour under iteration as thedomain itself. Here, for a transcendental entire function f , the ω -limit set ω ( z, f ) of apoint z ∈ C is the set of accumulation points of its orbit in (cid:98) C . For a wandering domain U of f , it follows by normality that ω ( z , f ) = ω ( z , f ) for z , z ∈ U , so in this case wecan write ω ( U, f ) without ambiguity.
Lemma 2.2.
Suppose that f is a transcendental entire function and that U is a wan-dering domain of f . Then the set { z ∈ ∂U : ω ( z, f ) (cid:54) = ω ( U, f ) } has harmonic measurezero relative to U . We need the following, which is [Six15, Lemma 3.1].
Lemma 2.3.
Suppose that ( E n ) n ∈ N is a sequence of compact sets and ( m n ) n ∈ N is asequence of integers. Suppose also that f is a transcendental entire function such that E n +1 ⊂ f m n ( E n ) , for n ∈ N . Set p n = (cid:80) nk =1 m k , for n ∈ N . Then there exists ζ ∈ E such that (2.1) f p n ( ζ ) ∈ E n +1 , for n ∈ N . If, in addition, E n ∩ J ( f ) (cid:54) = ∅ , for n ∈ N , then there exists ζ ∈ E ∩ J ( f ) such that (2.1)holds. To prove Theorem 1.3 we require the following, which seems to be new.
Lemma 2.4.
Let f be a transcendental entire function with a finite logarithmic asymp-totic value α ∈ C . Let r > be sufficiently small that f is a universal covering from acomponent T of f − ( B ( α, r )) to B ( α, r ) \ { α } . Then there exist R > , and a compo-nent V of T ∩ B (0 , R ) such that the following holds. Suppose that γ ⊂ C \ B (0 , R ) is acontinuum such that V lies in a bounded component of T \ γ . Then the complementarycomponent of f ( γ ∩ T ) containing α lies in B ( α, r ) . YNAMICAL SETS WHOSE UNION WITH INFINITY IS CONNECTED 5
Proof.
Let r and T be as in the statement of the lemma. Choose R > V of T ∩ B (0 , R ), such that f ( V ) contains an annulus of theform A := { z ∈ C : r − δ < | z − α | < r } , for some δ ∈ (0 , r ).Now, suppose that γ ⊂ C \ B (0 , R ) is a continuum such that V lies in a boundedcomponent of T \ γ . Let S be the component of C \ f ( γ ∩ T ) that contains α . We needto show that S ⊂ B ( α, r ). Suppose, therefore, that this is not the case. Then S containsboth α and a point ζ ∈ A . Since S is a domain, we can let Γ be a curve in S thatjoins α and ζ . Without loss of generality (replacing ζ with some other point of Γ ∩ A ifnecessary), we can assume that Γ ⊂ B ( α, r ).Let ζ (cid:48) ∈ V be a preimage of ζ , and let Γ (cid:48) be the component of f − (Γ) that contains ζ (cid:48) . Then Γ (cid:48) ⊂ T joins a point in V to a point in an unbounded component of T \ γ ,which is a contradiction. (cid:3) Results on (cid:98) I ( f ) , (cid:100) BO ( f ) , and (cid:100) BU ( f )Theorem 1.2 is a consequence of the following lemma. Lemma 3.1.
Suppose that f is a transcendental entire function with an unboundedFatou component in BO ( f ) (resp. BU ( f ) ). Suppose that U is a bounded domain thatmeets BO ( f ) (resp. BU ( f ) ). Then ∂U also meets BO ( f ) (resp. BU ( f ) ).Proof. We prove only the case of BO ( f ). The case of BU ( f ) is very similar, and isomitted.Suppose first that U ⊂ F ( f ). If U is not itself a Fatou component, then the resultfollows by normality. Hence we can assume that U is a Fatou component of f .If U is preperiodic, then it is easy to see that ∂U ⊂ BO ( f ). Hence we can assumethat U is wandering. The conclusion of the lemma then follows by Lemma 2.2.We can assume, therefore, that U meets J ( f ). Let V be the unbounded Fatou compo-nent in BO ( f ). It follows by Lemma 2.1 that there exists n ∈ N such that f n ( U ) ∩ V (cid:54) = ∅ .Hence f n ( ∂U ) ∩ V (cid:54) = ∅ , and the result follows. (cid:3) Proof of Theorem 1.2.
As in the case of Lemma 3.1 we prove only the case of BO ( f ).The case of BU ( f ), which is very similar, is omitted.Suppose that, with the hypotheses of the theorem, (cid:100) BO ( f ) was not connected. Thenthere would be disjoint open sets H , H ⊂ (cid:98) C such that (cid:100) BO ( f ) ⊂ H ∪ H and ∂H i ∩ BO ( f ) (cid:54) = ∅ , for i ∈ { , } . Without loss of generality we can assume that H is bounded and meets BO ( f ). Itfollows by Lemma 3.1 that ∂H meets BO ( f ), which is a contradiction, completing theproof. (cid:3) Theorem 1.3 is a consequence of the following, which clearly is analogous to Lemma 3.1.
Lemma 3.2.
Suppose that f is a transcendental entire function, and that f has a finitelogarithmic asymptotic value in J ( f ) . Suppose that U is a bounded domain that meets BU ( f ) . Then ∂U also meets BU ( f ) .Proof. Suppose first that U ⊂ F ( f ). If U is not itself a Fatou component, then theresult follows by normality. Hence we can assume that U is a Fatou component of f . It DAVID J. SIXSMITH is known [OS16, Theorem 1.1] that U must be wandering. The result then follows byLemma 2.2.We can assume, therefore, that U meets J ( f ). Let α ∈ J ( f ) be a finite logarithmicasymptotic value of f . Let r, R >
0, let T be a component of f − ( B ( α, r )), and let V bethe component of T ∩ B (0 , R ), such that the properties stated in Lemma 2.4 all hold.Let ( R n ) n ∈ N be a sequence of real numbers larger than R that tend to infinity. Let W be a bounded open disc containing any exceptional point of f . We can assume that W ⊂ B (0 , R n ), for n ∈ N .We now construct a point in ∂U ∩ BU ( f ). Set U := U . By Lemma 2.1, there exists n ∈ N such that B (0 , R ) \ W ⊂ f n ( U ). Hence there is a continuum E ⊂ ∂U suchthat V lies in a bounded component of T \ f n ( E ).Now consider f ( f n ( E ) ∩ T ). It follows by an application of Lemma 2.4 that thecomplement of f ( f n ( E ) ∩ T ) has a simply connected component containing α , andlying in B ( α, r ). Call this component U . Note that ∂U ⊂ f n +1 ( E ).Since U meets J ( f ) – recall that α ∈ J ( f ) – we can iterate the above construction.We obtain a sequence of integers ( n k ) k ∈ N and a sequence of continua ( E k ) k ∈ N such that(3.1) E k +1 ⊂ f n k +1 ( E k ) , f n k ( E k ) ⊂ C \ B (0 , R k ) , and f n k +1 ( E k ) ⊂ B ( α, r ) , for k ∈ N . It follows by Lemma 2.3 that there is a point ζ ∈ E and a sequence of integers( p k ) k ∈ N such that f p k ( ζ ) ∈ E k +1 , for k ∈ N . It follows from (3.1) that ζ ∈ ∂U ∩ BU ( f ),as required. (cid:3) Finally in this section, we use these results to prove Theorem 1.1.
Proof of Theorem 1.1.
We construct a transcendental entire function f with a finitelogarithmic asymptotic value in J ( f ), and a second finite logarithmic asymptotic valuelying in F ( f ) ∩ BO ( f ). It is easy to deduce from the second fact that f has an unboundedFatou component in BO ( f ). The result then follows by Theorem 1.2 and Theorem 1.3.Consider the family of functions f α,β ( z ) := 2 α √ π (cid:90) z e − w dw + β = α erf( z ) + β, for α, β ∈ C , α (cid:54) = 0 . Here erf( z ) denotes the error function; see [AS72, p.297].Clearly f α,β has no critical values. It can be seen that f α,β has two finite logarithmicasymptotic values, obtained as z tends to infinity along the real axis in the positive andnegative directions. It is a calculation to show that these asymptotic values are equal to ± α + β .We choose values for α and β so that α + β ∈ J ( f α,β ) and − α + β lies in a parabolicbasin of f α,β . First we let c be a complex solution to erf( z ) = 1. In particular, we set c = − . . . . + 1 . . . . i ; see [DLMF, Table 7.13.2]. We then let α = e c √ π β = c − α. It follows that f α,β ( α + β ) = f α,β ( c ) = α + β, and f (cid:48) ( α + β ) = 2 α √ π e − c = 1 . YNAMICAL SETS WHOSE UNION WITH INFINITY IS CONNECTED 7
Figure 1.
Two views of the Julia set (black) of f α,β . The asymptoticvalues ± α + β are denoted by black circles; in particular α + β is the topright circle, and is a parabolic fixed point.Hence α + β is a parabolic fixed point and so lies in the Julia set of f α,β . It can be seenfrom Figure 1 that − α + β lies in the parabolic basin of this point. This is exactly whatwe require. (cid:3) Remarks. (1) It follows by [RRRS11, Theorem 1.2] that every component of I ( f α,β ) is un-bounded and path-connected; in the terminology of [Ben17], f α,β is criniferous . DAVID J. SIXSMITH
It can be seen also that this function is postsingularly bounded . We refer to[Ben17] for a definition, and further information on functions with this property.(2) From the computer pictures, it appears that F ( f α,β ) is connected, although wehave not been able to prove this. If this were indeed the case, then it wouldfollow at once that BO ( f ) and (cid:100) BO ( f ) are connected.(3) Clearly, another approach to the proof of Theorem 1.1 would be to find a tran-scendental entire function f with an unbounded Fatou component in BO ( f ) andan unbounded Fatou component in BU ( f ); the result would follow by Theo-rem 1.2. While this seems possible, it is also likely to be more complicated thanthe example given here, because of the difficulty of constructing transcendentalentire functions with an unbounded Fatou component in BU ( f ).4. Results on spiders’ webs
Recall that if f ∈ B , then I ( f ) ⊂ J ( f ); see [EL92, Theorem 1]. It follows thatTheorem 1.4 is an immediate consequence of the following. Theorem 4.1.
Suppose that f is a transcendental entire function with a finite loga-rithmic asymptotic value α / ∈ F ( f ) ∩ I ( f ) . Then I ( f ) separates no finite point frominfinity.Proof. Suppose first that α ∈ J ( f ). It follows that (cid:100) BU ( f ) is connected, by Theorem 1.3.Hence I ( f ) is not a spider’s web. On the other hand, if α ∈ F ( f ), then either α ∈ BO ( f )or α ∈ BU ( f ). In these cases the result follows by Theorem 1.2. (cid:3) Next we prove Theorem 1.5.
Proof of Theorem 1.5.
One direction is immediate; it is easy to see from the definitionthat a set separates every point of C from infinity if it is a spider’s web.In the other direction, we consider first the case for I ( f ). Suppose that I ( f ) separatesa point of J ( f ) from infinity. In other words, there is a bounded open set U that meets J ( f ) and the boundary of which lies in I ( f ). Let W be a disc containing any exceptionalpoint of f . Suppose that R > W ⊂ B (0 , R ). By Lemma 2.1there exists n = n ( R ) ∈ N such that B (0 , R ) \ W ⊂ f n ( U ). Since I ( f ) is forwardinvariant, it follows that for all sufficiently large R >
0, there is a bounded simplyconnected domain G such that B (0 , R ) ⊂ G and ∂G ⊂ I ( f ).The fact [RS12, Theorem 1.1] that I ( f ) contains an unbounded component impliesthat I ( f ) contains a spider’s web. The remark [RS12, p.807] then implies that I ( f ) is aspider’s web.The case for A ( f ) is almost identical to that of I ( f ), and is omitted. If f has nomultiply connected Fatou components, then it follows from [Kis98, Theorem 1] that allcomponents of J ( f ) are unbounded. The result for J ( f ) then follows in much the sameway as that of I ( f ). (cid:3) The following, which was promised in the introduction, is now quite straightforward.
Corollary 4.2.
Suppose that f is a transcendental entire function. Then I ( f ) is aspider’s web if and only if (cid:100) BO ( f ) ∪ (cid:100) BU ( f ) is disconnected. YNAMICAL SETS WHOSE UNION WITH INFINITY IS CONNECTED 9
Proof.
One direction is immediate, and so we assume that (cid:100) BO ( f ) ∪ (cid:100) BU ( f ) is discon-nected. Then there is a bounded open set U such that U meets BO ( f ) ∪ BU ( f ) and ∂U ⊂ I ( f ). Arguing as in the proof of Lemma 3.1, we can deduce that U ∩ J ( f ) (cid:54) = ∅ .The result now follows by Theorem 1.5. (cid:3) Acknowledgment:
The author is grateful to Lasse Rempe-Gillen, Phil Rippon, GwynethStallard and John Osborne for useful conversations, and also to Ben and Pete Strulo fortheir help with Figure 1.
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Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK,ORCiD: 0000-0002-3543-6969
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