The escaping set of the exponential

THE ESCAPING SET OF THE EXP ONENTIA L LASSE REMPE Abstract. W e show that the set I ( f ) of p oints that con verge to infinity under it- eration of the exp onential ma p f ( z ) = exp( z ) is a connected subset of the co mplex plane. 1. Intr oduction If f : C → C is an en tire transcenden tal function, then its esc aping set I ( f ) is the set of p oints that tend to infinit y under iteration: I ( f ) = { z ∈ C : f n ( z ) → ∞} . F or the dynamically simplest en tir e functions, suc h as expo nen tial maps of the form f ( z ) = exp( z ) + a with a < − 1, the escaping set is the disjoint union of uncountably man y curv es to infinit y , each of whic h is a connected comp onen t of I ( f ). (In particular, I ( f ) is disconnected while I ( f ) ∪ {∞} is connected a nd path-connected.) Eremenk o [E] conjectured t ha t eve ry connected comp onen t of I ( f ) is unbounded for eve r y transcen- den tal en tire function f . Despite recen t progress (compare e.g. [R1, RS1, R 3 S]), this question is still v ery m uc h op en. In view of this, the escaping set is usually view ed v ery m uc h as a set that is lik ely to b e disconnected. Ho we ver, Ripp on and Stallard [RS1] prov ed that the escaping set of an entire function with a m ultiply-connected w andering domain is in fact connected. They hav e since exten ded this result to muc h la rger classes of en tire functions [RS2]. These examples are quite d ifferen t from the exp onen tia l maps men tioned ab o ve in that they do not b elong to the Er emenko-Lyubich class B := { f : C → C transcenden tal, en tire : sing ( f − 1 ) is bo unded } (where sing ( f − 1 ) denotes the set o f critical and asymptotic v alues of f ). W e note that, if f ∈ B , then I ( f ) is a subset of the Julia set J ( f ) [EL, Theorem 1]. Bergw eiler (p ersonal comm unication) a sk ed whether the escaping set of a function in B can b e connected, and more precisely whether this migh t b e t he case for the function f ( z ) = π sin( z ). While Mihaljevic-Brandt [M-B] has given a negat ive answ er to the latter, Ripp on and Stallard observ ed that, fo r the function f ( z ) = (cosh( z )) 2 , the escaping set is connected. Indeed, the union of the real axis with all its iterated preimages is pa th-connected (and clearly dense in I ( f )). In con tra st to this example, for t he exp onen t ial map f ( z ) = exp( z ) ev ery path- connected comp onen t of the escaping set is kno wn to b e a single curv e to ∞ tha t is relativ ely closed and no where dense in I ( f ) (see Prop osition 3.2). It may seem plausible 2000 Mathematics Subje ct Classi fi c ation. Primary 37F10; Seco ndary 30D0 5,37F10,5 4F15. Suppo rted b y E PSRC F ello wship EP/E 0528 51/1. 1 2 LASSE REMPE that these path- connected comp onen ts are also the connecte d comp o nen ts of I ( f ), but w e show that the situation is rather differen t. 1.1. Th eorem (Esc a ping set of the exp onen t ia l) . L e t a ∈ ( − 1 , ∞ ) and c onsider the function f ( z ) = exp( z ) + a . Then I ( f ) is a c onne cte d subset of the plane. The pro of is elemen tary; the main idea is to consider a coun table sequence of preimage comp onen ts o f the negativ e real axis t ha t w as studied by Dev aney [D] in his cons truction of an indecomp osable con t inuum. (See Fig ure 1.) Eac h of these comp onen ts is an arc tending to infinit y in b oth directions, but w e shall show that their union is connected. Theorem 1.1 then follo ws relativ ely easily . Basic notation. As usual, w e denote t he complex plane by C , and the Riemann sphere b y ˆ C = C ∪ {∞} . Closures and b oundaries will b e unders to o d to be tak en in C , unless explicitly stated otherwise. Ac knowledgm ents. I w ould lik e to thank W alter Bergw eiler, Mary Rees, Gwyneth Stallard and Phil Ripp on for in teresting discus sions. 2. The D ev aney continuum F or the res t of the article, fix a ∈ ( − 1 , ∞ ) and set f ( z ) = exp( z ) + a . Then f n ( x ) → ∞ for all x ∈ R . Let H + and H − denote the upp er and lo wer half planes, resp ectiv ely . Let S + denote the strip at imag ina ry part s b etw een 0 a nd π ; similarly S − is the strip at imaginary parts b etw een 0 and − π . F or σ ∈ { + , −} , let L σ : H σ → S σ b e the branc h of f − 1 taking v alues in S σ . L σ is a confomal isomorphism that exte nds to a homeomorphism betw een H σ \ { a } and S σ ; we denote this ex t ension also b y L σ . Define γ σ 0 := ( −∞ , a ), and inductiv ely γ σ k +1 := L σ ( γ σ k ) . Then each γ σ k , k ≥ 1, is an injectiv e curv e tending to infinit y in b oth directions. (Also, γ − k is the reflec tio n of γ + k in the real axis for all k .) W e define sets Γ σ and X σ b y Γ σ := [ k ≥ 0 γ σ k , X σ := Γ σ . See Figure 1 for a picture of the set X + . W e require the follo wing k ey fact [D, p. 631] 2.1. P rop osition (Hausdorff limit of γ ± k ) . L et σ ∈ { + , −} . The set X σ ∪ {∞} is the Hausdorff limit (on the Riemann spher e ˆ C ) of the se quenc e ( γ σ k ∪ {∞} ) . (In p articular, S k ≥ k 0 γ σ k is dense in X σ for al l k 0 .) Pr o of. Let z 0 ∈ X σ ∪ ∞ , and let U b e a neigh b orho o d of z 0 in ˆ C . W e need to sho w that γ k ∪ U 6 = ∅ f o r all suffi ciently larg e k . By definition of X σ , the set U con tains some z 1 ∈ Γ σ . Let D n denote the (Euclidean) disk o f radius 2 π a round f n ( z 1 ). It is elemen tary t o see — using the f a ct that f is THE ESCAPING SET OF THE EXPONENTIAL 3 Figure 1. The set X + expanding in a suitable right half plane — that there is n 0 with f j ( a ) / ∈ D n for all n ≥ n 0 and j < n . W e ma y assume that n 0 is c hosen sufficien tly larg e to ensure that also f n ( z 0 ) ∈ R f or n ≥ n 0 . Hence for n ≥ n 0 , t here is a branc h ϕ n : D n → C of f − n with ϕ n ( f n ( z 0 )) = z 0 . Clearly max z ∈ D n | ϕ ′ n ( z ) | → 0 as n → ∞ (again due to the expansion of f in a r igh t half plane). In particular, there is k 0 ≥ n 0 + 1 suc h that for k ≥ k 0 , the ima g e of ϕ k − 1 is con tained in U . W e then hav e ϕ k − 1 ( f k − 1 ( z 0 ) + π i ) ∈ γ k ∩ U , and the claim f o llo ws.  2.2. P rop osition (Γ ± connected) . The sets Γ + and Γ − ar e c o nne cte d. Pr o of. Let σ ∈ { + , −} . Supp ose tha t U ⊂ C is an o p en set with Γ σ ∩ U 6 = ∅ and Γ σ ∩ ∂ U = ∅ . W e need to sho w that Γ σ ⊂ U . Let z 0 ∈ U ∩ Γ σ . Then b y Prop osition 2.1, there is k 0 suc h tha t γ k ∩ U 6 = ∅ for all k ≥ k 0 . Since γ k is connected, in fact γ k ⊂ U . Th us Γ σ ⊂ X σ = [ k ≥ k 0 γ σ k ⊂ U . By c hoice of U , w e hence ha ve Γ σ ⊂ U , as desire d.  3. Proo f of t he Theorem By Prop osition 2.2, the union of Γ + and Γ − connects the horizontal line γ − 1 at imagi- nary part − π with γ + 1 at imaginary part π . Since the set I ( f ) is 2 π i -p erio dic, it follows that the set Y := [ σ ∈{ + , −} , k ∈ Z  Γ σ + 2 π ik  is a connected subset of I ( f ). Y con tains all p oin ts whose imagina r y parts are o dd m ultiples of π ; i.e. f − 1  ( −∞ , a )  . 3.1. Prop osition (Preimages of Y ) . Set Y 0 := Y and inductively Y j + 1 := Y j ∪ f − 1 ( Y j ) . Then Y j is c onne cte d for al l j . Pr o of. The pro of is b y induction on j . Note tha t Y j con tains Y for all j . Let k ∈ Z , and let L k : C \ [ a, ∞ ) → C b e the branc h of f − 1 that tak es v alues with imaginary parts b et w een 2 π k and 2 π ( k + 1). Set z k := (2 k + 1) π i . Then z k , f ( z k ) ∈ Y ⊂ Y j , and hence z k ∈ Y j ∩ L k ( Y j ). As L k is a con tinuous function (and Y j is con tained 4 LASSE REMPE in its domain of definition), it follows from the induction hypothesis that Y j ∪ L k ( Y j ) is connected. Hence Y j + 1 = [ k ∈ Z L k ( Y j ) ∪ Y j is connected, as claimed.  Pr o of of The o r em 1.1. The set S j ≥ 0 f − n ( − 1) ⊂ S j ≥ 0 Y j is dense in t he Julia set, and hence in the escaping set. Since S j ≥ 0 Y j is a connected subset of I ( f ), the claim follows.  W e contrast our res ult with the following fact, men tioned in the in tro duction. 3.2. Prop osit ion (P ath- connected comp o nen ts of I ( f )) . L et P b e a p ath-c onne cte d c omp onen t of I ( f ) , wher e again f ( z ) = exp( z ) + a , a ∈ ( − 1 , ∞ ) . Then P is r elatively close d and n owher e dense in I ( f ) . Pr o of. The pa t h- connected comp o nen ts of I ( f ) (in f act, of the escaping set of a n y ex- p onen tial map) are completely describ ed in [FRS, Corolla ry 4.3]. First of all, for a ny n ≥ 0, each connected componen t of f − n ( R ) is a path-connected comp onen t of I ( f ). Ev ery o ne of these is no where dense and closed in C , and in particular relativ ely closed in I ( f ). No w suppose t ha t z 0 ∈ I ( f ) nev er maps to the p ositiv e real axis under iterat ion. Let s = s 0 s 1 s 2 . . . b e the sequence of in tegers suc h that Im f n ( z 0 ) ∈  (2 s n − 1) π , (2 s n + 1) π  for all n . It fo llo ws from the assumption o n z 0 that s m ust contain infinitely many nonzero en tries (see [D, Theorem on p. 632]). Let K = K s b e the set of all points z ∈ J ( f ) with Im f n ( z ) ∈  (2 s n − 1) π , (2 s n + 1) π  for all n ≥ 0. Clearly K is closed and now here dense. It is know n [D K, SZ] that K ∩ I ( f ) is path-connected; in fact, K ∩ I ( f ) is the trace of an injectiv e curv e g s : [0 , ∞ ) → C or g s : (0 , ∞ ) → C with g s ( t ) → ∞ as t → ∞ . (This curv e is called a Devan ey hair or a dynamic r ay .) W e remark that , for certain addresses s , the limit set of g s ( t ) a s t → 0 will not consist of a single p oint [D J] (compare also [R2]) . Any es caping p oints in this limit set m ust necessarily lie on g s themselv es. By [FRS, Coro llary 4.3], the curv e P := K ∩ I ( f ) = g s is the path-connected comp o- nen t of I ( f ) con t aining z 0 . Since K is closed and no where dense , the claim follo ws.  Reference s [D] Rob ert L. Dev aney , Knaster-like c ontinua and c omplex dynamics , Er go dic Theory Dynam. Sys- tems 1 3 (199 3), no. 4, 6 27–63 4. [DJ] Rober t L. Dev aney and Xa vier Jar que, Inde c omp osable c ontinua in ex p onent ial dynamics , Con- form. Geo m. Dyn. 6 (2002), 1–12 . [DK] Robert L. Dev aney and Micha l Krych, Dynamics of exp( z ), Ergo dic Theo ry Dynam. Sys tems 4 (1984), no. 1 , 35– 52. [E] Alexandre ` E. 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Ma th. 32 (2007), 35 3–369 , a rXiv:math.DS/051 1588 . [RS1] Philip J. Rippo n and Gwyneth M. Stallard, On quest ions of Fatou and Er emenko , Pro c. Amer. Math. So c. 133 (2005 ), no. 4 , 11 19–11 26. [RS2] , Esc aping p oints of entir e functions of smal l gr owth , Math. Z. (to appea r), arXiv:080 1.3605 . [R 3 S] G ¨ un ter Rottenfußer, Jo hannes R ¨ uc kert, Las se Remp e, and Dierk Sc hleicher, Dynamic r ays of entir e functions , Prepr in t #2007/05 , Ins titute for Mathematical Sciences, SUNY Stony Bro ok, 2007, a rXiv:0704 .3213 , submitted for publica tion. [SZ] Dierk Schleic her and Joha nnes Zimmer, Esc aping p oints of exp onential maps , J. Lo ndon Math. So c. (2) 67 (2003), no. 2 , 380 –400. Dept. of Ma thema tical Sciences, U niversity of Liverpool, Liverpool L69 7ZL, UK E-mail addr ess : l .rempe @liver pool.ac.uk