Characterizing indecomposable plane continua from their complements

CHARA CTERIZING INDECOMPOSABLE PLANE C O NTINUA FR OM THEIR COMPLEM E NTS CLINTON P . CURR Y, JOHN C. MA YER, AND E. D. TYMCHA TYN Abstract. W e show that a plane con tin uum X is indecomposable iff X h as a sequence ( U n ) ∞ n =1 of not necessarily distinct complemen tary domains satis- fying the double-p ass co ndition : for an y sequence ( A n ) ∞ n =1 of op en arcs, with A n ⊂ U n and A n \ A n ⊂ ∂ U n , there is a sequence of shadows ( S n ) ∞ n =1 , where eac h S n is a shado w of A n , suc h that lim S n = X . Suc h an open arc divides U n int o disjoint subdomains V n, 1 and V n, 2 , and a shadow (of A n ) i s one of the sets ∂ V n,i ∩ ∂ U . 1. Introduction In this pap er, a c ontinuum is a co mpact, co nnec ted, no nempt y metr ic space . A contin uum is de c omp osable if it can b e written as the union of tw o of its pr op er sub c ontin ua ; other wise, it is inde c omp osable . Let C denote the complex plane and let C ∞ denote the Riemann spher e C ∪ {∞} . A plane domain is a subset of C ∞ which is confor mally isomo rphic to the op en unit dis k D ⊂ C ∞ (whic h is to say that it is ope n, co nnected, simply connected and its bounda r y is a nondegenerate sub c ontin uum o f C ∞ ). If X is a co nt inuum in C ∞ , the comp onents o f C ∞ \ X are called c omplementary domains and ar e pla ne domains. If W ⊂ C ∞ , we denote the bo undary of W b y ∂ W . W e say that a p oint x of a co nt inuum is bu rie d if it do es not lie on the b oundary of a n y complementary doma in. The spherica l metric on C ∞ is deno ted b y d , and H d denotes the Hausdor ff metric on the h yp erspace of sub c ontin ua of C ∞ [18, Section 4.1]. There a re several ways of recognizing intrinsically tha t a c ontin uum X is inde- comp osable. F or instance, X is indecomp osa ble if and only if every prop er sub con- tin uum of X is nowhere dense in X [10]. Also, a c o nt inuum X is indecomp osable if there ar e p oints a, b, c ∈ X such that no prop er sub contin uum of X contains any t wo of these p oints. In this pap er, we a re interested in re cognizing indecomp osa ble planar contin ua not by intrinsic prop erties, but by the re la tionship b etw ee n the contin uum and its ambien t space. Indecomp osable contin ua arise natura lly in dynamical systems [8, 7]. How ev er, in sp ecific dynamical systems , it is often difficult to reco gnize them. In complex analytic dyna mics , the Julia set of a rationa l ma p f : C ∞ → C ∞ is the set o f unsta- ble p oints under itera tion of f (see [15] for definitions). A long-standing question Date : Nov em ber 2, 2018. 2000 Mathematics Subje ct Classific ation. Prim ary: 54F15; Secondary: 37F20 . Key wor ds and phr ases. indecomposable contin uum, complemen tary domain, Julia set, com- plex dynamics, buried p oint . The third named author was supported in part by NSERC 0GP005616. W e thank the D epart- men t of Mathematics and Computer Science at N i pissing Universit y , Nor th Bay , Ontario, for the opportunity to work on this paper in pleasant surroundings at their annual top ology workshop. 1 2 C. P . CURR Y, J. C. MA YER, AND E. D. TYMCHA TYN in complex analytic dyna mics asks: Can the Julia set of a r ational function b e an inde c omp osable c ontinuum? Several author s hav e a ttack ed this question, amo ng them [1 4, 4] for poly nomials, [22] for bicritic al rational maps (rationa l maps with exactly t wo critical p oints), a nd [5] for Julia sets o f a cla ss o f rational functions with no bur ie d points. In this situation, it is muc h easier to ana lyze the co mple- men t of the Julia set, called the F atou set ; this motiv ates o ur interest in studying indecomp osability from the p oint of view of a contin uum’s complement. The se c ond na med author, with v a rious co-a uthors [14, 4 , 5 ], investigated the recognition of indecomp osa ble contin ua from their complemen t in the cas e tha t ∂ U = X fo r some complementary domain U of X . The tool used was prime end theory , and a characterizatio n was obtained in the context of X b eing a J ulia set, making use o f the dynamics . The characterizatio n of indeco mpo s able contin ua from their complements in the current pap er primarily addre s ses the cas e that X is not the b o undary of any of its complementary doma ins (which would imply that ther e are infinitely many co mplemen tary domains, each having b oundaries nowhere dense in X ). Even b etter, this characteriza tion also subsumes the first case and is en tirely top ological. T o state o ur characterization theorem, we need some definitions. These concepts are related to those which aro s e o riginally in prime end theory . Definition 1.1. Let U b e a pla ne domain. A cr osscut o f U is an op en arc A = ( a, b ) ⊂ U such that A = [ a, b ] is a clo sed arc which meets ∂ U exactly in the set { a, b } . A gener ali ze d cr osscu t of U is an op en arc A ⊂ U such that A \ A ⊂ ∂ U . Notice that the no tio n of a gener alized crosscut is strictly br oader than the no tion of a crosscut. It is easy to see that a generalized cross cut o f a domain U cuts U int o tw o nonempty disjoint sub domains V 1 and V 2 such that U = V 1 ∪ A ∪ V 2 . Definition 1. 2. Let U be a plane do ma in and A a generalized cro sscut of U . W e call each comp onent of U \ A a cr osscut neighb orho o d . If V is a cros scut neighborho o d deter mined b y a generalized cros scut A , we ca ll the contin uum S = ∂ V ∩ ∂ U a shadow of A . Thu s, a generalized crosscut A of a domain U has exa ctly t wo cr osscut neigh- bo rho o ds, a nd co ns equently t wo shadows whose union is ∂ U . Examples below show that one or b o th of these shadows can b e prop er sub contin ua of ∂ U o r, more surprisingly , all of ∂ U . Limits b elow are interpreted in the metric H d . Definition 1 .3. A sequence ( U n ) ∞ n =1 of (not necessarily distinct) complemen tary domains of a contin uum X sa tisfies the double-p ass c ondition if, for any sequence of generalized cr osscuts A n of U n , ther e is a sequence of shadows ( S n ) ∞ n =1 of ( A n ) ∞ n =1 such that lim n →∞ S n = X . In Section 3, we prov e the following theorem, whic h is the main theorem of this pap er. Theorem 1.4 (Characterization Theo rem) . A planar c ontinuum X is inde c om- p osable if and only if it has a se quenc e ( U n ) ∞ n =1 of c omplementary domains which satisfies the double-p ass c ondi tion. INDECOMPOSABILITY FROM THE COMPLEMENT 3 2. P ar tial and Prio r Resul ts 2.1. Brief Histo ry . The firs t partial reco gnition theore m for indecomp osable con- tin ua from the complement is that of Kur atowski [13]. Theorem 2.1 (Kuratowski) . If a plane c ontinuum X is the c ommon b oundary of thr e e of its c omplementary domains, then X is either inde c omp osable or the union of two pr op er inde c omp osable sub c ont inua. The following theorem of Rutt seems quite different, and is also only a pplicable if X is the bo unda ry of some complementary domain. Theorem 2.2 (Rutt, [19]) . If a nonde gener ate plane c ontinuum X is the b oundary of a c omplementary domain U , and if ther e is a prime end of U whose impr ession is ∂ U = X , t hen X is either inde c omp osable or the union of two pr op er inde c omp osable sub c ontinua. Without go ing in to detail (but see [4]), the impr ession of a prime end of U is the intersection of the s hadows of a sequence ( A n ) ∞ n =1 of cros scuts of U having the pro per t y that for each n , ( A m ) m>n is a pair wis e closure disjoint null sequence contained in o ne of the cros scut neighborho o ds of A n . The connectio n among the theor ems ab ov e is made explicit by a tec hnical the- orem of B ur gess. While the or iginal result is stated in terms of what Burg ess calls simple disks, the theorem can be equiv alently stated in terms of closed ba lls. F or a ∈ C ∞ and r > 0, define the b al l of ra dius r ab out a by B r ( a ) = { z ∈ C ∞ | d ( a, z ) < r } . Theorem 2. 3 (Bur gess, [2 , Theor em 9 ]) . L et H b e a close d set and X a c ontinuum in the plane. Su pp ose X 1 , X 2 , and X 3 ar e sub c ontinua of X and D 1 , D 2 , and D 3 ar e p airwi se disjoint close d b al ls with D i ∩ H = ∅ and ∅ 6 = D i ∩ X i = D i ∩ X for e ach i ∈ { 1 , 2 , 3 } . Then t her e do not exist thr e e distinct c omplementary domains of X ∪ H su ch that e ach of t hem interse cts e ach of the b al ls D i . Using this theor em, Burges s proves the following reco gnition theor em, which also applies when the contin uum is not the unio n of the b oundaries of its co mplemen tary domains. Corollary 2 .4 (Burgess, [2, Cor ollary to Theo r em 9]) . If t he plane c ont inuum X is the limit of a se quenc e of distinct c omplementary domains of X , then either X is inde c omp osable, or ther e is only one p air of inde c omp osable c ontinua whose union is X . As recog nition theorems, the ab ov e suffer from the weakness of their conclus io n. In [4, 5] dynamical considerations rule o ut that the Julia set o f a p olynomial can b e the unio n of tw o proper indecomp osable subc ontin ua . How ever, this is under the hypothesis that the Julia set is the b ounda ry of one of its complementary domains. The fo llowing definition and r ecognition theorem app ear in [5]. Since it represents a simplification of the pr o of in [5], we prove Theor em 2.6 making use of Theo rem 2 .3. Definition 2.5. An anticha in of c rosscuts of a plane domain U is a sequence ( H n ) ∞ n =1 of distinct pairwis e clos ure dis jo in t crosscuts of U suc h that, for each m , one cro s scut neig h b orho o d of H m contains a ll the cr osscuts ( H n ) n 6 = m . 4 C. P . CURR Y, J. C. MA YER, AND E. D. TYMCHA TYN Theorem 2.6. L et U ⊂ C ∞ b e a plane domain. L et z ∈ U . Supp ose ther e exists an ant ichain ( H n ) ∞ n =1 of cr osscut s of U such that lim n →∞ Sh( H n ) = ∂ U , whe r e Sh( H n ) is the shadow of the cr osscut neighb orho o d W n of U \ H n which misses z . Then ∂ U is inde c omp osable. Pr o of. F or a co nt radiction, s uppo s e ∂ U satisfies the hypo theses of the theor em, but may b e written as the union of prop er sub contin ua X 1 and X 2 . By pas sing to a subsequence, we may a ssume that ( H n ) ∞ n =1 conv erges to a p oint of ∂ U . Choo se disjoint closed balls D 1 and D 2 such that (1) z / ∈ D i , (2) D i ∩ H n = ∅ for all i ∈ { 1 , 2 } , n ∈ N , (3) X i int ersects the interior of D i , and (4) D i ∩ X j = ∅ if i 6 = j . Cho ose thr e e cr osscuts H 1 , H 2 , and H 3 so that the comp onent W i of C ∞ \ ( ∂ U ∪ H i ) missing z hits b oth D 1 and D 2 . Notice tha t, since the crossc uts are members o f an antic hain, W i ∩ W j = ∅ for distinct i and j in { 1 , 2 , 3 } . Let R 1 , R 2 , a nd R 3 be ar cs from z to ∂ U disjoint (except for z ) from each other and from D 1 ∪ D 2 , and such that each H i lies in a different complementary comp onent U i in U of X = ∂ U ∪ R 1 ∪ R 2 ∪ R 3 . Notice that W i ⊂ U i . Define X 3 = R 1 ∪ R 2 ∪ R 3 , and let D 3 ⊂ U b e a closed ball ab out z which is disjoint from D 1 and D 2 . Thus, each U i int ersects each of D 1 , D 2 , and D 3 ; this co nt radicts Theo rem 2.3, with H = ∅ in the statement.  2.2. Necessary Conditi on. In this sectio n we s how that for a plane con tin uum X to b e indecomp osa ble, it is necessar y that X have a sequence of complementary domains whose b oundaries conv erge to X . The pr o of requires a few additional facts and definitions. Definition 2 .7. The c omp osant , deno ted C ( p ), of a p oint p in a contin uum X is the union o f all the prop er subc ontin ua of X that contain p . Theorem 2.8 ([10]) . L et X b e a nonde gener ate inde c omp osable c ont inu um. Then the fol lowing hold: (1) X has c p airwise disjoint c omp osants. (2) Each c omp osant is dense in X . (3) Each c omp osant c an b e writt en as a c ountable incr e asing union of nowher e dense pr op er su b c ontinua of X , c onver ging to X in the H ausdorff metric. Definition 2.9. A connected topo logical space X is said to be u nic oher ent if, for any pair A and B of clo sed, connected subsets such that A ∪ B = X , the intersection A ∩ B is co nnected. Note that the plane itself and an op en or closed ball in the plane is unico herent [20]. Reca ll that B r ( a ) denotes the op en ball of radius r > 0 a bo ut center a . Theorem 2.1 0. L et X b e an inde c omp osable plane c ontinuum. Then ther e is a se qu enc e ( U n ) ∞ n =1 of (n ot ne c essarily distinct) c omplementary domains of X such that lim ∂ U n = X . Pr o of. This is clear if X is a po in t, so assume X is a nondeg enerate indeco mpo sable contin uum. T ake p, q , r ∈ X , each in a different comp osant of X . F or each n ∈ N , INDECOMPOSABILITY FROM THE COMPLEMENT 5 Figure 1. The Knaster buck ethandle c ontin uum (left); a union of tw o Knaster contin ua (right) meeting at a sequence o f p oints conv erging to their common endp oint. define Q n = the comp onent o f X \ B 1 /n ( p ) containing q R n = the comp onent o f X \ B 1 /n ( p ) containing r Notice that lim n →∞ Q n = lim n →∞ R n = X , by Theor em 2.8. Since Q n and R n are differe nt comp onents of X \ B 1 /n ( p ), they are sepa rated in C ∞ \ B 1 /n ( p ) b y C ∞ \ ( B 1 /n ( p ) ∪ X ). Also, Q n and R n are closed in the norma l s pa ce C ∞ \ B 1 /n ( p ), so there is a subset K n , closed in C ∞ \ B 1 /n ( p ), of C ∞ \ ( B 1 /n ( p ) ∪ X ) which s e pa rates Q n and R n . Since C ∞ \ B 1 /n ( p ) is homeomorphic to the closed unit disk in the plane, it is unicoherent, so a comp onent L n of K n is a c lo sed (in C ∞ \ B 1 /n ( p )) and connected subset o f C ∞ \ ( B 1 /n ( p ) ∪ X ) which s eparates Q n and R n in C ∞ \ B 1 /n ( p ) [20]. Moreov er, since L n ⊂ C ∞ \ X , it lies in a single complementary domain U n of X . The sequence ( U n ) ∞ n =1 formed in this wa y is the required sequence o f complementary domains . It is evident that lim n →∞ ∂ U n ⊂ X ; we aim to show that X ⊂ lim n →∞ ∂ U n . Cho ose ǫ > 0, and x ∈ X . Let N ∈ N such that, for all n ≥ N (1) Q n ∩ B ǫ ( x ) 6 = ∅ , (2) R n ∩ B ǫ ( x ) 6 = ∅ , a nd (3) B 1 /n ( p ) ∩ B ǫ ( x ) = ∅ . F or n ≥ N , choose q n ∈ Q n ∩ B ǫ ( x ) and r n ∈ R n ∩ B ǫ ( x ) for n ≥ N . The straig ht line s e g ment A n from q n to r n is a s ubset o f B ǫ ( x ) a nd, hence, of C ∞ \ B 1 /n ( p ), so A n m ust mee t L n , sinc e L n separates q n from r n in C ∞ \ B 1 /n ( p ). Since L n ⊂ U n , A n int ersects U n ∩ B ǫ ( x ). Since q n , r n are not in U n (they lie in X ), A n int ersects ∂ U n , and ∂ U n ∩ B ǫ ( x ) 6 = ∅ . This is true fo r all n ≥ N , so x ∈ lim inf n →∞ ∂ U n ⊂ lim n →∞ ∂ U n . This completes the pro o f.  3. The Chara cteriza tion Theorem W e saw in Subsection 2.2 that having a sequence of c o mplement ary domains whose bo undaries conv erge to X is a necessary condition for the plane co nt inuum X to b e indecomp osable. Exa mple 3.2 b elow sho ws that this co ndition is no t sufficient, even if the doma ins ar e distinct, and sugg ests that we m ust find a way to rule out that the sequence of complement ary domains “splits” into “halves” each of which conv erge to pr op er indecomp osa ble sub contin ua. 6 C. P . CURR Y, J. C. MA YER, AND E. D. TYMCHA TYN Figure 2. Tw o Knaster co nt inua meeting on an end interv al o f each. 3.1. Examples. Example 3.1. The Kna ster buck ethandle contin uum, depicted o n the left in Fig- ure 1, is a standard exa mple of an indecomp osable contin uum. It can b e viewed as a disk from which successively deep er and denser fjords are dug. Notice that, for an y gene r alized crosscut dr awn in its single co mplemen tary domain, infinitely many fjords lie in one cross cut neig hbo rho o d or the o ther , so one sha dow is dense. Notice that o ne comp osant of X is the o ne-to-one contin uous image of the half line [0 , ∞ ). The p oint of X corr esp onding to the p oint 0 of [0 , ∞ ) is called the endp oint of X . F or a mo re precise cons truction, see [1 3, V ol. I I, p. 205]. Example 3.2. The co ntin uum X depicted on the right in Figure 1 is an example of a contin uum which sa tisfies the hypotheses of Theorem 2.4 without b eing in- decomp osable. It is a symmetric union of tw o Knas ter buc k ethandle co n tinu a X 1 and X 2 int ersecting in a co unt able set which lies on a vertical line. The contin uum has infinitely many co mplemen tary domains ( U i ) ∞ i =1 . The decomp osability o f X can be detected from the co mplemen tary do mains a s follows: t o its complemen- tary domains ( U i ) ∞ i =1 , as so ciate the collection of cross c uts ( K i ) ∞ i =1 , wher e K i lies in U i on the v ertical axis o f sy mmetry . Each cr osscut ha s one shadow which is a sub c ontin uum of X 1 , and another which is a subco nt inuum of X 2 . Therefore, a ny conv ergent se quence of shadows must limit to a prop er s ubco n tinuum of X . Example 3. 3 . The c o nt inuum in Figur e 2 is the union of a pair of Knas ter contin ua X 1 and X 2 with distinct endpo in ts such that X 1 ∩ X 2 is the horizo nt al arc A b etw een the endp oints of X 1 and X 2 . This contin uum, like Example 3.1, has the pro per ty that every cros s cut has a dense shadow, despite the co n tinu um’s decomp osability . Let K b e a genera liz e d cr osscut such that one end lands on a p oint of X \ A a nd the other end w ig gles b etw een the Knaster cont inua a nd compactifies on A . Neither shadow of this gener alized cr osscut is dense, so the constant sequence consisting of this cross c ut fails the do uble pass condition. 3.2. Pro of of Characterization Theorem. In Definition 1.3, we defined the double-pass condition on a s e quence o f generaliz e d cro sscuts in a sequence of com- plement ary doma ins which is motiv ated b y Exa mple 3.2 and by the similarly- functioning co ndition of Co o k and Ingra m [6] intro duced for reco gnizing indeco m- po sable (c hainable) con tin ua in terms of refining op en cov ers. Her e we prove o ur INDECOMPOSABILITY FROM THE COMPLEMENT 7 main theo rem: the existence of a sequence of c omplement ary domains satisfying our double-pass condition is equiv a lent to indecomp osa bilit y . The following Lemma follows fro m [23, (A1.4 )], but we include a self-contained pro of here for conv enience. Lemma 3 .4. Supp ose that φ : U → D is a c onformal isomorphi sm, wher e U is a plane domain. Then the image of a nul l se quenc e ( K n ) ∞ n =1 of cr osscut s of U is a nul l se qu enc e of cr osscuts of D . Pr o of. By w ay of contradiction, let ( K n ) ∞ n =1 be a null seq uence of cr osscuts of U such that the image sequence ( A n ) ∞ n =1 = ( φ ( K n )) ∞ n =1 consists of cross cuts w ho se diameters ar e b ounded awa y from zer o. Then, by pas sing to a subsequence, we may a ssume that the image sequence acc um ulates o n a nondegenerate co nt inuum L ⊂ D . Since ( K n ) ∞ n =1 do es not acc um ulate on a subset of U , we see that L ⊂ ∂ D . Also, we may assume without loss of gener ality that ( K n ) ∞ n =1 conv erges to a po in t of x ∈ ∂ U . Let t ∈ L . There exists a c hain of cr osscuts ( A ′ n ) ∞ n =1 of D con verging to t which ma ps to a nu ll se q uence ( K ′ n ) ∞ n =1 of cr osscuts of U by φ − 1 (see [1 5, Lemma 17.9]). W e may assume that ( K ′ n ) ∞ n =1 conv erges to a p o int of ∂ U b y pas sing to a subsequence. Since ( A n ) ∞ n =1 accumulate o n t , all but finitely man y A n int ersect the crosscut neighborho o d of A ′ m corres p onding to t . Also, since ( K n ) ∞ n =1 forms a null sequence in U , we see that all but finitely many K n (th us A n ) lie entirely within the crosscut neighborho o d of K ′ m (th us A ′ m ) corr esp onding to t . How ev er, the cro s scut neighborho o ds of A ′ m form a null s equence, so ( A n ) ∞ n =1 form a null sequence.  W e say that a pair of subse ts E 1 and E 2 of ∂ D are un linke d if there exist interv als I 1 ⊃ E 1 and I 2 ⊃ E 2 such tha t | I 1 ∩ I 2 | ≤ 2. Lemma 3.5. Supp ose U ⊂ C ∞ is a plane domain, and let φ : U → D b e a c onformal isomorphism. L et B 1 and B 2 b e disjoint close d b al ls me eting ∂ U in their interiors, and let E i ⊂ ∂ D denote the en dp oints of al l of the cr osscut s of D which c onst itute φ (( ∂ B i ) ∩ U ) . If E 1 and E 2 ar e u nlinke d, then t her e is a gener alize d cr osscut K of U which sep ar ates B 1 ∩ U fr om B 2 ∩ U in U . Mor e over, if ∂ U is lo c al ly c onne cte d, K is a cr osscut of U . Pr o of. Let I 1 and I 2 be minimal closed in terv als of ∂ D such that I i ⊃ E i for i ∈ { 1 , 2 } a nd | I 1 ∩ I 2 | ≤ 2 . Note that ( ∂ B i ) ∩ U is the unio n of a null se quence o f crosscuts o f U , s o φ (( ∂ B i ) ∩ U ) is the union of a null sequenc e of cro sscuts of D by Lemma 3 .4. I 1 and I 2 are unlinked implies that neither φ ( B 1 ∩ U ) nor φ ( B 2 ∩ U ) separates the other in D . By connectedness o f D , there is then a uniq ue comp onent D of D \ φ (( B 1 ∪ B 2 ) ∩ U ) whic h meets b oth φ ( ∂ B 1 ∩ U ) and φ ( ∂ B 2 ∩ U ). The crosscuts on ∂ D form a n ull sequence, so it is not difficult to sho w that D ⊂ D is lo cally connec ted. The endp oints of I 1 and I 2 are each on ∂ D . Let K b e a cro sscut joining an endpo int of I 1 to an endp oint o f I 2 such that K sepa r ates the interiors of I 1 and I 2 , in ∂ D , and thus φ ( B 1 ) and φ ( B 2 ) in D . Then φ − 1 ( K ) is a genera lized cros s cut of U which separ ates B 1 ∩ U from B 2 ∩ U . F urther, if ∂ U is lo ca lly connected, φ − 1 extends to a cont inuous function φ − 1 : D → U [15, Theorem 17.14]. In this ca se, φ − 1 ( K ) is a true cr osscut of U , as φ − 1 ( K ) = φ − 1 ( K ) is an arc.  Now we hav e the to ols to prove o ur Charac ter ization Theorem 1.4. 8 C. P . CURR Y, J. C. MA YER, AND E. D. TYMCHA TYN Pr o of of The or em 1.4. First, supp ose that X is indecomposa ble. W e show X sat- isfies the double-pa s s condition. By Theo rem 2.10, there exists a sequence ( U n ) ∞ n =1 of complementary domains o f X such that lim n →∞ ∂ U n = X . Let ( K n ) ∞ n =1 be a sequence of generaliz e d crossc uts, with K n in U n for each n ∈ N . Let A n and B n be the shadows of K n , with H d ( A n , X ) ≤ H d ( B n , X ), where H d denotes the Hausdorff metric. W e claim that lim n →∞ A n = X . Since the hypers pa ce of subco nt inua of X is a compact metric space, it is sufficient to show that every co nv ergent subsequence of ( A n ) ∞ n =1 conv erges to X . Let ( n i ) ∞ i =1 be suc h that ( A n i ) ∞ i =1 conv erges to a contin uum A ⊂ X . By pas s ing to a subsequence, we may ass ume ( B n i ) ∞ i =1 also conv erges to a contin uum B ⊂ X . Since lim i →∞ ∂ U n i = X and A n i ∪ B n i = ∂ U n i , we hav e that A ∪ B = X . Since X is indecomp osa ble, not b oth A and B may b e prop er sub contin ua of X , so A = X or B = X . Since, H d ( A n i , X ) ≤ H d ( B n i , X ) for all i , we hav e A = X . This concludes the pro of of this implication. Now we prove the conv erse. Let X be a contin uum with a sequence ( U n ) ∞ n =1 of complementary domains s a tisfying the double- pa ss c ondition. Suppose, b y wa y o f contradiction, that X = X 1 ∪ X 2 , where X 1 and X 2 are pro pe r sub contin ua of X . W e can then find op en balls B 1 and B 2 such that (1) B 1 ∩ B 2 = ∅ , (2) B i ∩ X i 6 = ∅ for i = 1 , 2, and (3) B i ∩ X j = ∅ for i 6 = j . Since ( U n ) ∞ n =1 satisfies the double- pa ss condition, there exis ts a particular N ∈ N such tha t, fo r any genera liz e d crosscut K of U N , one shadow of K in tersects b o th B 1 and B 2 . W e fix U = U N , and let φ : U → D be a conformal isomor phism. Define E 1 and E 2 , as in Lemma 3.5, to b e the se ts of endpo in ts of the cro sscuts compris ing φ (( ∂ B 1 ) ∩ U ) and φ (( ∂ B 2 ) ∩ U ), r esp ectively . Ther e a re tw o cases: Either E 1 and E 2 are linked, or they a re not. The sec o nd cas e ca nnot o ccur , as Lemma 3.5 asserts the existence of a generaliz ed crosscut K 0 of U separa ting B 1 ∩ U a nd B 2 ∩ U , contrary to our assumption. Thu s, E 1 and E 2 are linked. Note that each of φ (( ∂ B 1 ) ∩ U ) and φ (( ∂ B 2 ) ∩ U ) consists of cr osscuts of D with endp oints in E 1 and E 2 , resp ectively . There are tw o cases: (1) either one o f φ (( ∂ B 1 ) ∩ U ) o r φ (( ∂ B 2 ) ∩ U ) sepa rates the other in D , or (2) neither φ (( ∂ B 1 ) ∩ U ) nor φ (( ∂ B 2 ) ∩ U ) separa tes the other in D . In each case, we co nstruct a crosscut A ⊂ D . This cr osscut will hav e the prop erty that φ − 1 ( A ) is a cro sscut of U w hich we show leads to a sepa ration o f one of X 1 or X 2 , a contradiction. In case (1), w itho ut lo ss of g enerality , φ (( ∂ B 1 ) ∩ U ) separ ates φ (( ∂ B 2 ) ∩ U ) in D . Since D is unico her ent, a comp onent o f φ (( ∂ B 1 ) ∩ U ) also se parates φ (( ∂ B 2 ) ∩ U ), so a crosscut in φ (( ∂ B 1 ) ∩ U ) do es. Let A b e this cr o sscut. Then φ − 1 ( A ) is a crosscut of U sepa rating B 2 ∩ U in U . F or cas e (2), we supp ose that neither φ (( ∂ B 1 ) ∩ U ) nor φ (( ∂ B 2 ) ∩ U ) separ a tes the other in D . Since E 1 and E 2 are linked, let e 1 and e ′ 1 be p oints o f E 1 separated in ∂ D by p oints of E 2 . Let K 1 and K ′ 1 be cr o sscuts of D in φ (( ∂ B 1 ) ∩ U ) that have e 1 and e ′ 1 , re s pec tively , as endp oints. Since φ (( ∂ B 2 ) ∩ U ) do es no t separate K 1 from K ′ 1 in D , there is an ar c C from a po in t of K 1 to a p oint of K ′ 1 in D \ φ ( B 2 ∩ U ). Let A ⊂ K 1 ∪ K ′ 1 ∪ C be a cr osscut of D from e 1 to e ′ 1 which then separates φ ( B 2 ∩ U ) in D . Because φ − 1 ( K 1 ) a nd φ − 1 ( K ′ 1 ) a re cr osscuts o f U , w e see that φ − 1 ( A ) is a crosscut of U . Mor eov er, φ − 1 ( A ) separa tes B 2 ∩ U in U . INDECOMPOSABILITY FROM THE COMPLEMENT 9 The pro of in cases (1) and (2 ) now pr o ceeds to gether. Let S 1 be a n irr educible arc which joins p oints of φ (( ∂ B 2 ) ∩ U ) which are sepa rated by A ; we may s tipulate that S 1 int ersects A exactly once, transversely . By applying φ − 1 to b oth A a nd S 1 , we o btain a cross cut A ′ of U and a compact arc S ′ 1 ⊂ U b etw een p oints of ∂ B 2 which intersects A ′ once tra nsversely . Let S ′ 2 be a compac t arc in B 2 which joins the endp oints of S ′ 1 . Then S ′ 1 ∪ S ′ 2 = S is a simple closed curve. Observe that S ∩ X 1 = ∅ , since S ′ 1 ⊂ U a nd S ′ 2 ⊂ B 2 . How ev er, the compact ar c A ′ joins p oints of X 1 and intersects S exactly once, transversely . Th us, some p oint of X 1 lies inside and another p oint outside of S , while X 1 ∩ S = ∅ , contradicting the co nnectedness of X 1 .  Examination of the pro o f of Theorem 1.4 gives a stronge r theorem fo r contin ua whose complementary domains have lo cally connec ted b oundaries . Definition 3.6. A sequence of complementary domains ( U n ) ∞ n =1 satisfies the cr oss- cut c ondition if, for e very sequence of cro s scuts ( A n ) ∞ n =1 , A n ⊂ U n , there exists a choice of shadows S n of A n such tha t lim S n = X . Example 3.3 showed that this is a str ic tly weaker condition than the double-pass condition. How ever, the following shows that, for a c e r tain class of c o nt inua, the t wo notions a re equiv alen t. Corollary 3.7. A planar c ontinuum X whose c omplementary domains have lo c al ly c onne cte d b oundaries is inde c omp osable if and only if it has a se quenc e ( U n ) ∞ n =1 of c omplementary domains which satisfies the cr osscut c onditio n. This follows fr om the pro of of Theor em 1.4, since the genera lized cr osscut of U constructed in the pro of with Lemma 3.5 is a cros s cut if ∂ U is lo cally connected. 4. Questions and Fur ther Resul ts W e clo se with a question ab out ratio nal Julia sets for which our Char acterization Theorem may prov e useful, and tw o theo r ems by the first author that will app ear in a subsequent pap er extending o ur r esults to surfaces. Question 4.1 . Let J = J ( R ) b e the Julia set of a rational function R : C ∞ → C ∞ and supp ose that J ha s bur ied p oints. Ca n J b e the union of tw o pro per indecomp osable sub contin ua? In par ticular, can J c o nt ain a prop er indec o mpo sable sub c ontin uum with interior in J ? Definition 4 .2. A surfac e is a connected Hausdor ff space with a countable basis each p oint of which has a neighbo rho o d homeo morphic to an op en ball in the plane. Let X be a co ntin uum in the surface S . As b efore, a comp onent of S \ X is called a c omplementary domain . Definition 4 .3. A co nnected top olog ical s pace X is multic oher en t of degree k if, for any pair of closed, connected sets A and B such that A ∪ B = X , the int ersection A ∩ B consis ts of at most k components. A complement ary domain in a surface, unlike in the planar cas e , need not b e simply connected. Using the notion of multicoherence and its co nsequences (see [21, Theorem 1 ] for the relev an t extensio n of the P hragm` en-Brouw er theorem), we can pr ov e the fo llowing theor em. W e omit the pro of, which is similar to the pro of of Theorem 2 .10. 10 C. P . CURR Y, J. C. MA YER, AND E. D. TYMCHA TYN Theorem 4.4. L et S b e a c omp act surfac e and X an inde c omp osable sub c ontinuum of S . Then ther e is a se quenc e ( U n ) ∞ n =1 of c omplementary domains of X su ch that lim ∂ U n = X . W e cla im in Theorem 4.4 that having a sequence o f complementary do ma ins conv erging to X is a necessa r y co ndition for contin uum X co nt ained in a surface S to b e indecomp o sable. W e saw in the plane a partial c o nv erse: given a s equence of distinct complement ary do mains ( U n ) ∞ n =1 such that lim ∂ U n = X , it follows that X is either indecomp osable or the union of t w o prop er indecomp osable sub contin ua (Theorem 2.4). In this connection, w e close with the following tw o theo rems gen- eralizing Burgess ’s Theo rem 2.4 and our Cha racteriza tio n Theorem 1.4 to c o nt inua in sur faces, pro ofs o f which will a ppea r subseque ntly in a pap er by the first-named author. Theorem 4.5. L et S b e a c omp act surfac e. Su pp ose c ontinuum X ⊂ S has a se qu enc e ( U n ) ∞ n =1 of distinct c omplementary domains with lim ∂ U n = X . Then either X is inde c omp osab le, or ther e is only one p air of inde c omp osable sub c ontinua whose union is X . Theorem 4. 6. L et S b e a c omp act surfac e. Su pp ose c ontinuum X ⊂ S has burie d p oints. Then X is inde c omp osabl e iff X has a se quenc e ( U n ) ∞ n =1 of distinct c omple- mentary domains satisfying the do uble-pass condition : for any se quenc e ( A n ) ∞ n =1 of gener alize d cr osscuts (suitably define d), with A n ⊂ U n , ther e is a se quenc e of shadows ( S n ) ∞ n =1 , wher e e ach S n is a shadow of A n , such that lim S n = X . 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Memoirs of the Americ an Mathematic al So ciety. 3, 1951. E-mail addr e ss , Clinton P . Curry: clintonc@uab .edu E-mail addr e ss , John C. May er : mayer@math.u ab.edu (Clint on P . Curry and John C. May er ) Dep ar tment of M a thema tics, University of Al- abama a t Birmingham, Birmingham , AL 35294-11 70 E-mail addr e ss , E. D. Tymc hat yn: tymchat@ma th.usask. ca (E. D. Tymc hat yn) Dept. of Ma thema tics and St atis tics, University of Saska tchew an, Saska toon, S aska tchew an, Canada S 7N 0W0