Any counterexample to Makienkos conjecture is an indecomposable continuum

ANY COUNTEREX AMPLE TO MAKIENKO’S CONJECTURE IS AN INDEC OMP OSAB LE CONTINUUM CLINTON P . CURR Y, JOHN C. MA YER, JO NA THAN MEDD A UGH, AND JAMES T. R OGERS, JR. Abstract. Makienko’s conjecture, a prop osed a ddition to Sul- liv a n’s dictionary , can be stated as follo ws: The Julia set of a rational function R : C ∞ → C ∞ has buried points if a nd only if no comp o nent of the F atou set is co mpletely inv ariant under the second itera te o f R . W e pr ov e Makienko’s conjecture for rational functions with Julia sets that are decomp os able co nt in ua. This is a very br oad collection of J ulia sets; it is not known if there exists a rational functions whose Julia set is an indecomp osable co ntin uum . Dedicated to Bob D ev aney on the o ccasion o f his 60 th birthda y . 1. Intr oduction Let R : C ∞ → C ∞ b e a rational function, whe re C ∞ denotes the Riemann sphere. The F atou se t of R , denoted F ( R ), is the domain of normalit y for the family of functions { R i | i ∈ N } . A comp onen t o f the F atou set is called a F atou c om p on e n t . The Julia set of R , denoted J ( R ), is the complemen t o f F ( R ). In t he case that the degree of R is not one, the Julia set is a non-empty , compact, p erfect subset of C ∞ . It is w ell-kno wn that J ( R n ) = J ( R ) and F ( R n ) = F ( R ) for any integer n ≥ 1. A set X is said to b e c ompletely invariant un der R prov ided that R ( X ) = X = R − 1 ( X ). Both J ( R ) and F ( R ) are completely in v ariant under R . In fact, J ( R ) is the smallest closed subset of C ∞ whic h is completely in v a r ian t unde r R and con ta ins at least three p oin ts [Bea91]. On the other hand, the F atou set ma y ha v e la r g e subsets that are Date : Nov ember 13, 20 18. 2000 Mathematics Subje ct Classific ation. Prima ry: 37F20; Secondary : 54F15. Key wor ds and phr ases. indecomp osa ble contin uum, Makienko’s conjecture, Makienko conjecture, Julia set, holomo rphic dyna mics, inv ariant F atou comp onent, complex dynamics, buried p oint, r esidual Julia set. W e thank the Departments of Mathematics at T ulane University and Nipissing Univ ersity (On tar io), and the Fields Institute a t the Universit y of T oro nt o for the opp ortunity to w ork on this pap er in pleasant s urroundings. 1 2 C. P . CURR Y , J. C. MA YER, J. MEDDA UGH, AND J. T. R OGERS , JR. completely in v ariant under R . F or example, if R is a p olynomial, then the basin of attraction of infinit y is a completely inv ar ia n t comp onen t of the F atou set. If R ( z ) = z − 2 , then the F atou set o f R has no completely inv a r ia n t comp onent, while t he F atou set of R 2 ( z ) = z 4 has t wo completely inv ariant comp onen ts. A p o int of the Julia set is said to b e burie d if it do es not b elong to the b oundary of a F ato u comp onen t. The set of all buried p oints of the Julia set of a map is called the r esidual Julia set . McMullen [McM88] presen ted the first ratio nal maps with non-empt y residual Julia sets. He sho w ed that f unctions o f the f o rm z 7→ z n + λ/z d ha ve Julia sets homeomorphic to the product o f a Cantor set and a Jordan curv e when- ev er 1 /n + 1 /d < 1 and λ is sufficien tly small. In t his case, there are uncoun tably comp onen ts of the Julia set wh ic h do not intersec t the b oundary of any F atou comp onen t. Later, John Milnor and T an Lei [ML93] and Dev aney , Lo ok, and Uminsky [DLU05 ] exhibited rational functions with Julia sets homeomorphic t o the Sierpinski carp et. It is w ell-kno wn that the residual Julia set is t hen a connected dense G δ subset. In terestingly , the kno wn examples of Julia sets which hav e non-empt y residual Julia sets are precisely the examples for whic h no F atou com- p onen t has a finite gra nd orbit. Pe t er M. Makienk o made the f o llo wing conjecture. Makienk o’s Conjecture. Let R : C ∞ → C ∞ b e a rationa l function. The Julia set J ( R ) has buried p oin ts if and only if the r e is no completely in v arian t comp o nen t of the F ato u set of R 2 . This conjecture w as for mulated as a p ossible entry in Sulliv a n’s dic- tionary in 1990 [EL90] as a parallel to a theorem of Abik o ff . Note t ha t one direction is easily prov ed with fa cts already g iven. Sp ecifically , if a F atou comp onen t F is completely in v ariant under R 2 , then ∂ F is also completely in v arian t, closed, and consists of more than three p oin ts, so ∂ F = J ( R ). The example z 7→ 1 z 2 illustrates wh y one must exam- ine the F atou set of R 2 . Ho w ev er, it is kno wn [Bea91 , Theorem 9.4.3] that a rational map may ha ve at most t wo completely inv ar ia n t F atou comp onen ts, in whic h case t he Julia set is a simple closed curv e. Makienk o’s conjecture has receiv ed atten tion in the past, w ith results b eing limited b y top ological considerations. Morosaw a [Mor97, Mor00] has pro v ed the conjecture in the cases tha t R is hyperb o lic or subhy p er- b olic. Qiao [Qia97] has pro ved t his conjecture under the assumption that J ( R ) is lo cally connected, as w ell as in the case that J ( R ) is not connected. This extends Moro saw a ’s results, since h yp erb olic and MAKIENKO’S CONJECTURE AN D I NDECOMPOSABILITY 3 subh yp erb olic rational functions with connected Julia sets hav e the prop erty that J ( R ) is lo cally connected. An interes t ing sp ecial case w a s prov ed by Sun and Y ang [SY03]. Let R b e a rational map with exactly tw o critical p o in ts and of degree at least three. Then either (1) R satisfies Makienk o ’s conjecture, or (2) J ( R ) is a Lake s of W ada con tinuum, and therefore either inde- comp osable or the union of t wo indecomp osable contin ua. In [CMTT06], the second alternativ e is impro v ed to J ( R ) b eing an indecomp o sable con tinuum. W e repro duce the theorem used to mak e this improv emen t as Theorem 12 b elo w. This collection of results indicates that any coun terexample to the conjecture m ust hav e a complicated Julia set. The main theorem of this pap er sa ys that the Julia set must b e extremely complicated. Theorem (Makienk o’s Conjecture for Decomp osable Julia Sets of Ra - tional Maps) . I f R i s a r ational function such that J ( R ) has no burie d p oin ts an d F ( R 2 ) has no c omple tely inva ri a nt c omp onents, then J ( R ) is an inde c omp osab le c ontinuum. Recall that a contin uum is de c omp osable if it can b e written as the union of tw o prop er subcontin ua; otherwise it is inde c om p osa ble . There are no kno wn ex a mples of Julia sets whic h are indecomp osable con- tin ua. In fact, whether or not there exists a rational function with an indecomp o sable contin uum as its Julia set is a w ell-know n unsolv ed problem [MR93]. 2. Main Re sul t F or this section, let R b e a rational function whic h is a coun terex- ample to Makienk o’s conjecture. Th en (1) J ( R ) is connected [Qia97], (2) J ( R ) 6 = C ∞ , (3) J ( R ) has no buried p oints, a nd (4) F ( R 2 ) has no completely inv aria n t comp onent. Therefore, J ( R ) is a con tinuum and w e will sho w that it is indecom- p osable. The argumen t will consist of t w o main par t s. In Subsection 2.1, w e mak e use t he dynamics of R to pr ov e t o p ological fa cts ab out J ( R ), ultimately that it is a contin uum whic h is irr e ducible ab out a finite set . In Subsection 2.2, we apply a decomp osition theorem of Kuratow ski to sho w that the Julia set m ust con ta in an indecomp osable sub contin uum with inte rior, a nd therefore m ust b e indecomp osable b y [CMTT06]. 4 C. P . CURR Y , J. C. MA YER, J. MEDDA UGH, AND J. T. R OGERS , JR. 2.1. Consequences o f the Dynamics. In this section w e pro ve our Irreducibilit y Theorem 3, from whic h it immediately follo ws that a coun terexample to Makienk o ’s conjecture cannot b e arcwis e connected. Lemma 1. Ther e exists a p erio dic F atou c omp onent U such that ∂ U = J ( R ) . Pr o of. If F is the collection of F a tou comp o nen ts, then S F ∈F ∂ F = J ( R ), since J ( R ) has no buried p oin t s. Note that F is coun table, so the Baire Category Theorem implies that some F 0 ∈ F has b oundary with in terior in J ( R ). Since R is top olo gically ex act, there exists k ∈ N suc h that R k ( ∂ F 0 ) = J ( R ), so R k ( F 0 ) has b oundary equal to J ( R ). By Sulliv an’s No W andering Domains Theorem, R k ( F 0 ) is even tually p erio dic. Let U b e the iterate which is p erio dic with p erio d n , and w e see that ∂ U = J ( R ).  Lemma 2. Ther e exists a F atou c omp onent V 6 = U such that R n ( V ) = U . Pr o of. Let U = U 1 , U 2 , . . . U n b e the F atou comp onen ts in the cycle of U . Supp ose that such a V do es not exist. Then U , and hence eac h U i , is completely inv a rian t under R n . Since the F atou set of a ratio nal function can ha v e at most t w o completely inv a rian t comp o nen ts [Bea91, Theorem 9.4.3], w e see that n = 2. Thus w e ha ve arrived at the con tradiction that U is completely inv aria nt under R 2 .  A con tinuum K is said to b e irr e ducible ab out S pro vided that S ⊂ K and no prop er sub contin uum of K con ta ins S . Theorem 3 (Irreducibilit y The o rem) . Supp ose R : C ∞ → C ∞ is a r ational function of de gr e e at l e ast 2 such that J ( R ) has no burie d p oin ts and F ( R 2 ) has no c ompletely invariant c omp one n t. Then J ( R ) is irr e ducible ab out a fin i te p oint set. Pr o of. By Lemma 1 , let U b e a p erio dic F atou component whose b ound- ary is J ( R ) . Without loss of generalit y , supp ose in fact that R ( U ) = U . Let V 6 = U b e a preimage of U . By wa y of con t r a diction, supp ose J ( R ) is not irreducible ab o ut any finite p o int set. W e will a rriv e at a con- tradiction b y first finding a con tin uum K ⊂ C ∞ suc h that (1) K do es not con tain J ( R ) , (2) K con tains all critical v alues of J ( R ) , and (3) K do es not separate U . W e will use K to construct arcs A 1 and A 2 . The arc A 1 will hav e the prop erty t ha t its endp oin ts lie in U and V , and the arc A 2 will b e in MAKIENKO’S CONJECTURE AN D I NDECOMPOSABILITY 5 the F atou set connecting the endp oin ts of A 1 , con tradicting the f a ct that U and V are distinct F atou comp onents . Let e K ⊂ J ( R ) b e a minimal sub con t inuum con t a ining a finite set consisting of (1) ev ery critical v alue in J ( R ), and (2) an accessible p oin t on ∂ F , for ev ery F atou comp onen t F (in- cluding U ) whic h meets the critical v a lue set. (Here, accessible means accessible from within F .) By the assumption that J ( R ) is not irreducible ab out a finite set, w e see that e K 6 = J ( R ). F or eac h F atou component F in tersecting the critical v alue set, there exists a n arc in F whic h contains C R ∩ F and meets J ( R ) exactly in a p oint of e K ∩ ∂ F . Let K denote the con tin uum comp osed of e K and these arcs. Notice that K ∩ J ( R ) = e K ( J ( R ) , and that K do es not separate U , as required. Let Q ′ b e a n ev enly co vered op en subset of C ∞ \ K whic h in tersects J ( R ). Since R ( ∂ V ) = J ( R ), there is a comp onen t Q of R − 1 ( Q ′ ) which in tersects ∂ V . Note that Q in tersects U a s w ell, since ∂ U = J ( R ). Let A 1 ⊂ Q b e an ar c joining a p oint of Q ∩ U to a p o in t of Q ∩ V . Since U and V eac h map to U , and R | Q is a homeomorphism, R ( A 1 ) ⊂ Q ′ is an arc whose endp oin ts lie in U . Connect the endp oin ts of R ( A 1 ) with an arc A ′ 2 ⊂ U \ K . Note that, since C ′ = R ( A 1 ) ∪ A ′ 2 is disjoin t from K , C ′ is con tained in a comp onen t of C ∞ \ K . That comp onent is a simply connected neigh b or ho o d of C ′ con taining no critical v a lues, so eac h comp onen t o f R − 1 ( C ′ ) maps homeomorphically onto C ′ . Let C denote the comp onent of R − 1 ( C ′ ) whic h con tains A 1 . Since R | C is a homeomorphism, we hav e tha t C is the union of A 1 and a preimage A 2 of A ′ 2 . Since A ′ 2 ⊂ U and F ( R ) is completely inv arian t, A 2 ⊂ F ( R ). Ho wev er, A 2 joins the endp oin ts of A 1 , whic h lie in U and V . This con t r adicts that U and V are distinct F atou comp onen ts.  Theorem 3 is a strong top ological constraint on the na ture of J ( R ). F or instance, any lo cally connected or path c onnected contin uum whic h is ir r educible ab o ut a finite set is necessarily a tree. This recov ers the results of Qiao [Qia 97] and Morosaw a [Mor97, Mor00] that J ( R ) cannot b e lo cally connected. F urthermore, this additio nally pro ves that J ( R ) cannot b e arcwise connected. 2.2. A pplying Kuratowski’s Decomposition. T he class of con- tin ua whic h are irreducible ab out a finite set con tains the class of inde- comp osable con tinua. Ho w ev er, b eing an indecomposable contin uum is 6 C. P . CURR Y , J. C. MA YER, J. MEDDA UGH, AND J. T. R OGERS , JR. generally m uch stronger than b eing irreducible ab out a finite set. W e will inv estigate the relationship in the presen t con text. W e fir st state a useful theorem from [CMR06]. See also [R98 ], where a similar t heorem first app eared, and [CMTT06 ] for v ariants . Theorem 4 ( [CMR06]) . L et R : C ∞ → C ∞ b e a r ational function. L et Y b e a c omp act subset of C ∞ and let X b e a nowher e dense c omp act subset of Y . Then R ( X ) is now h er e d ense in R ( Y ) . A crucial top olog ical c haracteristic of J ( R ) follow s immediately . Corollary 5. Int J ( R ) ( ∂ V ) 6 = ∅ . In the pr esen t situation, we hav e more top ological information about the contin uum J ( R ) – it is the b oundary of one complemen tar y domain U , and another complemen ta ry domain V has non-empt y in terior in its b oundary , relativ e t o J ( R ). These facts a r e sufficien t to prov e that J ( R ) mus t con tain an indecomp osable sub contin uum with non-empt y in terior, and therefore is itself an indecomp osable con t inuum. Definition 6. Supp ose X ⊂ C ∞ is the b oundary of tw o connected op en sets U, V ⊂ C ∞ . Then X is monostr a tic if it is the countable union of indecomp osable sub con tin ua and no where dense sub con tinua. Kuratow ski [Kur28] pro v ed that a contin uum whic h is the common b oundary of tw o regions admits a monotone decomp o sition to a circle if a nd only if it is not monostratic. Eac h elemen t of the decomp osition is the coun table union of indecomp osable sub con tinua and no where dense sub con tinua of the common b oundary . Thus, an y elemen t of the decomp osition whic h con tains in terior necessarily con tains an indecom- p osable contin uum. Lemma 7. L et U b e a simply c o n ne cte d op e n subset of C ∞ with non- de ge n er ate b oundary. L et V 6 = U b e a c om plementary domain of ∂ U . Then ∂ V is the c ommon b oundary of at le ast two r e gion s: V and Comp( C ∞ \ V , U ) , the c omp onent o f C ∞ \ V w h i ch c ontains U . Pr o of. By the Boundary Bumping Theorem, ∂ Comp ( C ∞ \ V , U ) ⊂ ∂ V . Also, observ e that ∂ V ⊂ U ⊂ Comp ( C ∞ \ V , U ) , and ∂ V ⊂ V ⊂ C ∞ \ Comp( C ∞ \ V , U ) . These three inclusions imply that ∂ Comp ( C ∞ \ V , U ) = ∂ V .  MAKIENKO’S CONJECTURE AN D I NDECOMPOSABILITY 7 Lemma 8. L et U ⊂ C ∞ b e a simply c onn e cte d op en set with non- de ge n er ate b oundary. L et V 6 = U b e a c omplementary domain o f ∂ U such that In t ∂ U ( ∂ V ) 6 = ∅ . Then, if ∂ V is a mono s tr atic c ontinuum, it c ontain s an inde c omp osable sub c ontinuum w i th interior in ∂ U Pr o of. If ∂ V is monostra t ic then by definition ∂ V = S ∞ i =1 X i ∪ S ∞ i =1 A i , where each A i is a nowhere dense sub con tinuum of ∂ V and each X i is an indecomp osable contin uum. Note that In t ∂ U ( ∂ V ) = S ∞ i =1 ( X i ∩ In t ∂ U ( ∂ V )) ∪ S ∞ i =1 ( A i ∩ In t ∂ U ( ∂ V )) is a Baire space, so it is not the union of a countable collection of nowhe re dense closed subsets of ∂ U . Since eac h A i is now here dense in ∂ V , and th us no where dense in ∂ U , some X i has interior in ∂ U .  Lemma 9. If X is a c ontinuum irr e ducible ab out { a 1 , . . . , a N } , then for e ach s ub c ontinuum Y ⊂ X the set X \ Y has at most N c omp onents. Pr o of. If Y ⊂ X is a con tinuum and X \ Y has mor e than N com- p onen ts, t hen at most N comp onen ts o f X \ Y can contain p oints of { a 1 , . . . , a N } , since comp o nen ts ar e disjoin t. By the Boundar y Bump- ing Theorem, the union of Y with those comp onen ts is then a prop er sub con tinuum of X con t aining { a 1 , . . . , a N } , contradicting that X is irreducible ab out { a 1 , . . . , a N } .  Lemma 10. L et X = ∂ U for some simply c onne cte d op en subse t of C ∞ with nonde gener ate b oundary. L et V 6 = U b e a c ompl e mentary domain of X such that In t X ( ∂ V ) 6 = ∅ . I f X is irr e ducible ab out a finite set and ∂ V is not monostr atic, then ∂ V c ontains an inde c omp osable sub c ontinuum with interior in X . Pr o of. Supp o se X is irreducible ab out a finite set A of car dina lity N . By Lemma 9, no sub con t in uum of X can separate X into more than N comp onen ts. In particular, X \ ∂ V consists of connected comp onents X 1 , . . . , X k , where k ≤ N . C ho ose x i ∈ X i ∩ ∂ V , and let B denote the finite set B = { x 1 , . . . , x k } ∪ ( A ∩ ∂ V ) . Let m : ∂ V → S 1 b e the monotone map guarantee d by Kuratows ki. W e now consider tw o cases: Does m − 1 ( m ( B )) ha ve interior in X ? Supp ose first that it do es. T hen m − 1 ( m ( B )) is the finite union of the contin ua { m − 1 ( m ( x )) | x ∈ B } whic h con tains an op en s ubset of X . The Baire Category Theorem implies that there exists p ∈ B suc h that M = m − 1 ( m ( p )) has interior in X . In particular, M has in terior in ∂ V , and according to the Kuratowsk i decomp osition, M = ( S ∞ i =1 M i ) ∪ ( S ∞ i =1 C i ) where eac h M i is indecomp osable, and eac h C i is a con t inuum of condensation of X . An argumen t ana la gous to that in 8 C. P . CURR Y , J. C. MA YER, J. MEDDA UGH, AND J. T. R OGERS , JR. the pro o f o f Lemma 8 sho ws that some M i has inte r ior in X , and the theorem is pro ve d. Otherwise, m − 1 ( m ( B )) is nowh ere dense in X . Set K = ∂ V \ m − 1 ( m ( B )) , whic h has non-empt y in terior in X since ∂ V do es. The set S 1 \ m ( B ) consists of finitely many disjoint op en interv als I 1 , . . . , I n , and K = m − 1 ( I 1 ) ∪ . . . ∪ m − 1 ( I n ). Since K con tains an op en subset of X , some m − 1 ( I q ) con ta ins an op en subset of X . Notice that I c q = S 1 \ I q is a closed interv a l con taining m ( B ). Since m is monotone, m − 1 ( I c q ) is a sub con tinuum of ∂ V con taining B . Set X ′ = X 1 ∪ . . . ∪ X k ∪ m − 1 ( I c q ) . Notice that X ′ is compact since I c q is closed. That m ( B ) ⊂ I c q implies t wo imp ortan t facts ab out X ′ . First, since { x 1 , . . . , x k } ⊂ m − 1 ( I c q ), w e see that X ′ is connected, sinc e each X i is a connected set which meets the connected set m − 1 ( I c q ). Second, X ′ 6 = X , since X 1 ∪ . . . ∪ X k ∪ m − 1 ( I c q ) do es not con ta in m − 1 ( I q ) (a s m − 1 ( I q ) has interior in X ). Th us, X ′ is a prop er sub con tinuum of X con ta ining B , contradicting the fact t ha t X is ir r educible ab out that set.  Theorem 11. L et X = ∂ U for a simply c onne cte d o p en subset U of C ∞ with nonde gener ate b oundary. L et V 6 = U b e a c omplementary dom ain of X such that Int X ( ∂ V ) 6 = ∅ . If X is irr e ducible ab out a fin ite set, then ∂ V c ontains an inde c omp osabl e c ontinuum with interior i n X . Pr o of. If ∂ V is monostra t ic, then this f ollo ws from Lemma 8. If ∂ V is not monostratic, then this follows fr om Lemma 10 .  Theorem 12 (Theorem 2.4 from [CMTT06]) . Supp ose J 6 = C ∞ is the c onn e cte d Julia set of a r ational map of de gr e e a t le ast two an d that J has no burie d p oints. I f K is an inde c o m p os a ble sub c ontinuum of J with nonempty interior in J , then K = J . Corollary 13 (Mak ienk o’s Conjecture for Decomposable Julia Sets of Ra tional Maps) . If R is a r ational function such that J ( R ) has no burie d p oints and F ( R 2 ) ha s no c ompletely invariant c om p on e nts, then J ( R ) is an inde c omp osab l e c ontinuum. Pr o of. By Lemma 1, there is a F atou comp onen t U so tha t ∂ U = J ( R ). By Lemma 2, let V 6 = U b e a preim age of U . Coro llary 5 giv es that ∂ V has in t erio r in J ( R ). Theorem 3 implies that J ( R ) is irreducible ab out some fin ite set. Theorem 11 sho ws that J ( R ) con tains an indecomp o sable sub con tinuum with interior, and T heorem 12 show s that J ( R ) is indecomp osable.  MAKIENKO’S CONJECTURE AN D I NDECOMPOSABILITY 9 3. Conclusions It has now b een demonstrated that a n y coun terexample to Makienk o conjecture must b e exceedingly complicated. In fact, it is not kno wn if the Julia set of a rational function can b e as complicated as required b y the conclusion of our theorem. W e restate here the question whic h app ears in [CMR06, CMTT06 , MR93, R9 8, SY03]. Question 14. Can the Julia set of a ratio na l function b e an indecom- p osable contin uum? Can the Julia set o f a rational function con tain an indecomp osable sub con tinuum with interior? Note that Theorem 12 indicates t ha t the answ ers to these tw o ques- tions are t he same for p olynomial Julia sets and fo r rational functions whose Julia set con ta ins no buried p oints . Sev eral authors hav e extended the study of Makienk o’s conjecture to tra nscenden tal and meromorphic functions. Dom ´ ınguez and F a g - ella [DF] surv ey the curren t state of affairs in this direction. Also see Ng, Zheng, a nd Choi [NZC06], who pro ve Makie nk o’s conjecture for lo- cally connected Julia sets of certain meromorphic functions. W e do not attempt to apply these tec hinques to the meromor phic case. In part ic- ular, our approach mak es use of Theorem 4, whic h fails dra ma t ically for entire and meromorphic functions. A particular p oin t of difficult y is summarized in the follow ing question. Question 15. Let f be a transcenden tal meromorphic function. If V is a F atou comp onen t suc h that ∂ V is nowh ere dense in J ( f ), do es it follo w that ∂ f ( V ) is also no where dense in J ( f )? It is also w or t h noting tha t o ur pro o f mak es use of the non-existence of w andering doma ins fo r rational functions, which in general is not true for transcenden tal functions. Reference s [Bea91] Alan F. Beardo n. It er ation of R ational F unctions. Springer- V erla g, 1991 [CMR06] Douglas K. Childers , Jo hn C. May er , and Ja mes T. Roger s , Jr. I nde- comp osable contin ua and the Julia s e ts of p olynomia ls. II. T op olo gy Appl. , 153 (1 0):1593 –1602 , 2006. [CMTT06] Douglas K. Childers, John C. Mayer, H. Murat T uncali, a nd E. D. Tymc hatyn. Indeco mp o s able contin ua and the J ulia sets o f ratio na l maps. I n Complex dynamics , v olume 396 o f Contemp. Math. , pages 1–20. Amer. Ma th. Soc., Providence, RI, 2006 . [DF] Patricia Dom ´ ınguez and N´ uria F agella. Residual Julia sets of r ational and tr a nscendental functions. In T rans c e nden tal Dyna mics a nd Com- plex Analy sis, volume 348 of L ondon Math. So c. L e ctur e Note Ser., Cambridge Univ. Press, Cambridge, 2007. 10 C. P . CURR Y , J. C. MA YER, J. MEDDA UGH, AND J. T. R OGERS , JR. [DLU05] Rob ert L. Dev aney , Daniel M. Lo ok, and David Uminsky . The escap e trichotom y for singularly p erturb ed rational maps. Indiana Univ. Math. J. , 54(6):1 621-1 634, 20 05. [EL90] A. E. Eremenko and M. Y u. L yubich. The dynamics of ana ly tic trans- formations. L eningr ad Math. J. 1(3):56 2-634 , 199 0. [Kur28] Casimir Kuratowski. Sur la structure des fr onti ` eres communes ` a deux r´ eg ions. F un d. Math. , 12(1):21 –42, 1928. [MR93] John C. Ma yer and James T. Roger s, Jr . Indecomp osa ble contin ua and the Julia sets of polynomials . Pr o c. Amer. Math. So c. 1 17(3): 795-8 02, 1993 [McM88] Curt McMullen. Automor phism of ra tional maps. In Holomorp hic func- tions and Mo du li I , volume 10 of Math. Sci. R es. In s t. Publ. , pag e s 31-60 . Springer-V erlag, New Y or k, 1988 . [ML93] John Milnor and T an Lei. A “Sierpinski carpet” as Julia set. App endix F in Geo metry and dy na mics of qua dratic rationa l maps. Exp erment . Math. , 2(1):37 - 83, 1993. [Mor97] S. Moros aw a. On the res idual J ulia sets of rationa l functions. Er go dic The ory Dynam. S yst ems , 17(1):205 -210, 1997. [Mor00] S. Morosawa. Julia sets o f subhyperb olic rational functions. Complex V ariables The ory Appl. , 4 1(2):151 -162, 2000. [NZC06] T uen W a i Ng, Jian Hua Zheng , a nd Y an Y u Choi. Residual Julia sets o f mero morphic functions. Math. Pr o c. Cambridge Philos. So c. , 141(1):11 3–12 6, 2006 . [R98] James T. Rogers, J r . Diophantine conditions imply c r itical po int s on the bo undaries of Siegel disk s of po lynomials. Comm. Math. Phys. , 19 5(1): 175–1 93, 1998. [Qia97] J. Qiao . T op olog ical complexity of J ulia sets. Sci. China Ser. A , 40(11):11 58–1 165, 1997 . [SY03] Y. Sun, C. C. Y ang. Buried p oints a nd La kes of W a da contin ua. D iscr et e Contin. Dyn. Syst. , 9 (2):379-3 82, 2003 E-mail addr ess , Clinton P . Curry : clin tonc@ uab.edu E-mail addr ess , J ohn C. May e r: m ayer@m ath.u ab.edu (Clin ton P . Curry and J o hn C. May er ) Dep ar tment of Ma thema tics, Uni- versity of A labama a t Birmingham, Birmingham, AL 35294-117 0 E-mail addr ess , J onathan Meddaugh: jm eddau gh@mat h.tulane.edu E-mail addr ess , J ames T. Rogers , Jr.: jim@ma th.tul ane.edu (Jonathan Meddaugh a nd J ames T. Rog ers, Jr.) D ep ar tment of Ma thema t- ics, Tulane U niversity, New Orleans , L A 70118