2 Genadi Levin and Sebastian van Strien of the Mandelbrot set, see [Y] (Actually, McMullen proved that whenever the critical orbit of an innitely reno

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1 Local connectivity of the Julia set of real polynomials Genadi Levin, Hebrew University, Israel Sebastian van Strien, University of Amsterdam, the Netherlands y December 31, 1994 and extended January 27, April 5 and November 14, Introduction and statements of theorems One of the main questions in the eld of complex dynamics is the question whether the Mandelbrot set is locally connected, and related to this, for which maps the Julia set is locally connected In this paper we shall prove the following Main Theorem Let f be a polynomial of the form f(z) = z` + c 1 integer and c 1 real Then with ` an even the Julia set of f is either totally disconnected or locally connected; f does not admit a measurable invariant line eld on its Julia set; In particular, the Julia set of z 2 +c 1 is locally connected if c 1 2 [?2; 1=4] and totally disconnected if c 1 2 R n [?2; 1=4] (note that [?2; 1=4] is equal to the set of parameters c 1 2 R for which the critical point c = 0 does not escape to innity) Hence the rst part of the main theorem answers a question posed by Milnor, see [Mil1] The second part of the Main Theorem extends the main result of McMullen in [McM] As usual, we say that f admits a measurable invariant line eld on its Julia set if there exists a measurable subset E of the Julia set of f and a measurable map which associates to Lebesgue almost every x 2 E a line l(x) through x which is f-invariant in the sense that l(f(x)) = Df(x) l(x) (So the absense of lineelds is obvious if the Julia set has zero Lebesgue measure) Part two of the Main Theorem was proved by McMullen for all real maps of the form given above which are innitely renormalizable If f(z) = z` +c 1 is quadratic (ie, ` = 2) and only nitely often renormalizable then the second part holds because then the corresponding parameter c 1 lies in the boundary levin@mathhujiacil y strien@fwiuvanl 1

2 2 Genadi Levin and Sebastian van Strien of the Mandelbrot set, see [Y] (Actually, McMullen proved that whenever the critical orbit of an innitely renormalizable map is `robust' then one has absense of lineelds Let us also mention that in the non-renormalizable quadratic case the Julia set has measure zero, as was proved by Lyubich and Shishikura [Ly2] If ` is large then this is false, see [SN]) Corollary 11 Let f be a polynomial of the form f(z) = z` + c 1 with ` an even integer and c 1 real Then there exists c C arbitrarily close to c 1 such that g(z) = z` + c 0 1 has a periodic attractor The corollary is in the direction of Fatou's conjecture which states that each polynomial can be approximated by a hyperbolic polynomial It follows immediately from the second part of the Main Theorem, because of the Lambda-Lemma, see [MSS] and also [McM] In the case that ` = 2 a much stronger statement is known: one can choose the coecients c 0 1 which are arbitrarily close to c 1 to be real This was proved in [Sw] and [GS1], and also announced in [Ly5] If the map is nitely often renormalizable and quadratic this already follows from the local connectivity result of Yoccoz [Y] Before giving the proofs, let us make some further comments First, we should emphasize that if the!-limit set!(c) of the critical point c = 0 is not minimal then it very easy to see that the Julia set is locally connected, see Section 3 Yoccoz [Y] already had shown that each quadratic polynomial which is only nitely often renormalizable (with non-escaping critical point and no neutral periodic point) has a locally connected Julia set Moreover, Douady and Hubbard [DH1] already had shown before that each polynomial of the form z 7! z` + c 1 with an attracting or neutral parabolic cycle has a locally connected Julia set As will become clear, the dicult case is the innitely renormalizable case In fact, using the reduction method developed in Section 3 of this paper, it turns out that in the non-renormalizable case the Main Theorem follows from some results in [Ly3] and [Ly5], see the nal section of this paper We should note that there are innitely renormalizable non-real quadratic maps with a non-locally connected Julia set, see [DH] and [Mil] Also maps with Cremer points have a non-locally connected Julia set, see [Mil] Hence, the results above really depend on the use of real methods On the other hand, Petersen has shown that quadratic polynomials with a Siegel disc such that the eigenvalue at the neutral xed point satises some Diophantine condition is locally connected, see [Pe] Of course, many questions remain: In principle, the methods of Yoccoz completely break down in the innitely renormalizable case and in the case of polynomials with a degenerate critical point The purpose of Yoccoz's methods is to solve the well-known conjecture about the local connectedness of the Mandelbrot set and therefore, some version of our ideas might be helpful in proving this conjecture For a survey of the results of Yoccoz, see for example [Mil] and also [Ly4]

3 Local connectivity of the Julia set of real polynomials 3 We should note also that Hu and Jiang, see [HJ] and [Ji1] have shown that for innitely renormalizable quadratic maps which are real and of so-called bounded type, the Julia set is locally connected Their result is heavily based on the complex bounds which Sullivan used in his renormalization results, see [Sul] and also the last chapter and in particular Section VI5 of [MS] (cf also [Ji2]) In fact, our methods enable us to extend Sullivan's result to the class of all in- nitely renormalizable unimodal polynomials independently of the combinatorial type! We should emphasize that these complex bounds form the most essential ingredient for the renormalization results of Sullivan [Sul]; in fact in McMullen's approach to renormalization, see [McM], these complex bounds play an even more central role In the previous proofs of the complex bounds see [Sul], and also Section VI5 of [MS], it is crucial that the renormalization is of bounded type and, moreover, the proof is quite intricate Therefore we are very happy that our methods give a fairly easy way to get complex bounds independently of the combinatorial type of the map (ie, only dependent of the degree of the map): Theorem A Let f be a real unimodal polynomial innitely renormalizable map Let f s(n) : V n! V n be a renormalization of this map to some interval V n R containing c Then this map has a polynomial-like extension F := f s(n) : 0 n! n with the modulus of n n 0 n is bounded from below by ; the diameter of n is at most C times the diameter of V n ; Here > 0 only depends on ` and C does not depend on anything In fact, is asymptotically like const=` as ` tends to 1 Moreover, if f is not s(n)=2-renormalizable then 0 n \!(c) V n The way we prove that such sets n exist is through cross-ratio estimates In fact, the estimates are similar to those that were made previously in [SN] In this way, we are able to get the `complex bounds' of Theorem A similar to those used by Sullivan in his renormalization result The shape of n is universal, up to rescaling, and in many cases for ` > 2 we can take n to be a Euclidean disc Note that our bounds are completely independent of the combinatorial type of the map We should note that Theorem A and its proof hold for any renormalization f s of (a maybe only nitely renormalizable map) f provided f 2s does not have an attracting or neutral xed point Also the result even holds when f is real analytic, see Section 12 In the non-renormalizable case we also have complex bounds Firstly, for each level for which one has a high return one has a polynomial-like mapping (Our denition of high case also includes what is sometimes called a central-high return, see the denition in Section 5) Theorem B Let f(z) = z` + c 1 with ` an even integer and c 1 real be a nonrenormalizable polynomial so that!(c) is minimal Assume W is the real trace of a central Yoccoz puzzle piece and F : [V i! W is the corresponding rst return map (on

4 4 Genadi Levin and Sebastian van Strien the real line) and assume that this map has a high return, ie, assume that F (V 0 ) 3 c where V 0 is the central interval Then there exist topological discs i and with i \ R = V i and \ R = W and a complex polynomial-like extension G: [ i i! of F such that the diameter of the disc is at most a universally bounded constant times the diameter of W In fact, if we only require that f is a quasi-polynomial-like mapping (see the definition above Proposition 41) then we have lower bounds for the modulus of n 0 Moreover, one has the following result which extends ideas of [Ly3] and [Ly5] (as was pointed out to us in an by Lyubich) Graczyk and Swiatek informed us that they also have a proof of a result similar to Theorem C The rst return property (stated in Theorem C) replaces the property of McMullen [McM] that the polynomial-like map is `unbranched' It is important in the proof of local connectivity of the Julia set and the absense of invariant lineelds Theorem C Let f(z) = z` + c 1 with ` an even integer and c 1 real be a nonrenormalizable polynomial so that!(c) is minimal If W is the real trace of a central Yoccoz puzzle piece and F : [V i! W is the corresponding rst return map (on the real line) Then after some `renormalizations' one can obtain an iterate ~ F : [ ~ V i! ~ W of F with ~W W such that there exist topological discs ~ i and ~ with ~ i \ R = ~V i and ~ \ R = ~W and a complex polynomial-like extension ~G: [ i ~ i! ~ of ~F Moreover, the diameter of the disc ~ is at most a universally bounded constant times the diameter of ~W ; when f j (c) 2 ~ 0 then f j (c) is an iterate of c under the map ~G; the modulus of the annulus ~ n ~ 0 is bounded away from zero by a constant which only depends on ` Let us say a few words about our proofs The main idea behind our proof of the Main Theorem is to construct generalized polynomial-like mappings F n : [ i i n! n which coincide on the real line with the rst return maps to certain Yoccoz puzzlepieces To do this we rst obtain real bounds to get Koebe space: these are based on a sophisticated version of the `smallest interval' argument They are a sharper version of those used before by Blokh, Lyubich, Martens, de Melo, Sullivan, van Strien, Swiatek and others Using those real bounds and the use of certain Poincare domains inside sets of the form C T = C n (R n T ) where T R is some interval This set (consisting of C with two innite slits) carries a Poincare metric This allows us to construct polynomial-like mappings and show that the diameter of these domains is comparable with that of the interval n \ R Next we compare these polynomial-like maps with those from the Yoccoz puzzle because the intersection of a Yoccoz puzzle-piece with the Julia is connected Next we show that

5 Local connectivity of the Julia set of real polynomials 5 the Julia set of the polynomial-like mappings of the Yoccoz puzzle coincides with the Julia set of the polynomial-like mappings F n, see Section 3 Since these domains get small, we are able to conclude local connectivity of the Julia set The paper is organized as follows In Sections 2-4 we show that the Main Theorem follows from Theorems A-C In Section 7, 8 and 9 we develop real bounds which will enable to estimate the shape of the pullbacks of certain discs or other regions We should emphasize that the real bounds in these sections hold for all unimodal maps with negative Schwarzian derivative In Sections 10 to 16 we apply these estimates to several cases The reader will observe that certain cases are proved by several methods For example, in Section 10 bounds are given in the innitely renormalizable case with ` 4 is proved, while this case also follows from the estimates (for a more general case) in Section 13 However, the domains in Section 10 are discs and those in Section 13 are considerably more complicated We believe that for future purposes it might be important to have good domains, and therefore even if it was sometimes not necessary for the proofs of our theorems, we have tried to treat each case in a fairly optimal way Finally, a short history of this paper since several others have partial proofs of Theorem A and the Main Theorem in the quadratic case Firstly, we were inspired by the papers of Hu and Jiang, see [HJ] and [Ji1] where it is shown that innitely renormalizable maps of bounded type (where Sullivan's bounds hold) have a locally connected Julia set The rst widely distributed version of our paper (dated December 31, 1994) included the proof of the Main Theorem in the quadratic case, the innitely renormalizable case, Theorem A (without doubling) and also some non-renormalizable cases Subsequently, Theorem B was included in the version of this paper of Januari 27, 1995 Graczyk and Swiatek distributed a preprint with a proof of Theorem A in the quadratic case on February 3, 1995 Lyubich and Yampolsky gave a proof of the Main Theorem and Theorem A in the quadratic case, in a draft dated February 22, 1995 The `quadratic' proofs of Graczyk, Swiatek, Lyubich and Yampolsky of Theorem A improve our estimates in certain cases because it sometimes allows one to obtain annuli with large moduli, but those proofs seem to heavily rely on the map being quadratic (In view of the estimates in [SN] such large moduli cannot be expected to exist in the higher order case) One advantage of our proof is that it shows that the small Julia set associated to some renormalization of a map, is contained in a denite sector which is actually based on the renormalization interval (In many cases, the small Julia set is contained in the disc based on the renormalization interval) After we told Lyubich about our methods to obtain local connectivity, he realized the relevance of his methods, see [Ly3], [Ly5], for proving local-connectivity in the non-renormalizable case In an dated February 10, 1995, he told us how to prove Theorem C using these methods, thus completing the proof of the Main Theorem in the non-renormalizable case To make this paper self-contained we added his proof in Section 14 in our paper, in the version of April 5, 1995 The proof of the non-existence of invariant lineelds was included November 14, 1995 The rst author would like to thank the University of Amsterdam where this work was started During a second visit to the University of Amsterdam in October 1995, the

6 6 Genadi Levin and Sebastian van Strien absense of lineelds was proved His research was partially supported by BSF Grant No , Jerusalem, Israel We thank Ben Hinkle for a useful comment, sending us a very detailed list of typos and pointing out a mistake We thank Misha Lyubich for telling us about his results in [Ly3] and [Ly5] and pointing out to us that they imply Theorem C We thank Edson Vargas for many discussions and explanations about the ideas in Section 4 of [Ly3] Finally, we thank Curt McMullen, Mitsu Shishikura and Greg Swiatek for some very helpful remarks 2 When do Julia sets of two polynomial-like mappings coincide? We shall use the fundamental notion of polynomial-like mapping [DH] or more precisely, we need its extension due to Lyubich and Milnor from [LM] Let D 0 ; D 1 ; : : : ; D i, and D be topological discs bounded by piecewise smooth curves and such that the closures D 0 ; : : : ; D i are contained in the interior of D, and such that each the discs D 0 ; : : : ; D i are pairwise disjoint Then we call R: D 0 [ D 1 [ : : : [ D i! D by `-polynomial-like if Rj D j is a univalent map onto D for each j = 1; : : : ; i and Rj D 0 is a `-fold covering of D 0 onto D If i = 0 in this denition, we obtain a polynomial-like map in the original sense of Douady-Hubbard The lled Julia set of R is said to be the set F R [ i j=0d j of the points z such that R k (z) is dened for all k = i; 2; : : : The Julia set J R R An equivalent denition of the lled Julia set F R is: 1\ F R = R?k (D): k=1 We shall use an extension of the Straightening Theorem due to Douady and Hubbard, [DH] This extension was also used in Lemma 71 of [LM], for the case that i = 1 Lemma 21 Let R: D 0 [ : : : [ D i! D be a `-polynomial-like map Then R is quasiconformally conjugate to a polynomial in neighborhoods of the lled Julia set F R and lled Julia set of the polynomial Proof: Let us rst pick a point x 0 2 D n (D 0 [ : : : D i ) and choose closed simple curves 0 ; : : : ; i : [0; 2]! C such that i (0) = i (2) = x 0, the curves i only meet at x 0 and i surrounds D i Moreover, we choose the function i to be smooth and so that d dt i(0) and d dt i(2) are two vectors based at x 0 having an angle =(i + 1) If, for example, i = 1 then 0 [ 1 is a gure eight Next pick a curve in C nd and a point x 1 2 Moreover, choose a smooth function dened on a neighbourhood N of x 0 such that (x 0 ) = x 1 and so that maps i \ N dieomorphically to \ (N) for

7 Local connectivity of the Julia set of real polynomials 7 each i = 0; 1; : : : ; i In local coordinates this map will have an expression of the form z 7! z i+1 plus higher order terms, ie, this map will have a critical point of order i+1 Now let A j be the open annulus between j and D j and let A be the open annulus between and D Moreover, nd a smooth map ~R: A j! A which extends to the closure of these sets so that it agrees with R j and with on the neighbourhood N of z 0 Choose this extension so that : A 0! A is a `-covering and : A j! A is a dieomorphism for j = 1; : : : ; i This map ~R becomes an extension of R if we dene it equal to R on D 0 [ : : : [ D i Next choose r > 1 so that the circle centered at the origin with radius r > 1 surrounds A 0 [ : : : [ A i We can extend ~R smoothly to a map ^R: C! C so that ^R(z) = z`+i for jzj r and so that ^R coincides with ~R on A 0 [ : : : [ A i The map ~R on the annulus fz ; jzj < rg n (A 0 [ : : : [ A i ) is a `-covering map to the annulus fz ; jzj < r`+i g n A Now we use the standard trick from the Straightening Theorem Take a standard conformal structure (ie, the Beltrami coecient = 0) on the basin of 1 of R and extend this structure to a L 1 function : C! fz ; jzj < 1g which is invariant under ^R Since ^R is conformal near innity and on D 0 [ : : : [ D i, there are only a bounded number of points in each orbit of ^R where this map is not conformal It follows that the supremum of j(z)j is bounded away from one, and by the Measurable Riemann Mapping Theorem, it follows that there exists a quasiconformal homeomorphism h: C! C with h(1) = 1 which has as its Beltrami coecient Since is invariant under ^R, it follows that h ^R h?1 is an holomorphic (` + i)-covering Hence ^R is quasiconformally conjugate to a polynomial map P (of degree (` + i)) tu A corollary is: Corollary 21 The Julia set J R is the limit set for the preimages of any point z 2 D (except, in the case that i = 0, for the point zero where zero is the `-multiple xed point of R) We can use all this to show that the Julia set of two polynomial-like mappings coincide In the applications of this we shall later on use for one of these the polynomiallike mapping of the Yoccoz puzzles Proposition 21 (cf [Ji1], [McM]) Let [ [ [ R 1 : D 0 1 D 1 1 : : : D i 1! D 1 ; [ [ [ R 2 : D 0 2 D 1 2 : : : D i 2! D 2 be two `-polynomial-like mappings, such that the critical point c of these maps coincide That is, c 2 D 0 1 \ D 0 2 is the unique and `-multiple critical point for both R 1 and for R 2 Moreover, assume that the following conditions hold:

8 8 Genadi Levin and Sebastian van Strien 1 R 1 (z) = R 2 (z) whenever both sides are dened, so that R 1 and R 2 are extensions of the same map R 2 Let C be the component of D 1 \ D 2 which contains R(c) Then also c 2 C, and there exist precisely i other points c 1 ; : : : ; c i so that c j 2 D j 1\D j 2 and R(c j ) = R(c), and, furthermore, c 1 ; : : : ; c i 2 C Under these conditions, the Julia sets of R 1 and R 2 coincide: J R1 = J R2 : If, additionally, c 2 J R1, (and, hence, c 2 J R2 ), then there exists a component of a preimage R?n 2 (D 2 ), which contains c and is contained in D 1 Proof: For k = 1; 2, let Ck; 0 : : : ; Ck i be the components of R?1 k (C), such that cj 2 C j k when j 6= 0, and c 2 Ck 0 Firstly, R k : C j k! C is a covering, which is just one-to-one if j 6= 0, and R k : Ck 0! C is a `-branching covering In particular, boundaries are mapped to boundaries Since R 1 = R 2 on the common domain of denition, we get that, in fact, C j 1 = C j 2 := C j, for every j Secondly, because of 2), each component C j has a point c j in common with the component C Since C j is connected and is contained in both D 1 and D 2, it belongs to a component of D 1 \ D 2 containing c j, ie, C j C Now consider a map R: C 0 [ C 1 [ : : : [ C i! C, which is one-to-one on every C j ; j 6= 0, and `-to-one on C 0 Take a point x 2 C Then R?1 (x) is a subset of C and it consists of l + i points (counting with multiplicities) That is, for any x 2 C, the sets R?1 1 (x) and R?1 2 (x) coincide and belong to C (21) Starting with x 0 2 C, we apply the corollary to Lemma 21 and (21) to get J R1 = J R2 := J If c 2 J, then consider a component K of J containing c Since K D 1, there exists a component of a preimage R?n 2 (D 2 ), which contains K and is contained in D 1 tu In the sequel we will use a particular case of Proposition 21 separately: Let us state it Proposition 22 Let [ [ [ R 1 : D 0 1 D 1 1 : : : D i R 2 : D 0 2 [ D 1 2 1! D 1 ; [ [ : : : D i 2! D 2 be two `-polynomial-like mappings, such that the critical points of R k coincide, this point c 2 D 0 1 \ D 0 and is a `-multiple critical point of both R 2 1 and R 2 Moreover, we assume that the following conditions hold: 1 R 1 (z) = R 2 (z) whenever the both parts are dened, so that R 1 and R 2 are extensions of a map R

9 Local connectivity of the Julia set of real polynomials 9 2 For k = 1; 2, all topological discs D k ; Dk; 0 : : : ; Dk i line R and satisfy R k (z) = R k (z) are symmetric wrt the real 3 Denoting I k = D k \ R and I j k = D j k \ R, one has I 2 I 1, I j 2 I j 1, and, for j = 1; : : : ; i, the (real) map R k : I j k! I k is one-to-one Under these conditions, the Julia sets of R 1 and R 2 coincide If, additionally, c 2 R lies in the Julia set of R 1 (and, hence of R 2 ), then there exists a component of a preimage R?n 2 (D 2 ), which contains c and is contained in D 1 3 Local connectivity of the Julia set Let us now prove that the complex bounds from Theorems A-C imply that the Julia set is locally connected To do this we use the Yoccoz partition If f is non-renormalizable then consider the Yoccoz partition generated by taking preimages of the external ray through the orientation reversing xed point of f and some equipotential If f is renormalizable start with a region bounded by external rays through the endpoints of a central renormalization interval and some equipotential; the preimages of this region we again call a Yoccoz partition Since intersection of these pieces with the Julia set is connected, it suces to show that the diameter of the pieces tends to zero Proposition 31 Let G(j): [ i i (j)! (j) be a sequence of polynomial-like mappings associated to a real polynomial f(z) = z` + c 1 with!(c) minimal such that the critical point c = (j) does not escape the domain of G(j) under iterations of G(j) and (j) is based on the intersection of Yoccoz puzzle piece with the real line (As before, we assume G(j): 0 (j)! (j) is `-to-one, and that each other map G: i (j)! (j) is an isomorphism If f is innitely renormalizable, then G(j) are polynomial-like in the sense of [DH]) Assume moreover that when f i (c) 2 0 (j) then f i (c) is an iterate of c under G(j) (we call this the rst return condition); the modulus of the annuli (j) n 0 (j) is uniformly bounded away from zero; the diameter of (j) tends to zero as j! 1 Then the Julia set of f is locally connected Remark 31 The proof of this proposition also works for a quasi-polynomial-like mapping For a denition of this notion see the denition of Proposition 41 Proof: Let us rst observe that we may assume that (j) also satises the rst return condition Indeed, consider the rst return map of G(j) to 0 (j) This is again a

10 10 Genadi Levin and Sebastian van Strien polynomial-like mapping ~G(j): [ i ~ i (j)! 0 (j) Obviously, if f i (c) 2 0 (j) then f i (c) is an iterate of c under ~G(j) In addition, the modulus of 0 (j) n ~ 0 (j) is equal to the modulus of 1` mod ((j) n 0 (j)) So we may replace G(j) by the rst return map and therefore in the remainder of the proof we can and will assume that the rst return condition even holds on (j) Now we use arguments similar to those in [Ji1] If z is in the Julia set but!(z) does not hit a critical piece then the Julia set is locally connected at z because of the contraction principle So choose a point z from the Julia set of f so that!(z) hits every critical piece Fix for the moment j and let P (j) be an open piece of the Yoccoz puzzle based on (j) \ R Extend G?1 to the slit region C (j)\r There exists a large integer N such that the full preimage G(j)?N (P (j)) is inside the domain of denition [ i i (j) of G, see Proposition 22 Note that G?N (P (j)) consists of nitely many (open) Yoccoz pieces Let us consider the pieces of G?N (P (j)) inside the central domain 0 (j), ie, P 0 (j) = G(j)?N (P (j)) \ 0 (j): Since!(z) hits every critical piece, there exists a minimal k = k(j) such that f k (z) 2 P 0 (j) In particular, the point f k (z) belongs to one of the Yoccoz pieces inside 0 (j) Let B j be the branch of f?k which maps a neighborhood of f k (z) to a neighbourhood of z Let us drop j in the notation for now Claim 1 The map B extends to a holomorphic map on Proof of the claim Assume the contrary We then get that for some minimal r < k that f?r () (along the same orbit) meets the critical value c 1 This means that the branch f?r follows the points c r+1 = f r (c 1 ) 2, c r = f r?1 (c 1 ),, c 2 = f(c 1 ), c 1 Among these iterations of c 1, let us mark all those c j1 ; c j2 ; : : : ; c jm, where j 1 < j 2 < : : : < j m, which hit the domain Because of the rst return assumption there exists integers k(1) < k(2) < : : : such that c j1 = G k(1) (c), c j2 = G k(2)?k(1) (c j1 ) = G k(2) (c),, c jm = G k(m) (c) It follows, that f?r = f?(s?1) G?(k(m)?1), where f?(s?1) is the branch from V to ^U corresponding to the restriction of G on 0 (so Gj 0 = f s?1 f) Hence, f?(r+1) () G?r(m) () 0 and f k?r?1 (z) 2 f?(r+1) (Pn) 0 = (Gj 0)?1 G?k(m)+1 (Pn) 0 Pn 0 This contradicts the minimality of k and proves the claim Let P j (z) = B j (P (j)) We want to show that the Euclidean diameters of P j (z) tends to zero as j! 1 For this, let us consider a domain Mj 0 (j) bounded by a core curve of the annulus (j) n 0 (j) Then max y2@m 0 n jf k (z)? yj= min y2@m 0 n jf k (z)? yj C(m), n = 1; 2; : : : (see eg [McM]) Dene E j = B j (Mj) 0 Since the modulus of the annulus (j) n Mj 0 is m=2, by Koebe's distortion theorem, max y2@ej jz? yj= min y2@ej jz? yj C 1 (j), j = 1; 2; : : : If we assume by contradiction that diamp j (z) d > 0 for j = 1; 2; : : :, then min y2@ej jz? yj d=2c 1 = r > 0, ie, the disc D z (r) E j Hence, f k (D z (r)) Mj, 0 for j! 1 and k(j)! 1 This is a contradiction with the nonnormality of the family f n at z 2 J(f) Thus, T j>0 P j (z) = fzg and so J(f) is again locally connected at z tu If f(z) = z` + c 1 is real and!(c) is minimal then by Theorems A-C the conditions of the previous result are satised and local connectivity of the Julia set follows in this

11 Local connectivity of the Julia set of real polynomials 11 case So let us turn to the case that!(c) is not minimal Proposition 32 Assume that f(z) = z` + c 1 is real and!(c) is not minimal Then J(f) is locally connected and zero Lebesgue measure In particular, the Main Theorem holds in this case Proof: If!(c) is not minimal then it contains a point x whose forward orbit avoids some critical piece P N (here we use that the traces of the critical pieces on the real line tend to zero in diameter because the map is real and so preimages of the xed point accumulate onto the critical point) In particular, this forward orbit lies in a hyperbolic set Therefore the Yoccoz puzzle-pieces P n (x) containing x shrink down in diameter to zero The puzzle-pieces P n (x) are mapped by some iterates of f onto a xed critical Yoccoz piece P N It follows that if!(c) is not minimal then there is a xed critical Yoccoz piece P N and a sequence of critical pieces P nk with n k! 1 so that each map f nk?n : P nk! P N is `-covering Let a be the xed point with more than one external argument (ie, in the real case the orientation reversing xed point) We will distinguish two subcases depending on whether or not the xed point a is in!(c) Let us rst assume that a =2!(c) Then there exists a xed neighbourhood U of P N such that U np N is free from the postcritical set!(c) (all Yoccoz pieces are closed, and the intersection of P N with J consists of nitely many preimages of a) Hence the extension of the map f nk?n : P nk! P N to a map onto U is still ` to one Hence there is a neighbourhood U k of P nk such that U k n P nk is an annulus whose modulus which does not depend on k and which does not contain any iterates of c The pieces P nk have what is called the unbranched property and as in the proof of Proposition 31 one concludes that J(f) is locally connected in this case So consider the case that a 2!(c) Again we want to construct puzzle pieces with the unbranched property Let a 1 be the unique real preimage of a dierent from a Since f is not renormalizable, there is a sequence a n tending to a, so that f(a n+1 ) = a n ; n = 1; 2; : : :, and, moreover, a n+1 is the unique real preimage of a n by the branch of f?1 xing a (all a n are inside of [c 2 ; c 1 ], of course) Since the multiplier of a is negative, a 3 ; a 5 ; : : : are all between a 1 and a while a 2 ; a 4 ; : : : are on the other side of a Fix some large n 0 and let c k be the rst visit inside [a n0 ; a] by the orbit of c Let n n 0 be maximal so that c k 2 [a n ; a] There are two cases to distinguish 1 n n Then c k 2 [a; a n ] is the rst visit inside I n = [a; a n?2 ] This means that we can pull-back univalently I n to c 1 by an inverse branch of the map f k?1 Moreover, f n?3 (I n ) = [a 1 ; a] is the rst cover of c (from the iterates of I n ) and f n?3 (c k ) 2 f n?3 ([a n ; a n+2 ]) = [a 3 ; a 5 ], which lies properly inside [a 1 ; a] Hence there exists critical pieces P n Q n which are mapped by f k?1+n?3 onto the puzzle pieces based on [a 3 ; a 5 ] respectively [a 1 ; a] (as a ` to one mapping with a unique branch point) Hence the modulus of Q n np n is independent on n Moreover, by construction if f kn (x) is the rst iterate of x inside P n then there a univalent branch of f?kn from Q n to a neighbourhood of x

12 12 Genadi Levin and Sebastian van Strien 2 n = n 0 If I n = [a n ; a], then f n?1 (I n ) = [a 1 ; a], but this time f n?1 (c k ) = c k+n?1 can be any point of [a 1 ; a 3 ] So, if it is close to a 1, we don't have a safe space The idea is to take a small portion around a 1 (containing c k+n?1 ) and continue to iterate until the rst hit of c Let b 3 ; b 5 ; : : : be symmetric to a 3 ; a 5 ; : : : with respect to c, so that f: [a 1 ; b 2i+1 ]! [a; a 2i ] is one to one, i = 1; 2; : : : If c k+n?1 is far from a 1, more precisely, if it is in [a 3 ; b 5 ], we have a constant safe space (the latter interval is inside the interior of [a; a 1 ]) So let c k+n?1 be in [b 5 ; a 1 ] Let j > 1 be maximal so that c k+n?1 2 [b 2j+1 ; a 1 ], ie, c k+n?1 2 [b 2j+1 ; b 2j+3 ] First, let us pull the interval [b 2j?1 ; a 1 ] back to c k : we nd an interval I n = [a n ; a ] which contains c k and so that f n?1 : I n! [b 2j?1 ; a 1 ] is one to one Again, c k is the rst visit of I n, hence, we can pull back univalently I n to around c 1 On the other hand, f[b 2j?1 ; a 1 ] = [a; a 2j?2 ], and further f 2j?3 [a; a 2j?2 ] = [a; a 1 ] covers c rst time Thus f 2j?2 [b 2j?1 ; a 1 ] = [a; a 1 ] cover c rst time But now already f 2j?2 (c k+n?1 ) 2 f 2j?2 ([b 2j+1 ; b 2j+3 ]) = f 2j?3 ([a 2j+2 ; a 2j ]) = [a 5 ; a 3 ] which lies inside the interior of [a 1 ; a] Again this produces puzzle pieces around c with the unbranched property As in Proposition 31 the local connectivity of J(f) again follows Let us now prove that J(f) has zero Lebesgue measure in the non-minimal case Since the set of points X which do not enter some critical piece of f, is uniformly expanding (with respect to some convenient metric), X has zero Lebesgue measure So if J(f) has positive Lebesgue measure, then there exists a Lebesgue density point z of J(f) with the property that!(z) 3 c Now take some sequence of critical puzzle pieces P nk with the unbranched property and and so that a) f n k maps P nk as a ` to one onto some xed puzzle piece P N and in fact, there exists neighbourhoods A k and U of these pieces such that f n k: A k! U is ` to one; b) if f m k(x) is the rst entry of the orbit of x in A k then f?m k extends univalently from A k to a neighbourhood of x As we saw above such puzzle pieces exist Let m k be the smallest integer such that f m k(z) 2 P nk So the puzzle piece P nk +m k (z) is mapped by f m k onto P nk and there exists a neighbourhood V k of P nk +m k (z) which is mapped univalently onto A k In particular, f n k?n +m k : V k! U is still ` to one Since P nk +m k (z) shrinks to zero in diameter (by the local connectivity of J(f)), and since z is a density point of J(f) it follows that the relative density of J(f) in P N is arbitrarily close to one Since J(f) is closed, this implies that J(f) P N which is impossible tu 4 Absence of invariant lineelds In this section we shall prove Theorem 41 Any polynomial f(z) = z` + c 1, where ` is even, and c 1 is real, does not admit an invariant line eld on its Julia set This result lls out the gap, which was left after McMullen [McM] proved the above theorem for all real innitely renormalizable polynomials of the above form So, we can

13 Local connectivity of the Julia set of real polynomials 13 restrict ourselves to the case of a non-renormalizable (or nitely often renormalizable) real polynomial Note that for ` = 2 the above theorem follows from Yoccoz's result [Y] on the local connectivity of the Mandelbrot set at parameters corresponding to non-renormalizable maps combined with McMullen's result mentioned above Let us start with the case that!(c) is non-minimal In that case the Julia set of f(z) = z` + c 1 has Lebesgue measure zero, and therefore certainly does not carry a measurable invariant lineeld, see Proposition 32 In fact, in that case we even have a stronger result: Theorem 42 Assume that f(z) = z` +c 0 1 is real and that!(c0 1) is non-minimal Then the Mandelbrot set M = fc 1 ; the Julia set of z 7! z` + c 1 is connected g is locally connected at c 0 1 Proof: Since f is real and!(c 0 1) is not minimal, f is only nitely renormalizable In fact, one can assume that f is not renormalizable Hence, it suces to show that any map g of the form z 7! z` + ^c 0 1 with the same combinatorics is conformally conjugate and therefore the same (As usual, by the combinatorics we mean the lamination given by the rational external rays) To see that such a conjugacy exists, rst notice that there is a shrinking sequence of critical pieces P n so that any such piece P n is mapped by some iterate f kn onto a xed critical piece P N in `-to-one fashion Moreover, Claim 1: There exists a puzzle piece P N and an annular neighbourhood A N consisting of nitely many puzzle pieces, a sequence of critical puzzle pieces P ni and integers k i so that f k i?1 maps a neighbourhood of f(p ni ) dieomorphically onto P N [ A and f k i (c) 2 P N n A Proof of Claim 1: To prove the ideas and notation from the proof of Proposition 32 If a =2!(c) then there exists a neighbourhood N which is disjoint from!(c) and so the above assertion is clear If a 2!(c) then let P N be the puzzle piece based on [a 3 ; a 5 ] and let U be the puzzle piece based on [a 1 ; a] If n n in the proof of Proposition 32 then there exists a critical puzzle piece P n such that f n?3+k?1 : f(p n )! P N is a dieomorphism Moreover, there exists a dieomorphic extension of this map onto U P N Now in this case f n?3+k (c) can lie anywhere in [a 3 ; a 5 ] and in particular close to the boundary of P N But if it lies to close to one of the boundary points of the interval [a 3 ; a 5 ] then we have to repeat the argument and proceed as in the second case of the proof (taking things to large scale again) If n = n 0 then we proceed as in case 2 to get an surrounding the boundary of P N (ie with inner boundary equal to P N ) To get an annulus which N we have to repeat the argument again This completes the proof of Claim 1

14 14 Genadi Levin and Sebastian van Strien From the previous claim it follows in particular that the distortion of f kn?1 : f(p n )! P N is uniformly bounded because of the denite extension to P n [ A Because P N is a given nite puzzle piece, these properties of f are also inherited by the polynomial g (the modulus of the corresponding annulus could be quite a lot smaller, but still is positive since all the boundary of the set A is dened by external rays) Hence the property stated above carry over to g since it has the same combinatorics as f In particular, any such polynomial g has locally connected Julia set Hence, there is a natural conjugacy h between f and g, which is homeomorphism and conformal outside the Julia set of f Let us prove h is quasiconformal on the Julia set (since the Julia set has measure zero as we have shown in Proposition 32, it will imply that h is a conformal conjugacy) To prove quasiconformality, we approximate h by a sequence of K-quasiconformal maps h n, where K is the same for all n, and h n tends to h uniformly on a neighbourhood of the Julia set (This idea is used in [GS1] and also in [Ka]) Fixing n, we shall now show how to construct a K-quasiconformal map h n from P n onto h(p n ), which is equal to h on the boundary and with K an absolute constant First we should notice that the point d(f) = f Nn (c) belongs to the xed domain D(f) = P N n A which lies properly inside P N, and the point d(g) = g Nn (c) belongs to the xed domain D(g) = h(p N ) n h(a), which also is a proper inside h(p N ) There exists an absolute K (dilatation) such that for any point d(f) in D(f), and for any point d(g) in D(g), there exists a K-quasiconformal map H from P N onto h(p N ) such that it coincides with h on the boundary; H takes the point d(f) to the point d(g), and H is, say, conformal in a disc around d(f) of a xed radius Therefore the map g?nn H f Nn from P n to h(p n ) is a well dened K-quasiconformal homeomorphism, and is equal to h on the boundary Here K is universal In the part I of the plane where h n is not yet dened, we set h n = h The map h n is a homeomorphism because h n = h on the boundaries of all these pieces The remainder I consists of the set R of points in the Julia set, which are not in the above univalent preimages, together with open sets outside the Julia set So, it is enough to show h n is K-quasiconformal (with the same K) on R This follows from the fact that R is removable for any quasiconformal map More exactly, R is removable for extremal lengths (which is the same) To see this, we use the following fundamental theorem of L Ahlfors and A Beurling, see [AB] R is removable in the above sense, if R has absolute measure zero But Claim 2: R does have absolute measure zero: Proof of Claim 2: By denition, R is the set of points in the Julia set whose forward orbit avoid a denite (open) central piece P of the Yoccoz puzzle In order to nd suitable annuli around points of R, it is convenient to consider a non-standard Yoccoz puzzle So pick a periodic orbit of f, which contains a point inside P, and dene a `non-standard' Yoccoz puzzle corresponding to the external rays associated to this

15 Local connectivity of the Julia set of real polynomials 15 periodic orbit and the equipotential which was used for the original Yoccoz puzzle Let Y 0 ; Y 1 ; : : : ; Y m be the corresponding (open) pieces on some level of the new Yoccoz puzzle, such that the central piece Y 0 lies inside P It is easy to see that given the piece P, such level of the new structrure exists Notice that R is a compact subset of the open set [ i>0 Y i Thus, we can associate to each Y i with i > 0, a narrow annulus A i inside Y i with outer boundary the boundary of Y i such that R is contained in [ i>0 (Y i n A i ) Choose all these annuli A i of the same positive modulus m Consider all branches of f?1 corresponding to these pieces Y i, i > 0 The images of Y i under these inverse branches lie again inside some set Y j because of the Markov property of puzzles These inverse branches are univalent since we do not consider Y 0 A composition of j such branches gives some set Y j in [ i>0 Y i (we call this the j-th level) Since all compositions are univalent, each component Y of such a set Y j contains an annulus of modulus m whose outer boundary coincides with the boundary of Y and which does not intersects R Hence there is a level l 1, and nitely many pieces of this level, which lie in [ i>0 Y i, and such that their union contains R It gives the rst set of annuli enclosing R For the same reason there is a level l 2 > l 1, and nitely many pieces of this level, which lie inside the chosen pieces of level l 1, and again contain the compact R It gives the second set of annuli enclosing R which, in addition, are contained properly inside the rst set of annuli And so on To prove that R has absolute measure zero, we argue as in Section 2 of [Mil] (the situation is invariant under maps which are univalent outside R) Thus we have proved that R is a Cantor set with absolute measure zero and thus the proof of Claim 2 is completed Thus, R is removable and therefore h n is a K-quasiconformal homeomorphism Since P n shrinks to c, the sequence h n tends to the homeomorphism h uniformly It follows that h is a K-quasiconformal homeomorphism tu To prove the result in the case that!(c) is minimal, let us start with the denition of a quasi-polynomial-like mapping, which is a convenient generalization of the notion of (generalized) polynomial-like mapping [DH], [LM] This generalization is useful because sometimes it is easier to construct a quasi-polynomial-like mapping then a polynomial-like mapping, see Theorem B and the Section 14 A QPL-mapping (quasi-polynomial-like mapping) with non-escaping critical point c is a nite collection of maps (G k : k! ; k = 0; 1; : : : ; i) where 0 ; 1 ; : : : ; i ; are open topological discs (we do not require the discs k to be disjoint as is done in for example [LM]) with k for k = 0; 1; : : : ; i; the map G 0 : 0! is a `-covering with the unique critical point c 2 0 ; for every k = 1; : : : ; i, the map G k : k! is a conformal isomorphism;

16 16 Genadi Levin and Sebastian van Strien dene the map G: i[ ( j n [ k6=j k )! j=0 such that G = G j on j n [ k6=j k, j = 0; 1; 2 : : : ; i; then all iterates G n (c), n = 0; 1; 2 : : : of the critical point c under this map G are well dened Remark 1 Given a QPL-mapping with non-escaping critical point, we can also dene its Julia set as the set of all points x 2, for which all iterates G n (x); n = 0; 1; : : : are well dened (ie, are in the set [ i j=0( j n [ k6=j k )) In particular, the closure of the orbit fg n (c)g 1 n=0 of the critical point c belongs to the Julia set (by condition (4) from the denition of a QPL-mapping) So, the Julia set is a non-empty compact forward invariant under the map G Proposition 41 Let (G k : k! ; k = 0; 1; : : : ; i) be a QPL-mapping with nonescaping critical point associated to a polynomial f(z) = z` + c 1 (ie, each G k is an iterate of f, and the critical point of the QPL-mapping is c = 0) Assume there exists a topological disc 0 containing the central domain 0 such that, if f r (c) 2 0 for some r > 0, then f r (c) is an iterate of c under the QPL-mapping Let x 2 J(f) and f k (x) 2 0 where k 0 is minimal Then a branch F of f?k has a univalent extension from a neighbourhood of f k (x) to the domain 0 Proof: See Claim 1 in the proof of Proposition 31 tu Given QPL-mapping (G k : k! ; k = 0; 1; : : : ; i) with non-escaping critical point c, we dene its rst return map to the central domain 0 as follows For every point x 2 0 from the!-limit set! G (c) of the critical point, such that the iterates G(x); : : : ; G m (x) are dened, G m (x) 2 0, and m = m(x) > 0 is minimal and nite with this property, the pull-back (x) of 0 by the map G along the points x; G(x); : : : ; G m (x) is uniquely determined The collection of maps fg m(x) : (x)! 0 g is the desired rst return map This denition makes sense because of Proposition 42 Let (G k : k! ; k = 0; 1; : : : ; i) be a QPL-mapping with nonescaping `-multiple critical point c, such that the set! G (c) is minimal Then: (A) the rst return map to 0 is again a QPL-mapping (G 0 k: 0 k! 0 ; k = 0; 1; : : : ; i 0 ) with non-escaping critical point c and such that the!-limit set of c under the corresponding map G 0 is minimal (B) the central (branched) map G 0 0: 0 0! 0 extends to an `-fold covering map eg 0 0: e 0 0! with unique critical point c, so that 0 0 e and mod ( e 0 0 n 0 0) 1` mod ( n 0):

17 Local connectivity of the Julia set of real polynomials 17 Proof: Let us divide the proof into steps 1 For any point x 2 0 \! G (c), the rst return time m = m(x) > 0 to 0 is nite, since the set! G (c) is minimal Moreover, the map G?m extends to univalent maps from to itself, except for the domain (c), and, for the Euclidean diameter of the image we have diam G?m () < A m ; where A > 0 and 0 < < 1 depend only on the initial conguration of the domains 1 ; : : : ; i ; This is because G?m :! is a composition of m uniform contractions G?1 k :! k, k = 1; : : : ; i (in the hyperbolic metric of ) 2 Assume that the number of dierent domains (x) corresponding to the points x as above is innite Then, by Step 1, a sequence of such domains tends to some point a Since this point belongs to! G (c), we get, for some x, that (x) is a proper inside of (a) This is a contradiction Thus, the number of dierent domains (x) is nite In other words, the rst return map is a nite collection of maps (G 0 k: 0 k! 0 ; k = 0; 1; : : : ; i 0 ), which satises conditions (1), (2), and (3) of the denition of QPL-mapping 3 The `-fold covering G 0 0: 0 0! 0 has an inverse, which is a composition of a univalent map F from Step 1 and the map G?1 0 Since F extends univalently to, it proves part (B) of the theorem 4 To end the proof of the part (A), we need to check condition (4) of the denition of a QPL map So, by contradiction, let us assume that G n (c) is contained in two dierent domains 0 k and 0 j It means that some further iterate of c under G belongs to two dierent domains of the given QPL-mapping, which contradicts the denition of a QPL map tu We shall derive Theorem 41 from the following statement, which could be applied to complex polynomial maps as well Proposition 43 Let G(n) = (G k (n): k (n)! (n); i = 0; 1; : : : ; i(n)) be a sequence of QPL-mappings associated to a polynomial of the form f(z) = z` + c 1, such that: 1 the!-limit set of the critical point c = 0 under the map f is minimal; 2 if f r (c) 2 0 (n), then f r (c) is an iterate of c under the map G(n); 3 the sequence of moduli of the annuli (n) n 0 (n) is uniformly bigger than a positive constant m; 4 the sequence of the domains (n) shrinks to the critical point c Then the Julia set of f carries no invariant line eld

18 18 Genadi Levin and Sebastian van Strien Proof: We will follow the main idea of Theorem 103 in [McM] (absence of invariant line eld for innitely renormalizable quadratic polynomial with complex apriori bounds) Let us rst outline the dierences with the proof in [McM] First, our renormalizations are generalized (even quasi-) polynomial-like maps, so that the small Julia sets are not connected, and the number of the components in the domain of denition of these maps can increase To overcome this problem, we consider the dynamics only on the central domains The second problem is that the central domain 0 (n) can become smaller and smaller compared to the range (n), so that the range of the limit dynamics can be the whole plane (after rescaling 0 (n) to a denite size) To avoid this, we shall rescale 0 (n) and (n) by dierent factors The third dierence is that in our setting the critical value of the `limit dynamics' can escape to the boundary of the range For this, we extend the dynamics passing to the rst return maps Fourthly, in [McM] a contradiction against the existence of a measurable invariant lineeld (dened on a set E) is obtained through a univalent map from the range of the renormalization to a neighbourhood of a point of density of the set E In our setting we cannot argue like that, and instead we use two consecutive rst return maps and apply each time Proposition 41 So let us start with the proof of Proposition 43: 1 Let us x for a moment a QPL-mapping G = G(n), and consider the rst return to its central domain By Proposition 42, we obtain in this way another QPL-mapping G 0 = G 0 (n) = (G 0 k: 0 k! 0 ; k = 0; 1; : : : ; i 0 ) (we drop the index n) Its central branch G 0 0: 0 0! 0 extends to an `-covering G 0 0: e 0 0! (we keep the same notation for the extension), where e Hence, mod ( 0 n 0 0) > m 0 = m` : Observe also the following property of the new map If f r (c) 2 0, for some r > 0, then f r (c) is an iterate of the new QPL-mapping G 0 2 Now, let us replace the initial sequence of the QPL-mappings G(n) by the sequence of the rst return maps G 0 (n) replacing the notations as well (so forget about the initial sequence) Thus, we have a sequence of QPL-mappings G(n) = (G k (n): k (n)! (n); k = 0; 1; : : : ; i(n)), with the following properties: 21 each G(n) is associated with the polynomial f; 22 the!-limit set of c under G(n) is minimal; 23 each iterate of c under the polynomial f entering the domain (n) is, in fact, an iterate of c under the map G(n); 24 mod ((n) n 0 (n)) > m 0, for all n; 25 the domains (n) shrink to c 3 Let us assume by contradiction that f admits an invariant line eld, ie, there is a f-invariant Beltrami dierential supported on J(f) Then we can x a point x 2 J(f) of almost continuity of We may assume from the beginning that! f (x) contains c since the set of the points of J(f) without this property has Lebesgue measure zero