BIACCESSIBLILITY IN QUADRATIC JULIA SETS II: THE SIEGEL AND CREMER CASES SAEED ZAKERI Abstract. Let f be a quadratic polynomial which has an irrationally indifferent fixed 8 point α. Let z be a biaccessible point in the Julia set of f. Then: 9 9 In the Siegel case, the orbit of z must eventually hit the critical point of f. • 1 In the Cremer case, the orbit of z must eventually hit the fixed point α. n Sie•gel polynomials with biaccessible critical point certainly exist, but in the Cremer case it a is possible that biaccessible points can never exist. J As a corollary,we conclude that the set of biaccessible points in the Julia set of a Siegel 5 or Cremer quadratic polynomial has Brolin measure zero. 1 ] S §1. Introduction. Let f be a polynomial map of the complex plane C. A fixed point D z = f(z) is called indifferent if the multiplier λ = f′(z) has the form e2πiθ, where the rotation h. number θ belongs to R/Z. We call z irrationally indifferent if θ is irrational so that λ is on at the unit circle but not a root of unity. m Let z be anirrationallyindifferent fixed point off. When f isholomorphically linearizable [ about z, we call z a Siegel fixed point. On the other hand, when z is nonlinearizable, it is 1 called a Cremer fixed point. v In this paper we only consider quadratic polynomials. Such a polynomial, which we can 0 put in the normal form 5 1 1 f : z z2 +c, (1) 0 7→ 8 has two fixed points (1 √1 4c)/2 which are distinct if and only if c = 1/4. If c / [1/4, ), 9 ± − 6 ∈ ∞ / so that the two fixed points have distinct real parts, then by convention the fixed point which h is further to the left is called α and the other fixed point 1 α is called β. The corresponding t a − multipliers are λ = 2α and 2 λ = 2β, with λ < 2 λ . Evidently only the α-fixed point m − | | | − | can be indifferent. The critical value parameter c is then given by : v i c = λ(2 λ)/4. X − r a Therefore, the set of all quadratic polynomials which have an indifferent fixed point is a cardioidinthec-planeparametrized byλontheunit circle. Theset ofquadraticpolynomials with an irrationally indifferent fixed point is then a dense subset of this cardioid. We call a quadratic polynomial f in (1) Siegel or Cremer if the α-fixed point is irrationally indifferent and has the corresponding property. By the theorem of Brjuno-Yoccoz [Yo], f is a Siegel polynomial if and only if θ = 1 logf′(α) satisfies the Brjuno condition: 2πi ∞ logq n+1 X < + , (2) q ∞ n n=1 where the q appear as the denominators of the rational approximations coming from the n continued fraction expansion of θ. 1 Stony Brook IMS Preprint #1998/1c January 1998 2 SAEED ZAKERI Recall that the filled Julia set of f is K(f) = z C : The orbit f◦n(z) is bounded n≥0 { ∈ { } } and the Julia set of f is the topological boundary of the filled Julia set: J(f) = ∂K(f). Both sets are nonempty, compact, connected and the filled Julia set is full, i.e., the com- plement C K(f) is connected. Every connected component of the interior of K(f) is a \ topological disk called a bounded Fatou component of f. In the Siegel case, the component S of the interior of K(f) containing the fixed point α is called the Siegel disk of f on which the action of f is holomorphically conjugate to the rigid rotation z e2πiθz. 7→ Since f(z) = f( z) by (1), the Julia set J(f) is invariant under the 180◦ rotation τ : z − 7→ z. If U is an open Jordan domain in the plane such that U τ(U) = , it follows that f − ∩ ∅ is univalent in some Jordan domain V containing the closure U. According to Sullivan, every bounded Fatou component must eventually map to the im- mediate basin of attraction of an attracting periodic point, or to an attracting petal for a parabolic periodic point, or to a periodic Siegel disk for f (see for example [Mi1]). On the other hand, by [Do1] a polynomial of degree d 2 can have at most d 1 nonrepelling ≥ − periodic orbits. It follows that in the Siegel case, every bounded Fatoucomponent eventually maps to the Siegel disk S centered at α. In the Cremer case, however, we simply conclude that K(f) has no interior, so that K(f) = J(f). §2. Accessibility. Given a quadratic polynomial f as in (1) with connected Julia set, there exists a unique conformal isomorphism ϕ : C K(f) C D, \ → \ called theBo¨ttcher map, withϕ( ) = andϕ′( ) > 0, which conjugatesf to thesquaring ∞ ∞ ∞ map: ϕ(f(z)) = (ϕ(z))2. (3) The ϕ-preimages of the radial lines and circles centered at the origin are called the external rays and equipotentials of K(f), respectively. The external ray R , by definition, is t ϕ−1 re2πit : r > 1 , { } where t R/Z is called the angle of the ray. From (3) it follows that ∈ f(R ) = R . t 2t (mod 1) More generally, let us consider an arbitrary compact, connected, full set K C, and let ϕ : C K C D be the unique conformal isomorphism with ϕ ( ) = and⊂ϕ′ ( ) > 0 K \ → \ K ∞ ∞ K ∞ given by the Riemann Mapping Theorem. We can define the external rays R for K in a t similar way. We say that R lands at p ∂K if lim ϕ−1(re2πit) = p. A point p ∂K is called accessible if there existts a simple a∈rc in C K wrh→ic1h sKtarts at infinity and termi∈nates at \ p. According to a theorem of Lindel¨of (see for example [Ru], p. 259), p is accessible exactly when there exists an external ray landing at p. We call p biaccessible if it is accessible through at least two distinct external rays. By a theorem of F. and M. Riesz [Mi1], K p \{ } is disconnected whenever p is biaccessible. It is interesting that the converse is also true. More precisely, if there are at least n > 1 connected components of K p , then at least n \{ } distinct external rays land at p (see for example [Mc], p. 85). BIACCESSIBLILITY IN QUADRATIC JULIA SETS II: THE SIEGEL AND CREMER CASES 3 In the case K = K(f) is the filled Julia set of a quadratic polynomial f in (1), note that τ(R ) = R . Hence if R lands at some p J(f), then R lands at τ(p) = p. t t+1/2 t t+1/2 ∈ − §3. Arithmetical conditions. As it is suggested by the theorem of Brjuno-Yoccoz, the behavior of the orbits near the indifferent fixed point is intimately connected to the arith- metical properties of the rotation number θ. There are certain classes of irrational numbers which are of special interest in holomorphic dynamics and in this paper we will be working with some of them. Let 1 θ = 1 a + 1 1 a + 2 ... be the continued fraction expansion of θ, where all the a are positive integers, and i p 1 n = q 1 n a + 1 1 a + 2 ... + 1 a n be the n-th rational approximation of θ. We say that θ is of constant type (we write θ ) if sup a < + . • ∈ CT n n ∞ θ is Diophantine (we write θ ) if there exist positive constants C and ν such that • ∈ D for every rational number 0 p/q < 1, we have θ p/q > C/qν. This condition is ≤ | − | equivalent to sup (logq /logq ) < + . n n+1 n ∞ θ is of Yoccoz type (we write θ ) if every analyticcircle diffeomorphism with rotation • ∈ H number θ is analytically linearizable. (An explicit arithmetical description of is given by Yoccoz although H it is not easy to explain; see [Yo].) A closely related condition, which we denote by ′, is defined as follows: θ ′ H ∈ H if and only if every analytic circle diffeomorphism with rotation number θ, with no periodic orbit in some neighborhood of the circle, is analytically linearizable [PM1]. θ is of Brjuno type (we write θ ) if θ satisfies the condition (2). • ∈ B We have the proper inclusions ′ and . It is not hard to show that , hence , ′ and , are setsHof⊂fulHl measuCreTin⊂RD/Z⊂. H ⊂ B D H H B §4. Basic results. Very little is known about the topology of the Julia set of f in the Siegel or Cremer case or the dynamics of f on its Julia set. The following theorem summarizes the basic results in the Cremer case: Theorem 1. Let f in (1) be a Cremer quadratic polynomial, so that θ / . Then ∈ B (a) The Julia set J(f) cannot be locally-connected [Su]. (b) Every neighborhood of the Cremer fixed point α contains infinitely many repelling peri- odic orbits of f [Yo]. (c) The critical point 0 is recurrent, i.e., it belongs to the closure of its orbit f◦n(0) n>0 { } [Ma]. (d) The critical point 0 is not accessible from C J(f) [Ki]. \ 4 SAEED ZAKERI See also [Sø] for the so-called “Douady’s nonlanding Theorem” which partially explains why the Julia set of a generic Cremer quadratic polynomial fails to be locally-connected. In the Siegel case, we know a little bit more, but still the situation is far from being fully understood. Theorem 2. Let f in (1) be a Siegel quadratic polynomial, so that θ . Let S denote the ∈ B Siegel disk of f. Then (a) If θ , then the boundary ∂S is a quasicircle which contains the critical point 0 ∈ CT [Do2]. The Julia set J(f) is locally-connected and has measure zero [Pe]. (b) If θ , then 0 ∂S [He1]. ∈ H ∈ (c) For some rotation numbers θ , the entire orbit of 0 is disjoint from ∂S [He2]. ∈ B\H In this case, J(f) cannot be locally-connected [Do2]. (d) For every θ , the critical point 0 is recurrent. ∈ B Part (b) was proved by Herman for θ , but his proof works equally well for θ . We ∈ D ∈ H will include a very short proof for the latter case in section §5. The proof of part (d) goes as follows: If 0 ∂S, then by classical Fatou-Julia theory, every point in ∂S is in the closure ∈ of the orbit of 0 [Mi1], and recurrence follows. If 0 / ∂S and 0 is not recurrent, then by ∈ [Ma] the invariant set ∂S is expanding, i.e., there is a constant λ > 1 and a positive integer k such that (f◦k)′(z) > λ for all z ∂S. It follows that the same inequality holds over | | ∈ some neighborhood U of ∂S with a slightly smaller λ > 1. We may as well assume that 1 U S is invariant. Take a small disk V ⋐ U S. Since f◦k is holomorphically conjugate U∩S to∩the rigid rotation z e2πikθz, there exist∩s a sequence n| such that f◦knj converges j 7→ → ∞ uniformly to the identity map on V as j . But this is impossible since for all z V, (f◦knj)′(z) > λnj . → ∞ ∈ | | 1 → ∞ Comparing the two theorems, we notice that the Cremer case and the Siegel case with 0 / ∂S share many properties. This is a general philosophy which is partially explained ∈ by the theory of “hedgehogs” introduced recently by Perez-Marco [PM1] (see section §5 below). Inspired by this similarity, one expects the following to be true: Conjecture. Let f be a Siegel quadratic polynomial and 0 / ∂S. Then ∈ (i) Every neighborhood of ∂S contains infinitely many repelling periodic orbits of f. (ii) The critical point 0 is not accessible from C K(f). \ By an argument similar to [Ki], one can show that (i) implies (ii) (see also Proposition 3). §5. Hedgehogs. Let f be a Siegel or Cremer quadratic polynomial as in (1). Let U be a simply connected domain with compact closure which contains the closure of the Siegel disk S in the linearizable case, or the indifferent fixed point α in the nonlinearizable case. Suppose that f is univalent in a neighborhood of the closure U. Then there exists a set H = H with the following properties: U (i) α H U, ∈ ⊂ (ii) H is compact, connected and full, (iii) ∂H ∂U is nonempty, ∩ (iv) ∂H J(f), ⊂ BIACCESSIBLILITY IN QUADRATIC JULIA SETS II: THE SIEGEL AND CREMER CASES 5 (v) f(H) = H. NotethatH hasnonempty interiorifandonlyifαislinearizable. Inthiscaseourassumption that f is univalent on U implies that the critical point is off the boundary of the Siegel disk. Clearly H S. Such an⊃H is called a hedgehog for the restriction f : U C. See Fig. 1(a) for the U | → Cremer case and (b) for the Siegel case. (We would like to emphasize that the topology of a hedgehog is infinitely more complicated than anything we can possibly sketch!) The existence of such totally invariant sets is proved by Perez-Marco [PM1]. Note that in the Siegel case, one can get totally invariant sets H with the above properties (i)-(v) even if ∂U intersects the closure S. But in this case the existence of H is not hard to show because we can simply take H as S or a compact invariant piece with analytic boundary inside the Siegel disk (see Fig. 1(c) and (d)). U U U U H H S H H=S (a) (b) (c) (d) Figure 1. Hedgehogs turn out to be useful because of the following nice construction: Uniformize the complement C H by the Riemann map ψ : C H C D and consider the induced \ \ → \ map g = ψ f ψ−1 which is defined (by (v) above) and holomorphic in an open annulus z C : 1 ◦< z◦ < r . Use the Schwarz Reflection Principle to extend g to the annulus {z ∈C : r−1 <|z| < r}. The restriction of g to the unit circle T will then be a real-analytic {diff∈eomorphism|w|hose}rotation number is exactly 1 logf′(α) = θ R/Z (see [PM1]). This 2πi ∈ allows us to transfer results from the more developed theory of circle diffeomorphisms to the less explored theory of indifferent fixed points of holomorphic maps. Using the above construction, it is not hard to prove the following fact (see [PM2]): Proposition 1. Let p be a point in a hedgehog H which is biaccessible from outside of H. Then p ∂S in the Siegel case and p = α in the Cremer case. ∈ In fact, let us assume that we are in the Siegel case and p / ∂S. Then one can find a simple arc γ in C H which starts and terminates at p and does∈not encircle the indifferent \ fixed point α. Let D be the bounded connected component of C (H γ). Evidently D is \ ∪ disjoint from S. The topological disk D′ = ψ(D) is bounded by the simple arc ψ(γ) and an interval I on the unit circle. (The fact that ψ(γ) actually lands from both sides on the unit circle follows from general theory of conformal mappings; see for example [Po], page 29.) Since g has irrational rotation number on the unit circle T, for some integer N we have N g◦i(I) = T. By choosing γ close enough to H, we can assume that g,g◦2, ,g◦N are Si=0 ··· all defined on D′ and N g◦i(D′) contains an entire outer neighborhood of T. It follows Si=0 that N f◦i(D) covers an entire deleted neighborhood of H. Therefore, some iterate f◦i(D) Si=0 intersects ∂S. Since f◦i is univalent on D S, it follows that D ∂S = , which contradicts ∪ ∩ 6 ∅ 6 SAEED ZAKERI our assumption. The proof in the Cremer case is similar. The construction of the circle maps associated with hedgehogs as described above gives short proofs for some interesting facts. As the first example, we prove that there are no periodic points on ∂S when the critical point 0 is off this boundary. This fact will be used in the proof of Theorem 3 (see [PM1] for a general proof in the case of indifferent germs; the fact that we are working with polynomials makes the proof even shorter). First we need the following lemma: Lemma 1. Let f be a Siegel quadratic polynomial as in (1) whose critical point 0 is off the boundary ∂S of the Siegel disk. Then the closure S is full and f acts injectively on it. It is reasonable to speculate that the closure of any bounded Fatou component for a quadratic polynomial is full. This is known to be true except when the polynomial has a periodic Siegel disk S with the critical point on its boundary ∂S. In this case, we do not know if ∂S can separate the plane into more than two connected components (a so-called “Lakes of Wada” example in plane topology [H-Y]). Proof. Since f(z) = f( z) for all z, if f is not injective on S, there must be a pair of − symmetric points p and p = τ(p) in ∂S. Since J(f) has a 180◦ rotational symmetry, − f−1(S) = S τ(S). So p and p also belong to ∂(τ(S)). Consider the connected component ∪ − V of C (S τ(S)) which contains the critical point 0. Since V is open and ∂V J(f), it \ ∪ ⊂ follows from the Maximum Principle that V has to be a bounded Fatou component of f. This contradicts the fact that 0 J(f). ∈ Let us now assume that S is not full and let U be a bounded component of C S. Since \ ∂U ∂S J(f), it follows again from the Maximum Principle that U has to be a bounded ⊂ ⊂ Fatou component of f, hence it eventually maps to S, i.e., f◦n(U) = S for some n 1. ≥ Therefore f◦n−1(U) = τ(S). But the boundary of f◦n−1(U) is a subset of ∂S, which implies that the common boundary ∂S ∂(τ(S)) is nonempty. This contradicts the fact that f ∂S ∩ | is injective. Proposition 2. Let f be a Siegel quadratic polynomial whose critical point 0 is off the boundary ∂S. Then there are no periodic points on ∂S. Proof. By the above lemma S is full and f acts injectively on it, so we can find a Jordan domain U containing S such that f is univalent. Consider a hedgehog H = H for the |U U restriction f . Clearly H S. Suppose that there is a periodic point on ∂S which is U | ⊃ necessarily repelling. Then there exists a rational external ray R landing at this point, hence f◦n(R) = R for some n 1 (see for example [Mi1]). Consider the induced map ≥ g = ψ f ψ−1 as described above, and look at the arc γ = ψ(R). It is a standard fact that γ has ◦to l◦and at some point p T [Po] and g◦n(p) = p. But this contradicts the fact that ∈ the rotation number of g is irrational. In the second application, we prove Theorem 2(b): We want to show that θ implies ∈ H 0 ∂S. If not, by Lemma 1 S is full and f is univalent. Consider a Jordan domain ∈ |S U, a hedgehog H and the induced circle map g as in the above proof. Since the rotation U number of g belongs to , g is analytically linearizable. The linearization can be extended holomorphically to an anHnulus neighborhood of the unit circle T. Pulling this neighborhood BIACCESSIBLILITY IN QUADRATIC JULIA SETS II: THE SIEGEL AND CREMER CASES 7 back, we find a larger domain containing S on which f is linearizable, which contradicts the definition of a Siegel disk. As a final application, we prove the following: Proposition 3. Let f be a Siegel quadratic polynomial whose critical point 0 is off the boundary ∂S. If θ ′, then every neighborhood of ∂S contains infinitely many repelling periodic orbits. Hen∈ce,Hthe critical point 0 is not accessible from C K(f). \ Proof. Consider the hedgehog construction as in the proof of Proposition 2 or the above proof for Theorem 2(b). If there are no periodic orbits in some neighborhood of ∂S, it follows that g has no periodic orbit in some neighborhood of T either. Since the rotation number of g is θ ′, g has to be linearizable. Now we can get a contradiction as in the ∈ H above proof for Theorem 2(b). So every neighborhood of ∂S must contain infinitely many periodic orbits. The fact that this implies nonaccessibility of 0 follows easily by an argument similar to [Ki]. §6. Wakes. To see the behavior of rays near infinity, it will be convenient to add a circle at infinity T R/Z to the complex plane to obtain a closed disk c topologized in the ∞ natural way. W≃e denote the point lim re2πit on T simply by (cid:13)e2πit. The action of f r→∞ ∞ ∞· in (1) on the complex plane extends continuously to c by (cid:13) f( e2πit) = e4πit, (4) ∞· ∞· which is just the doubling map on T . Note that the symmetry f(z) = f( z) also extends ∞ − to c if we define e2πit = e2πi(t+1/2). (cid:13) −∞· ∞· Definition. Let f be a quadratic polynomial as in (1) with connected Julia set. Let z = α be a biaccessible point in J(f) with two distinct rays R and R′ landing on it. We call 6 (R,R′) a ray pair. By the Jordan Curve Theorem, R R′ z cuts the plane into two open ∪ ∪{ } topological disks. By the wake W of the ray pair (R,R′) we mean the connected component of C (R R′ z ) which does not contain the fixed point α. The other component is called \ ∪ ∪{ } the co-wake and it is denoted by Wˇ . Point z is called the root of W. The angle a(W) of the wake is just the (normalized) measure of W T . Clearly a(W)+a(Wˇ ) = 1 (see Fig. 2(a)). ∞ ∩ Since distinct external rays are disjoint, it follows that any two wakes with distinct roots are either disjoint or nested. In the following lemma we collect basic properties of wakes (compare with [G-M] or [Mi2]): Lemma 2. Let z J(f) be a biaccessible point, z = α, and let W be a wake with root z. ∈ 6 (a) If z = 0, then a(W) > 1/2 if and only if W contains the critical point 0. 6 (b) If a(W) = 1/2, then z must be the critical point 0. Conversely, if there is any ray R landing at 0, then R′ = τ(R) also lands at 0 and the two rays span a wake W with a(W) = 1/2. (c) Let a(W) < 1/2 and f(z) = α. Then f(W) is a wake or a co-wake with root f(z), 6 depending on whether α / W or α W. In either case, f : W f(W) is a − ∈ − ∈ → conformal isomorphism and a(f(W)) = 2a(W). Proof. Let W be the wake of a ray pair (R,R′). 8 SAEED ZAKERI (a) Let 0 W and a(W) < 1/2. Consider the symmetric region τ(W) whose angle is ∈ equal to a(W). W and τ(W) intersect since both contain 0 (see Fig. 2(b)). On the other hand, W τ(W) T = because a(W) < 1/2. Since W and τ(W) are both homeomorphic ∞ ∩ ∩ ∅ to closed disks, it follows that the ray pairs (R,R′) and (τ(R),τ(R′)) must intersect, which is a contradiction. Therefore a(W) > 1/2 if 0 W. ∈ ˇ On the other hand, let a(W) > 1/2. Then the angle of the co-wake W has to be less than ˇ 1/2, so by the above argument 0 / W, or 0 W. This proves (a). ∈ ∈ a(W) τ(R ) R W τ(R ) U z R R R z α z α W α * * * W 0 −α τ(R ) 0 −α -z 0 −α -z τ(R ) W R R (a) (b) (c) Figure 2. (b) If a(W) = 1/2, then R′ = τ(R). Hence z = τ(z) by continuity, which means z = 0. The converse is trivial. (c) If a(W) < 1/2, then the ray pairs (R,R′) and (τ(R),τ(R′)) cut the plane into simply connected domains W, τ(W) and an open set U which is either a simply connected domain or the disjoint union of two simply connected domains depending on whether z = 0 or z = 0. 6 By (a), 0 / W τ(W). Consider the ray pair (f(R),f(R′)) landing at f(z), and let W′ ∈ ∪ be the corresponding wake. The pull-back of W′ by f either consists of the disjoint union W τ(W) or the open set U (see Fig. 2(c)). In the first case, f maps W to W′ isomorphically ⊔ and α / W. In the second case, however, we must have α W, α τ(W), and both W − ∈ − ∈ ∈ and τ(W) map isomorphically to the co-wake Wˇ ′. The fact that a(f(W)) = 2a(W) simply follows from (4). §7. The main theorem. Now we are in a position to state and prove the main theorem of this paper: Theorem 3. Let f be a quadratic polynomial as in (1) which has an irrationally indifferent fixed point α. Let z be a biaccessible point in the Julia set of f. Then: In the Siegel case, the orbit of z must eventually hit the critical point 0. • In the Cremer case, the orbit of z must eventually hit the fixed point α. • (Compare [S-Z] where this same result for the Cremer case is proved by a somewhat different argument.) In the Siegel case, if the critical point 0 is accessible, then exactly two rays land there (see the proof of Lemma 2(b)). This happens, for example, when θ , since in this case ∈ CT by Theorem 2(a) the Julia set is locally-connected. On the other hand, for some rotation BIACCESSIBLILITY IN QUADRATIC JULIA SETS II: THE SIEGEL AND CREMER CASES 9 numbers θ ′, the critical point is not accessible so that there are no biaccessible ∈ B ∩ H points in the Julia set (see Corollary 1). In the Cremer case, if the fixed point α is accessible, then infinitely many rays land there. In fact, if R lands at α, then t is irrational and every R lands at α also. However, there t 2nt is no known example where one can decide whether α is accessible or not. Theorem 3 can be viewed from a more general perspective: Let f be any quadratic poly- nomial with connected Julia set J(f). There is a unique measure µ of maximal entropy log2, called the Brolin measure, which is supported on J(f). In fact, µ coincides with the harmonic measure induced by the radial limits of the inverse of the B¨ottcher map (see for example [E-L]). It is a standard fact that µ is f-invariant and ergodic. For z J(f), let ∈ v(z) denote the number of external rays which land at z. (In Milnor’s terminology [Mi2], this is called the valence of z.) For 0 n define J = z J(f) : v(z) = n . It fol- n ≤ ≤ ∞ { ∈ } lows from elementary plane topology that the union J is at most countable (see [Po], Sn≥3 n page 36). On the other hand, the fact that almost every external ray (with respect to the Lebesgue measure on R/Z) lands shows that µ(J ) = 0. Putting these two facts together, 0 we conclude that J(f) = J J up to a set of µ-measure zero. Note that v(f(z)) = v(z) 1 2 ∪ unless z is the critical point. Therefore, if we neglect the grand orbit of the critical point which has µ-measure zero, it follows that both J and J must be f-invariant subsets of the 1 2 Julia set. Ergodicity of µ then shows that up to a set of µ-measure zero, either J(f) = J or 1 J(f) = J . Forexample, inthe first part ofthis paper [Za], it isshown thatfor anyquadratic 2 polynomial f with locally-connected Julia set, J(f) = J must be the case unless f is the 1 Chebyshev map z z2 2. For this map the Julia set is the closed interval [ 2,2] and every 7→ − − point is the landing point of exactly two rays except for the endpoints 2 where unique ± rays land, so that J(f) = J . Theorem 3 in particular proves that if f has an irrationally 2 indifferent fixed point, then J = J up to a set of Brolin measure zero, thus covering some 1 non locally-connected cases. It is conjectured that J = J is true for every non-Chebyshev 1 quadratic polynomial. The proof of Theorem 3 is based on the following lemma: Lemma 3. Let f be a Siegel or Cremer quadratic polynomial as in (1). Assume that there exists a biaccessible point in J(f) whose orbit never hits the critical point 0 or the fixed point α. Then there exists a ray pair which separates α from 0. Proof. Let z J(f) be such a biaccessible point and (R,R′) be a ray pair which lands at ∈ z. Consider the associated wake W with root z. Since z = 0, we have a(W ) = 1/2 by 0 0 6 6 Lemma 2(b). If a(W ) > 1/2, then 0 W by Lemma 2(a) and (R,R′) separates α from 0. 0 0 ∈ Let us consider the case where a(W ) < 1/2. If α W , then (R,R′) must separate α 0 0 − ∈ − from 0 because by Lemma 2(a), 0 / W . It follows that the symmetric ray pair (τ(R),τ(R′)) 0 ∈ separates α from 0. If, however, α / W , then by Lemma 2(c), W = f(W ) is a wake with 0 1 0 − ∈ root z = f(z) with angle a(W ) = 2a(W ). 1 1 0 Now we can replace W by W in the above argument. If either a(W ) > 1/2 or a(W ) < 0 1 1 1 1/2 and α W , we can find a ray pair separating α from 0. Otherwise, we consider the 1 − ∈ new wake W = f(W ) with angle a(W ) = 22a(W ). Since each passage W W implies 2 1 2 0 i i+1 7→ doubling the angles, this process must stop at some stage, and this proves the lemma. Proof of Theorem 3. It will be more convenient to consider the Cremer case first. Suppose that the orbit of z never hits α. Since the critical point is not accessible by Theorem 1(d), 10 SAEED ZAKERI Lemma 3 gives us a ray pair (R,R′) landing at some point p J(f) which separates α ∈ from 0. Let W be the corresponding wake with root p and consider the co-wake Wˇ . The restriction of f to the closure of Wˇ is univalent since otherwise this closure would intersect the closure of the symmetric domain τ(Wˇ ), which is impossible since a(Wˇ ) < 1/2. To work with a Jordan domain in the plane we cut off Wˇ along an equipotential curve and call the resulting domain U (see Fig. 3(a)). R V W U 1 R W HV p W2 p2 p1 α p α p * n * W 0 n S 0 −α −α W R W W R (a) (b) Figure 3. Now we consider a hedgehog H for the restriction f : U C as given in section §5. U U | → Notethatpistheonlypoint oftheJuliaset ontheboundaryofU andthatH J(f)hasto U ⊂ intersect this boundary. Therefore, we simply have p H . Let us consider a slightly larger U ∈ Jordan domain V U with compact closure such that f is still univalent. The hedgehog H for the restric⊃tion f : V C has to contain p a|lVso and reach the boundary of V. V V | → Since p is biaccessible from outside of the Julia set, it follows that H p is disconnected. V \{ } Therefore, p is biaccessible from outside of H . This contradicts Proposition 1, and finishes V the proof of the theorem in the Cremer case. Let us now assume that we are in the Siegel case. If the orbit of z eventually hits the critical point 0, there is nothing to prove. Otherwise, since this orbit trivially cannot hit the fixed point α S, we are again in the situation of Lemma 3. Therefore, there exists a ray ∈ pair (R,R′) landing at a point p J(f) which separates α from 0. In particular the critical ∈ point 0 is off the boundary ∂S of the Siegel disk. Then the same argument as in the Cremer case with an application of Proposition 1 shows that p must belong to ∂S. As before, let W be the wake of the ray pair (R,R′), with root p. Then by construction W contains the critical point 0 while the co-wake Wˇ contains the Siegel disk S and has its boundary touching S only at p. The point p is not periodic by Proposition 2. Hence the successive images p = f◦n(p) ∂S are all contained in Wˇ for n 1. Therefore each wake n ∈ ≥ W corresponding to the ray pair (f◦n(R),f◦n(R′)), with root point p , is also contained n n in Wˇ (see Fig. 3(b)). In particular, none of these wakes contains the critical point. Hence a(W ) = 2a(W ) < 1/2 for all n by Lemma 2(c), which is clearly impossible. The contra- n+1 n diction shows that the orbit of z must eventually hit the critical point. 2 By Proposition 3, we have the following corollary: