´ A CASE OF THE DYNAMICAL ANDRE-OORT CONJECTURE D. GHIOCA, H. KRIEGER, AND K. NGUYEN Abstract. We prove a special case of the Dynamical Andr´ e-Oort Conjecture formulated by Baker and DeMarco [3]. For any integer d ≥ 2, we show that for a rational plane curve C parametrized by (t, h(t)) for some non-constant polynomial h ∈ C[z], if there exist infinitely many points (a, b) ∈ C(C) such that both z d +a and z d +b are postcritically finite maps, then h(z) = ξz for a (d − 1)-st root of unity ξ. As a by-product of our proof, we show that the Mandelbrot set is not the filled Julia set of any polynomial g ∈ C[z]. 1. Introduction Motivated by the results on unlikely intersection in arithmetic dynamics from [3, 13, 14], Baker and DeMarco [4, Conjecture 1.4] recently formulated a dynamical analogue (see Conjecture 1.4) of the Andr´e-Oort Conjecture, characterizing the subvarieties of moduli spaces of dynamical systems which contain a Zariski dense subset of postcritically finite points. A morphism f : P1 −→ P1 of degree larger than 1 is said to be postcritically finite (PCF) if each of its critical points is preperiodic; that is, has finite forward orbit under iteration by f . We prove in Section 2 the following supporting case of the dynamical Andr´e-Oort Conjecture. Theorem 1.1. Let d ≥ 2 be an integer, and let h ∈ C[z] be a non-constant polynomial. If there exist infinitely many t ∈ C such that both maps z 7→ z d + t and z 7→ z d + h(t) are PCF maps, then h(z) = ζz, where ζ is a (d − 1)-st root of unity. As a by-product of our proof we show that the Mandelbrot set is not a filled Julia set for any polynomial h ∈ C[z]; more generally, we show that no multibrot set is a filled Julia set. The d-th multibrot set Md is the set of all t ∈ C with the property that the orbit of 0 under the map z 7→ z d + t is bounded (with respect to the usual archimedean absolute value); when d = 2 we obtain the classical Mandelbrot set. The filled Julia set Kh of a polynomial h(z) is the set of all z ∈ C such that |hn (z)| is bounded independently of n ∈ N, where hn (z) is the n-th iterate of the map z 7→ h(z). We prove: Theorem 1.2. For each d ≥ 2, there does not exist a polynomial h(z) whose filled Julia set is Md . Theorem 1.2 was widely believed to be true in the complex dynamics community, and there are various ways to attack the anecdotal result; for example, exploiting the selfsimilarity of Julia sets, or considering the Hausdorff dimension of boundaries of Fatou components. To our knowledge, however, there is no published proof, nor there was known for specialists a clear argument for proving Theorem 1.2. We provide two proofs for Theorem 2010 Mathematics Subject Classification. Primary 37F50; Secondary 37F05. Key words and phrases. Mandelbrot set, unlikely intersections in dynamics. The first author was partially supported by an NSERC Discovery Grant. The second author was partially supported by NSF grant DMS-1303770. 1 2 D. GHIOCA, H. KRIEGER, AND K. NGUYEN 1.2. In Section 3, we use classical and relatively elementary methods in complex dynamics to deduce the result. In particular, we prove that landing pairs of parameter rays on the d-th multibrot set cannot be preserved under a polynomial map (see Proposition 3.5). In the proof of Theorem 1.2 from Section 3 we also use an old result of Bang [6] regarding the existence of primitive prime divisors in arithmetic sequences. In Section 4, we provide a primarily algebraic proof of Theorem 1.2, which is shorter, but relies on the deep no-wandering-domains theorem of Sullivan [34] and the delicate classification theory of invariant curves of P1 × P1 done by Medvedev and Scanlon [21]. More precisely, we prove that if a polynomial g(z) has a Fatou component which is the image of the open unit disk under another polynomial map, then the Julia set of g is a circle, and so g(z) = az k + b for some a, b ∈ C (according to [29]). Since the main hyperbolic component of Md is easily seen to be the image of the open unit disk under a polynomial map, we obtain the conclusion of Theorem 1.2. We also discuss in Section 4 additional related problems, and we state a general question regarding possible algebraic relations between the Fatou components (respectively between the Julia sets) of two polynomials (see Question 4.3). Given the contrasting approaches and backgrounds required for our two proofs, and also given the intrinsic interests in both approaches, we include both for the benefit of the reader. The connection between Theorem 1.1 and Theorem 1.2 is made in Section 2 through a result on unlikely intersections in dynamics proven in [14, Theorem 1.1]. Namely, we use [14] to show that under the assumptions of Theorem 1.1, the set P := {t ∈ C : z d + t is a PCF map} is totally invariant under the polynomial h. When deg(h) ≥ 2, we deduce a contradiction to Theorem 1.2; if deg(h) = 1, a simple complex-analytic argument yields h(t) = ζt for some (d − 1)-st root of unity ζ (see Proposition 2.2). Motivating our study is a common theme for many outstanding conjectures in arithmetic geometry: given a quasiprojective variety X, one defines the concepts of special points and special subvarieties of X. Then the expectation is that whenever Y ⊆ X contains a Zariski dense set of special points, the subvariety Y must be itself special. For example, if X is an abelian variety and we define as special points the torsion points of X, while we define as special subvarieties the torsion translates of abelian subvarieties of X, we obtain the Manin-Mumford conjecture (proved by Raynaud [26, 27]). The Andr´e-Oort Conjecture follows the same principle outlined above. We present the conjecture in the case when the ambient space is the affine plane. In this case, the conjecture was proven first by Andr´e [1] and then later generalized by Pila [23] to mixed Shimura varieties (in which the ambient space is a product of curves which are either a modular curve, an elliptic curve, or the multiplicative group). Theorem 1.3 (Andr´e [1], Pila [23]). Let C ⊂ A2 be a complex plane curve containing infinitely many points (a, b) with the property that the elliptic curves with j-invariant equal to a, respectively b are both CM elliptic curves. Then C is either horizontal, or vertical, or it is the zero set of a modular polynomial. Motivated by the Andr´e-Oort Conjecture, Baker and DeMarco [4] formulated a dynamical version of the Andr´e-Oort Conjecture where the ambient space is the moduli space of rational maps of degree d, and the special points are represented by the PCF maps. The postcritically finite (PCF) maps are crucial in the study of dynamics, but also analogous to CM points in various ways. The PCF points form a countable, Zariski dense subset in the dynamical moduli space, and over a given number field, there are only finitely many non-exceptional ´ A CASE OF THE DYNAMICAL ANDRE-OORT CONJECTURE 3 PCF maps of fixed degree (see [7] for proof, and definition of non-exceptional). Additionally, Jones [18] has shown that the arboreal Galois representation associated to a rational map defined over a number field K will have small (of infinite index) image for PCF maps despite the image being generally of finite index, analogous to the situation for the `-adic Galois representation associated to an elliptic curve over K, according to whether the curve has CM or not. We state below a special case of [4, Conjecture 1.4], which is much closer in spirit to Theorem 1.3 and avoids some of the technical assumptions stated in [4, Conjecture 1.4]. Conjecture 1.4. Let C ⊂ A2 be a complex plane curve with the property that it contains infinitely many points (a, b) with the property that both z 2 + a and z 2 + b are PCF maps. Then C is either horizontal, or vertical, or it is the diagonal map. The Andr´e-Oort Conjecture fits also into another more general philosophy common for several major problems in arithmetic geometry which says that each unlikely (arithmetic) intersection occuring more often than expected must be explained by a geometric principle. Typical for this principle of unlikely intersections is the Pink-Zilber Conjecture (which is, in turn, a generalization of the Manin-Mumford Conjecture). Essentially, one expects that the intersection of a subvariety V of a semiabelian variety X with the union of all algebraic subgroups of X of codimension larger than dim(V ) is not Zariski dense in V , unless V is contained in a proper algebraic subgroup of X; for more details on the Pink-Zilber Conjecture, see the beautiful book of Zannier [38]. Taking this approach, Masser and Zannier [19, 20] proved that given any two sections S1 , S2 : P1 −→ E on an elliptic surface π : E −→ P1 , if there exist infinitely many λ ∈ P1 such that both S1 (λ) and S2 (λ) are torsion on the elliptic fiber Eλ := π −1 (λ), then S1 and S2 are linearly dependent over Z as points on the generic fiber of π. For a general conjecture extending both the Pink-Zilber and the Andr´e-Oort conjectures, see [24]. At the suggestion of Zannier, Baker and DeMarco [3] studied a dynamical analogue of the above problem of simultaneous torsion in families of elliptic curves. The main result of [3] is to show that given two complex numbers a and b, and an integer d ≥ 2, if there exist infinitely many λ ∈ C such that both a and b are preperiodic under the action of z 7→ z d + λ, then ad = bd . The result of Baker and DeMarco may be viewed as the first instance of the unlikely intersection problem in arithmetic dynamics. Later more general results were proven for arbitrary families of polynomials [13] and for families of rational maps [14]. The proofs from [3, 13, 14] use powerful equidistribution results for points of small height (see [5, 10, 12, 36, 37]). Conjecture 1.4 is yet another statement of the principle of unlikely intersections in algebraic dynamics. Theorem 1.1 yields that Conjecture 1.4 holds for all plane curves of the form y = h(x) (or x = h(y)) where h ∈ C[z]. One may attempt to attack the general Conjecture 1.4 along the same lines. However, there are significant difficulties to overcome. First of all, it is not easy to generalize the unlikely intersection result of [14, Theorem 1.1] (see also Theorem 2.1). This is connected with another deep problem in arithmetic geometry which asks how smooth is the variation of the canonical height of a point across the fibers of an algebraic family of dynamical systems (see [14, Section 5] for a discussion of this connection to the works of Tate [35] and Silverman [31, 32, 33]). It is also unclear how to generalize Theorem 1.2; in Section 4 we discuss some possible extensions of Theorem 1.2 (see Question 4.2). Acknowledgments. We thank the American Institute of Mathematics (AIM) and the organizers of the workshop “Postcritically finite maps in complex and arithmetic dynamics” hosted by the AIM where the collaboration for this project began. We are grateful to Patrick Ingram who asked Conjecture 1.4 during the aforementioned workshop. We thank Thomas 4 D. GHIOCA, H. KRIEGER, AND K. NGUYEN Tucker for several useful discussions regarding this project. Last, but definitely not least, we are indebted to Laura DeMarco for her very careful reading of a previous version of our paper and for suggesting numerous improvements. 2. Proof of Theorem 1.1 In this Section, we deduce Theorem 1.1 from Theorem 1.2 combined with two other results. The first of the two results we need (see Theorem 2.1) is proven in [14, Theorem 1.1] in higher generality than the one we state here, and it is a consequence of the powerful equidistribution results of Yuan and Zhang [36, 37] combined with a complicated analysis of the variation of the canonical height in an algebraic family of morphisms. The result we will use in the proof of Theorem 1.1 is [14, Theorem 1.1] stated for polynomial families over the base curve P1 . Theorem 2.1. Let f ∈ Q[z] be a polynomial of degree d ≥ 2, and let P, Q ∈ Q[z] be nonconstant polynomials. We let ft (z) := f (z) + P (t) and gt (z) := f (z) + Q(t) be two families of polynomial mappings, and we let a, b ∈ Q. If there exist infinitely many t ∈ Q such that both a is preperiodic for ft and b is preperiodic for gt , then for each t ∈ Q we have that a is preperiodic for ft if and only if b is preperiodic for gt . The second result we need for the proof of Theorem 1.1 is the result below, proved in Section 3 by analyzing the coefficients of the analytic isomorphism constructed in [11] between the complement of Md and the complement of the closed unit disk. Proposition 2.2. Suppose µ(z) = Az + B is an affine symmetry of C satisfying µ(Md ) = Md . Then A = ξ and B = 0, where ξ is a (d − 1)-st root of unity; moreover, all (d − 1)-st roots of unity provide a rotational symmetry of Md . Using Theorems 1.2 and 2.1, and Proposition 2.2 we can prove now Theorem 1.1. In terms of notation, we always denote by ∂S the boundary of a set S ⊆ C. Proof of Theorem 1.1. Fix d ≥ 2, and suppose h(z) ∈ C[z] is a non-constant polynomial so that both ft (z) = z d + t and gt (z) = z d + h(t) are PCF for infinitely many t ∈ C. First of all, it is immediate to deduce that each parameter t such that ft is PCF (i.e., 0 is preperiodic for z 7→ z d + t) is an algebraic number. Then Theorem 2.1 yields that for all t ∈ C, the map z 7→ ft (z) is PCF if and only z 7→ gt (z) is PCF. The closure of the set of parameters t ∈ Md which correspond to a PCF map z 7→ z d + t contains the boundary of Md ; more precisely, this boundary can be topologically identified by the PCF points as follows: ∂Md consists exactly of those c ∈ C such that every open neighborhood of c contains infinitely many distinct PCF parameters. Using this characterization of ∂Md and the above conclusion of Theorem 2.1 that ft (z) is PCF iff gt (z) is PCF, we see that by continuity of h and the open mapping theorem, ∂Md is totally invariant for h; that is, ∂Md = h−1 (∂Md ). Since C \ Md is connected, and h maps with full degree on a neighborhood of ∞, C \ Md is also totally invariant for h. So, h−1 (Md ) = Md , and if h is linear, then h is an affine symmetry of Md ; in this case Proposition 2.2 yields the desired conclusion. Suppose from now on that h has degree at least 2, and denote by hn the n-th iterate of h under composition. Recall the definition of the filled Julia set of h Kh := {z ∈ C : |hn (z)| is bounded uniformly in n}, and then the Julia set of h is Jh := ∂Kh ; by Montel’s theorem (see [22]), Jh is also the minimal closed set containing at least 3 points which is totally invariant for h. Therefore ´ A CASE OF THE DYNAMICAL ANDRE-OORT CONJECTURE 5 Jh ⊂ ∂Md . Since C \ Md is connected, Kh ⊂ Md . On the other hand, since C \ Md is totally invariant for h and contains a neighborhood of ∞, every point of Md is bounded under iteration by h, and so by definition, Md ⊂ Kh . Therefore Md = Kh , contradicting Theorem 1.2. 3. The d-th multibrot set is not a filled Julia set Our goal in the Section is to provide a complex-dynamical proof of Theorem 1.2. We first recall a few basic facts and definitions from complex dynamics; see [11, 22]. Throughout this section we denote by D the closed unit disk in the complex plane, and we denote by S 1 its boundary (i.e., the unit circle in the complex plane). Let f : C → C be a polynomial of degree d ≥ 2. If the filled Julia set Kf is connected, there exists an isomorphism (called a B¨ ottcher coordinate of f ) φf : C \ Kf → C \ D so that φf conjugates f to the d-th powering map. Writing φf (z) , z→∞ z one can show that α is a (d − 1)-st root of the leading coefficient of f , and φf is unique up to choice of this root. When f is monic, we always normalize φf so that α = 1. Fix d ≥ 2, and let fc (z) := z d + c (with the exception of the d-th multibrot set, we will suppress dependence on d in notation). We note that the d-th multibrot set Md is defined to be the set of parameters c ∈ C such that the Julia set Jfc is connected; equivalently, Md is the set of parameters for which the forward orbit of 0 under fc is bounded. As described in [11], if c 6∈ Md , then the B¨ ottcher coordinate φfc will not extend to the full C\Kfc , but is guaranteed to extend to those z which satisfy Gfc (z) > Gfc (0), where Gfc is the dynamical Green’s function log |fcn (z)| Gfc (z) = lim . n→∞ dn In particular, the B¨ ottcher coordinate is defined on the critical value c. Therefore we have a function Φ(c) := φfc (c), and one can show (see also [3]) that this function is an analytic isomorphism C \ Md → C \ D, with α = lim Φ(c) = 1. c Using a careful analysis of the coefficient of Φ allows us to compute the affine symmetry group of Md and thus prove Proposition 2.2. lim c→∞ Proof of Proposition 2.2. By definition of the d-th multibrot set, it is clear that Md is 2π invariant under rotation by an angle multiple of d−1 . Suppose now that µ(z) = Az + B fixes Md ; it follows that µ fixes C \ M. Denote by Ψ(z) the inverse of Φ(z). The automorphism m(z) = Φ ◦ µ ◦ Ψ fixes ∞ and so is a rotation m(z) = λz with λ ∈ S 1 . Now, Ψ has local expansion about ∞: ∞ X Ψ(z) = z + bm z −m ; m=0 as computed by Shimauchi in [30], where bm = 0 for 0 ≤ m < d−2 and bd−2 6= 0. Expanding the equality Ψ(m(z)) = µ(Ψ(z)) shows A = λ is a (d − 1)-st root of unity; for d > 2 we also have B = 0 as desired. For d = 2, one computes b0 = −1/2 and b1 6= 0, and again expanding we see that A = λ = ±1 and B = λ/2 − 1/2. Since z 7→ −z − 1 is clearly not a 6 D. GHIOCA, H. KRIEGER, AND K. NGUYEN Figure 1. Examples of parameter rays for d = 2. symmetry of the Mandelbrot set, then there are no nontrivial affine symmetries of M2 and so, Proposition 2.2 is proved. From now on we fix a polynomial h(z) with complex coefficients having connected Julia set. The goal of Theorem 1.2 is to show that the filled Julia set of h does not equal Md . As before, we let φh be a B¨ ottcher coordinate for h. Definition 3.1. Fix θ ∈ [0, 1]. The external ray Rh (θ) := φ−1 {re2πiθ : r > 1} h is the dynamic ray corresponding to θ; the external ray R(θ) := Φ−1 {re2πiθ : r > 1} is the parameter ray corresponding to θ. Sometimes, by abuse of language, we will refer to a (dynamic or parameter) ray simply by the corresponding angle θ. The following result is proved in [11]. Theorem 3.2 (Douady-Hubbard). All parameter rays R(θ) with θ ∈ Q will land; that is, the limit limr→1 Φ(re2πiθ ) exists and lies on the boundary of the d-th multibrot set. If θ is rational with denominator coprime to d, then there is at most one θ0 ∈ [0, 1] such that θ0 6= θ, and R(θ) and R(θ0 ) land at the same point. We call such a pair (θ, θ0 ) a landing pair. The period of the pair is the period of θ (and θ ) under the multiplication-by-d map modulo 1; call this map τd . Any pair of period n will land at the root of a hyperbolic component of period n. Note that by definition, any θ of period n under τd will have denominator dividing dn − 1. As an important example of the above facts, the only period 2 angles under the doubling map τ2 are θ = 1/3 and θ0 = 2/3, which land at the root point of the unique period 2 hyperbolic component, namely at c = − 43 (see Figure 1). We make the obvious but important remark that since h is (continuously) defined on the entire complex plane, any pair of dynamic rays Rh (θ) and Rh (θ0 ) which land on the same 0 ´ A CASE OF THE DYNAMICAL ANDRE-OORT CONJECTURE 7 point will be mapped under h to a (possibly equal) pair of rays that also land together. The impossibility of such a map on the parameter rays of the d-th multibrot set which preserves landing pairs is the heart of the proof of Theorem 1.2. Finally, note that external rays cannot intersect, and that removing any landing pair Rh (θ), Rh (θ0 ) will decompose C \ Kh into a union of two disjoint open sets. Thus we can make the following definition. Definition 3.3. Suppose Rh (θ) and Rh (θ0 ) land at the same point. The open set in C \ (Kh ∪ Rh (θ) ∪ Rh (θ0 )) which does not contain the origin is the wake of the rays. We recall the following classical classification of fixed points from complex dynamics (for more details, see [22]). Definition 3.4. If fc (α) = α, we call λ = fc0 (α) the multiplier of the fixed point. If |λ| < 1, then we say that α is attracting. If λ is a root of unity, we say the fixed point is parabolic. Denote by Hd the main hyperbolic component of Md : Hd := {c ∈ C : z d + c has an attracting fixed point}. See [11] or [22] for more on hyperbolic components of Md and fixed point theory. Proposition 3.5. Let d, n ≥ 2 be integers, let θ = dn1−1 and θ0 = dnd−1 . Then θ and θ0 form a landing pair, and their common landing point lies on the boundary of the main hyperbolic component Hd . The proof follows the ideas of [9, Proposition 2.16] (which yields the case d = 2); however, for d > 2 the combinatorics are more delicate, so we prove the Proposition for all d ≥ 2. Before proceeding to the proof of Proposition 3.5 we introduce the notation and state the necessary facts regarding the combinatorics of subsets of S 1 required for our arguments. We denote by τd the d-multiplication map on R/Z; by abuse of notation, we denote also by τd the induced map on S 1 , i.e. τd e2πα = e2πdα . Following the notation of [9] and [16], we make the following definitions. Definition 3.6. Given r = m n ∈ Q (not necessarily in lowest terms), we say that a finite set X ⊂ S 1 is a degree d m/n-rotation set if τd (X) = X, and the restriction of τd to X is conjugate to the circle rotation Rr via an orientation-preserving homeomorphism of S 1 . p Writing m n = q in lowest terms, we say X has rotation number p/q. A useful equivalent definition is the following: X = {θi } ⊂ R/Z, indexed so that 0 ≤ θ0 < θ1 < · · · < θn−1 < 1, is a degree d m/n-rotation subset of S 1 if it satisfies τd (θi ) ≡ θi+m mod n mod 1 for all 0 ≤ i < n − 1. Note that by Corollary 6 of [16], if X has n = kq elements, where 1 ≤ k ≤ d − 1. m n = Definition 3.7. Given a set S = {θ0 , . . . , θn−1 } satisfying 0 ≤ θ0 < θ1 < · · · < θn−1 < 1, p q is in lowest terms, any such 8 D. GHIOCA, H. KRIEGER, AND K. NGUYEN suppose that S is a degree d m/n-rotation set. The deployment sequence of S is the ordered set of d integers {s1 , . . . , sd−1 }, where si denotes the total number of angles θj in i the interval [0, d−1 ). By [16, Theorem 7], rotation subsets are determined by the data of rotation number and deployment sequence, as shown in the following statement. Theorem 3.8 (Goldberg [16]). A rotation subset of the unit circle is uniquely determined by its rotation number and its deployment sequence. Conversely, given r = pq in lowest terms and a deployment sequence 0 ≤ s1 ≤ s2 ≤ · · · ≤ sd−1 = kq, there exists a rotation subset of S 1 with this rotation number and deployment sequence if and only if every class modulo k is realized by some sj . Now we can proceed to the proof of Proposition 3.5. Proof of Proposition 3.5. Let r = n1 . The boundary of Hd contains exactly d − 1 parameters ci (1 ≤ i ≤ d − 1) so that fci has a fixed point of multiplier e2πir . For each ci , denote by αi the unique parabolic fixed point of fci , which necessarily attracts the unique non-fixed critical point 0 (see [22]). Consequently, the Julia set of fci (which we denote by Jfci ) is locally connected, so all external rays land, and we have a landing map Li : R/Z → Jfci , with 2πiθ ), Li (θ) := lim φ−1 fc (ρe ρ→1+ i where φfci is the normalized uniformization map φfci : C \ Kfci → C \ D. Then the landing map defines a continuous semiconjugacy, so that fci (Li (θ)) = Li (τd (θ)). L−1 i (αi ), Define Si := the set of angles whose rays land at αi . Since fci has multiplier e2πi/n at αi , Si is a rotation subset of S 1 with rotation number r = n1 (see Lemma 2.4 of [17]). Writing Si = {0 ≤ θi,0 < θi,1 < · · · < θi,kn−1 < 1}, with k cycles, each pair (θi,j mod kn , θi,j+1 mod kn ) cuts out an open arc ai,j in S 1 . Since the map preserves cyclic orientation and the collection Si , any arc ai,j with length less than 1 d is mapped homeomorphically onto another arc and expands by a factor of d. Since n > 1, no arc is fixed, so for each arc, some iterate of the arc cannot be mapped homeomorphically onto its image, and has length ≥ d1 . For such an arc, the map τd maps the arc onto S 1 , and so the sector S which is bounded by the corresponding rays in the dynamical plane has image containing the set [ T := C \ R(θi,j ). 0≤j≤kn−1 Since the critical point is contained in an attracting petal, both the critical point 0 and the critical value ci are contained in the Fatou set, so in T . Since S maps onto T , fci (S) contains the critical value, and so S contains the critical point. Since the sectors are disjoint, we conclude that there is a unique orbit of arcs, and so a unique cycle of elements of Si , i.e. k = 1. Note also that the unique sector corresponding to an arc of length ≥ d1 is the only sector which can contain a fixed ray, and therefore each element si,j of the deployment sequence satisfies si,j ∈ {0, n}. ´ A CASE OF THE DYNAMICAL ANDRE-OORT CONJECTURE 9 We know by [28] that the angles of parameter rays landing on ci is a subset of Si for each i; by τd -invariance, then, the Si are disjoint. Therefore, by Theorem 3.8, the sets S1 , . . . , Sd−1 are precisely the d − 1 rotation subsets with rotation number r = n1 and deployment sequences containing only the values 0 and n. Noting that the subset 1 d dn−1 F := , ,..., n dn − 1 dn − 1 d −1 has rotation number n1 and deployment sequence {n, n, . . . , n}, we see that there exists an i such that F is the set of angles of dynamic rays landing at αi for fci ; without loss of generality, F = S1 . It remains to show that dn1−1 and dnd−1 land together on c1 in parameter space. The pair of angles in S1 landing on c1 in parameter space are precisely the angles of S1 whose dynamic rays are characteristic; that is, their wake separates the critical point 0 from the k k+1 critical value c1 (see [28]). Since τd maps an arc [ dnd−1 , ddn −1 ] ⊂ R/Z homeomorphically n−1 onto its image if and only if 0 ≤ k < n − 1, the wake in the dynamical plane of R( ddn −1 ) n and R( dnd−1 ) = R( dn1−1 ) contains the critical point. Therefore the wake of R( dn1−1 ) and R( dnd−1 ) contains the critical value, and the conclusion follows. Proof of Theorem 1.2. Suppose h is a polynomial with filled Julia set Kh = Md . It is clear that such a h(z) must have D = deg(h) ≥ 2 (since the filled Julia set of a linear polynomial is either the empty set, or a point, or the whole complex plane). Therefore we have a B¨ottcher coordinate φh on C \ M as described above. Lemma 3.9. Suppose that h(z) is a polynomial of degree D ≥ 2 with Kh = Md . Then the B¨ ottcher coordinate φh can be chosen to be the map Φ : C \ Md → C \ D, and h can be chosen to be monic. Proof. Since Md is connected, any such polynomial will have B¨ottcher coordinate which extends to the full C \ Kh . Therefore φh (Φ−1 (z)) is an automorphism of C \ D which fixes ∞; consequently, it is a rotation by some λ on the unit circle. Thus φh (z) = λΦ(z) for all z ∈ C \ Md . Note then that φh (z) lim = λ, z→∞ z so λD−1 is the leading coefficient of h, call it aD . Denote by h(z) the polynomial whose coefficients are complex conjugates of those of h. Since Md is invariant under complex conjugation, then Md is also the filled Julia set of h(z). By the classification theorems of polynomials with the same Julia set (see [29], noting ` that Jh = ∂Md cannot be a line segment or a circle), some iterates hk and h then differ by an affine symmetry of Md ; by degree, k = `, and we may replace h by hk to assume h and h differ by an affine symmetry of Md . By Proposition 2.2, the affine symmetry group k of Md is the set of rotations by (d − 1)-st roots of unity {ζd−1 : 0 ≤ k < d − 1}, we have k h(z) = ζd−1 h(z) for all z. Therefore the leading coefficient aD of h(z) has the property that ` ad−1 ∈ R; replacing h(z) by the appropriate ζd−1 h(z) leaves the Julia set invariant, and has D real leading coefficient. Since the new aD also has modulus 1, it is ±1; if d is odd, we can k choose ζd−1 so that h is monic. Suppose that d is even and aD = ±1. The intersection Md ∩ R is an interval [α, β] with α < 0 and β > 0. The interval (0, β) is contained in the interior of Md . We let {αn }n be a sequence of roots of hyperbolic components of Md such that limn→∞ αn = α; note 10 D. GHIOCA, H. KRIEGER, AND K. NGUYEN that no αn is contained in the interior of Md . Now we claim that h(β) = β. Indeed, since both C \ Md and Md are invariant under h, we conclude that h(β) is on the boundary of Md (since β ∈ ∂Md ); so, if h(β) 6= β, then h(β) < 0. Moreover, because h(x) ∈ / Md for any x > β, we conclude that h(β) = α. But then h((0, β)) is an interval contained in the interior of Md , whose closure contains α; thus h((0, β)) contains an interval (α, γ) with α < γ < 0. So, h((0, β)) is contained in the interior of Md but contains infinitely many points αn , which are not in the interior of Md . This contradiction shows that indeed h(β) = β. Hence h((β, +∞)) is an infinite interval containing β; thus limx→+∞ h(x) = +∞ which yields that aD cannot be negative, and so, aD = 1, as desired. It follows that we have the following commutative diagram: Φ C \ Md C\D z 7→ z D h C \ Md Φ C\D Since R( dm1−1 ) and R( dmd−1 ) land together for all m ≥ 2, so do their forward iterates k k D dD under h; that is, R( dm −1 ) and R( dm −1 ) land together for all m ≥ 2, k ≥ 0. Recall the following classical result of Bang [6]. Theorem 3.10 (Bang [6]). Let a ≥ 2 be an integer. There exists a positive integer M such that for each integer m > M , the number am − 1 has a primitive prime divisor; that is, a prime p such that p | (am − 1) and p - (ak − 1) for all k < m. According to Bang’s theorem, choose an integer M > 2 such that for all m ≥ M , dm − 1 has a primitive prime divisor p which does not divide D. For any k, there exists a unique integer m(k) such that dm(k) − 1 ≤ Dk (d2 − 1) < dm(k)+1 − 1. We choose K sufficiently large so that for all k ≥ K, then m(k) ≥ M . For any k ≥ K, Dk dD k write m := m(k); as noted above, the parameter rays R( dm −1 ) and R( dm −1 ) land together, say at c. Since these are periodic rays with period dividing m, fc (z) has a periodic point α of period n | m at which the dynamic rays of the same angles land (see [11] for a proof of Dk dD k n this). Therefore dm −1 and dm −1 lie in the same cycle of R/Z under multiplication by d ; note this cycle has maximal length m/n. So for some 0 ≤ r < m/n, k Dk rn+1 D ≡ d dm − 1 dm − 1 mod 1, and so Dk (1 − drn+1 ) ≡ 0 mod (dm − 1). Choose a primitive prime divisor p of dm − 1 which is coprime to D. Primitivity and the congruence above then imply rn + 1 = m (also note that r < m/n). Since n | m, the only possibility is n = 1. So α is a fixed point, and c must lie on the main hyperbolic component Hd . However, by choice of m = m(k) and the fact that (d − 1) divides d2 − 1 and dm+1 − 1, we have 1 Dk d ≤ ≤ 2 . d2 − 1 dm − 1 d −1 ´ A CASE OF THE DYNAMICAL ANDRE-OORT CONJECTURE 11 Since R( d21−1 ) and R( d2d−1 ) land together on the main hyperbolic component, any ray in m m −1 −1 their wake cannot possibly do the same. We conclude that Dk = dd2 −1 or Dk = d · dd2 −1 , m 2 which is impossible since p | (d − 1) and p - D · (d − 1). Therefore we have a contradiction, and Theorem 1.2 is proved. 4. Unlikely Algebraic Relations Among Certain Sets in Dynamics In this Section, we provide a second proof of Theorem 1.2 (see Corollary 4.5), namely that for each d ≥ 2, there exists no polynomial h(z) of degree at least 2 such that Md is the filled Julia set Kh of h. Our method is motivated by some natural questions (see Questions 4.2 and 4.3) in complex dynamics which can be viewed as examples of unlikely relations between sets associated to algebraic dynamical systems (such as the d-th multibrot set, or the Julia set or Fatou components). In each case the expectation is that a polynomial relation between two Julia sets, or two Fatou components, or a polynomial relation on the d-th multibrot set is induced by a linear relation between those sets. Firstly, we note that arguing identically as in the proof of Theorem 1.1 we can strengthen Theorem 1.2 as follows. Theorem 4.1. There exists no polynomial h ∈ C[z] of degree greater than 1 such that h−1 (Md ) = Md . Proof. Since Md is a bounded subset of C, we get that Md ⊆ Kh . On the other hand, since Jh is the smallest closed subset of C totally invariant under h which contains at least 3 points, we conclude that Jh ⊂ Md . Since C \ Md is connected and Md is closed, we conclude that Kh ⊆ Md and therefore Md = Kh , which contradicts Theorem 1.2 (or Corollary 4.5). We ask whether one could weaken the hypothesis of Theorem 4.1 even further. Question 4.2. Let d ≥ 2, and let h ∈ C[z] such that h(Md ) = Md . Is it true that h(z) must be a linear polynomial of the form ξz for some (d − 1)-st root of unity ξ? Essentially, Conjecture 4.2 predicts the polynomial relations in the parameter space. It is natural to ask what are the polynomial relations also in the dynamical space; so we formulate below two questions which were motivated by Theorem 4.1. Question 4.3. Let P (z), f (z), g(z) ∈ C[z] be polynomials of degrees at least 2. (a) Assume P (Jf ) = Jg . Is it true that Jg is the image of Jf under a linear polynomial? (b) Assume there is a Fatou component Uf of f , and a Fatou component Ug of g such that P (Uf ) = Ug . Is it true that Jg is the image of Jf under a linear polynomial? The philosophy behind Question 4.3 is to ask that whenever there is a polynomial relation between the Julia sets of the polynomials f and g, or between a Fatou component of f and a Fatou component of g, this relation forces a strong rigidity on the dynamical systems corresponding to f and g. Indeed, the conclusion that Jg = µ(Jf ) for some linear polynomial µ yields that g and µ ◦ f ◦ µ−1 have the same Julia sets; hence a polynomial relation between corresponding dynamical components of f respectively of g yields a geometric relation between the corresponding Julia sets. Note that Schmidt and Steinmetz [29] classified the polynomials with the same Julia sets; apart from the exceptional classes of monomial mappings and the Chebyshev polynomials, any two polynomials with the same Julia set must have a common iterate. Hence there is a rigid relation between the two dynamical systems of f and of g knowing that there is a polynomial relation between either the two Julia sets or between two Fatou components. We also note that Question 4.3 (a) is connected with the Dynamical Manin-Mumford Conjecture (see [39] and [15]) which, in 12 D. GHIOCA, H. KRIEGER, AND K. NGUYEN a special case, asks for classifying the polynomial relations between the sets of preperiodic points of two rational maps. Motivated by Question 4.3 we found an alternative proof of Theorem 1.2 (independent from the one presented in Section 3), which in particular answers Question 4.3 positively if f is a monomial. Theorem 4.4. Let P (z), h(z) ∈ C[z] with degree at least 2. Suppose that either P (S 1 ) = Jh , or P (D) is a Fatou component of h. Then Jh is a circle. Theorem 4.4 gives a positive answer to Question 4.3 when f is linearly conjugate to a monomial (i.e. Jf is a circle), and suggests a general approach to be pursued for an answer to Question 4.3. We have the following immediate corollary: Corollary 4.5. There is no polynomial h(z) ∈ C[z] having degree at least 2 such that Kh = Md . Proof. Assume there is such an h. Recall the main hyperbolic component: Hd := {c ∈ C : z d + c has an attracting fixed point}. Let α be an attracting fixed point with multiplier λ satisfying |λ| < 1. From dαd−1 = λ, and c = α − αd , we have that Hd is exactly the image of the disk of radius ( d1 )1/(d−1) centered at the origin under the polynomial z − z d . Since Hd is a connected component of the interior of Md , it is a Fatou component of h. Theorem 4.4 implies that Jh = ∂Md is a circle, a contradiction. We will prove Theorem 4.4 by studying the set of solutions (u, w) ∈ S 1 × S 1 of equations of the form r ◦ q(u) = q(w), where r, q ∈ C(z) are non-constant. Using a simple trick of Quine [25], one embeds r(S 1 ) into an algebraic curve in P1 (C) × P1 (C) as follows. Define η : P1 (C) → P1 (C) × P1 (C) by η(z) = (z, z) for z ∈ C, and η(∞) = (∞, ∞). Let (x, y) denote the coordinate function on P1 × P1 . Then η(S 1 ) is contained in the closed curve C defined by xy = 1. For any non-constant rational map r(z), let Cr denote the closed curve: Cr := {(r(x), r(y)) : (x, y) ∈ C}. The key observation is that η(r(S 1 )) is contained in Cr . Note that Cr is the image of C under the self-map (r, r) of P1 × P1 , so Cr is irreducible. We have the following simple lemma: Lemma 4.6. Let r(z), q(z) ∈ C(z) be non-constant. Assume the equation r ◦ q(u) = q(w) has infinitely many solutions (u, w) ∈ (S 1 )2 . Then the curve Cq is invariant under the self-map (r, r) of P1 × P1 . Proof. The given assumption implies that the irreducible curves Cq and Cr◦q have an infinite intersection. Hence Cq = Cr◦q . Note that Cr◦q is exactly the image of Cq under (r, r). If r(z) ∈ C[z] is a polynomial of degree D ≥ 2, then we may use the results of Medvedev and Scanlon from [21] to classify the curves invariant under the map (r, r) of P1 × P1 . To do so, we require the following definition: Definition 4.7. We call a polynomial f (z) exceptional if f is conjugate by a linear map to either ±CD (z) for a Chebyshev polynomial CD (z), or a powering map z 7→ z D . ´ A CASE OF THE DYNAMICAL ANDRE-OORT CONJECTURE 13 We remind the readers that the Chebyshev polynomial of degree D is the unique polynomial CD (z) of degree D such that CD (z + z1 ) = z D + z1D . Note that by the classification theory of Julia sets in [29], the Julia set of a polynomial is a line segment or a circle if and only if the polynomial is exceptional. Proposition 4.8. Let r(z) ∈ C[z] be a non-exceptional polynomial of degree D ≥ 2, and let q(z) ∈ C[z] be non-constant. Then the curve Cq is not invariant under (r, r). Proof. We assume that Cq is invariant under (r, r): note that Cq is not a vertical or horizontal line. By Theorem 6.26 of [21, p. 166], there exist polynomials π(z), ρ(z), H(z) and a curve B in P1 × P1 satisfying the following conditions: (i) r ◦ π = π ◦ H, r ◦ ρ = ρ ◦ H. (ii) Cr is the image of B under the self-map (π, ρ) of P1 × P1 . (iii) B is periodic under the self-map (H, H) of P1 × P1 . Furthermore, there is a nonconstant polynomial ψ(z) ∈ C[z] commuting with an iterate of H(z) such that B is defined by the equation y = ψ(x) or x = ψ(y). Assume B is defined by y = ψ(x) (the case of the equation x = ψ(y) can be treated similarly). Since Cq is the image of B under (π, ρ), we have: Cq = {(π(x), ρ(ψ(x))) : x ∈ P1 (C)}. Thus (∞, ∞) is the only point in Cq whose first coordinate is ∞. On the other hand, we recall the definition of Cq : Cq := {(q(x), q(y)) : (x, y) ∈ C} = {(q(x), q(1/x)) : x ∈ P1 (C)} since C is the closed curve defined by xy = 1. So (∞, q(0)) is the only point in Cq whose first coordinate is ∞, a contradiction. Proof of Theorem 4.4. Let D = deg(g) ≥ 2 (note also that Question 4.3 is trivial if g were a linear polynomial). We first suppose P (S 1 ) = Jg . It follows that g ◦ P (S 1 ) = g(Jg ) = Jg = P (S 1 ). By Lemma 4.6, the curve CP is invariant under the self-map (g, g) of P1 × P1 . By Proposition 4.8, g must be linearly conjugate to ±CD (z) or z D , and so Jg is a line segment or circle. The image of S 1 under a polynomial cannot be a line segment. To see why, assume there were a polynomial Q(z) mapping S 1 onto the interval [0, 1]. The imaginary part of Q, which is harmonic and vanishes on S 1 , must vanish on D by the maximum modulus principle. Hence Q(D) ⊆ R, a contradiction since Q(D) has a non-empty interior. So Jg is a circle, as desired. We now assume that P (D) is a Fatou component of g. Because P is a polynomial, we conclude that P (D) is a bounded Fatou component. By the No-Wandering-Domain Theorem of Sullivan [34], there exist m ≥ 0 and n > 0 such that g m+n (P (D)) = g m (P (D)). Replacing g by an iterate, we may assume g 2 (P (D)) = g(P (D)). It follows that g 2 ◦ P (D) = g ◦ P (D). Note that for every open, bounded subset S of C, S − S is infinite: translating, we may assume 0 ∈ S, and then every ray originating from 0 will contain a point in S − S. Now let S := g ◦ P (D) = g 2 ◦ P (D). Since S = g ◦ P (D) = g 2 ◦ P (D), we have that g ◦ P (S 1 ) ∩ g 2 ◦ P (S 1 ) contains the infinite set S − S. Applying Lemma 4.6, we see that the curve Cg◦P is invariant under (g, g). By Proposition 4.8, g is linearly conjugate to ±CD (z) or z D . Note that Jg cannot be a line segment since the only Fatou component of such a g is unbounded. Therefore Jg is a circle, completing the proof of Theorem 4.4. 14 D. GHIOCA, H. 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