PRIMITIVE TUNING VIA QUASICONFORMAL SURGERY WEIXIAO SHEN AND YIMIN WANG Abstract. Using quasiconformal surgery, we prove that any primitive, postcritically-finite hy- perbolic polynomial can be tuned with an arbitrary generalized polynomial with non-escaping critical points, generalizing a result of Douady-Hubbard for quadratic polynomials to the case of higher degree polynomials. This solves affirmatively a conjecture by Inou and Kiwi on surjectivity of the renormalization operator on higher degree polynomials in one complex variable. 1. Introduction Quasiconformal surgery is a powerful technique in the theory of holomorphic dynamics of one variable. The process often consists of two steps. First, we construct a quasi-regular map with certain dynamical properties, by modifying part of a holomorphic dynamical system, or extending the definition of a partially defined holomorphic dynamical system. Then we prove existence of a rational map which is (essentially) conjugate to the quasi-regular mapping. Successful applications of this technique include Douady-Hubbard’s theory of polynomial-like mappings ([5]) and Shishikura’s sharp bounds on the number of Fatou cycles ([20]), among others. Tuning is an operator introduced by Douady-Hubbard [5] to prove existence of small copies in the Mandelbrot set. Recall that the Mandelbort set consists of all complex numbers c for which 2 2 Pc(z) = z + c has a connected Julia set. Let Pa(z) = z + a be a quadratic polynomial for which the critical point 0 has period p. Douady-Hubbard proved that there is a homeomorphism τ = τa, from M into itself with τa(0) = a and with the property that the p-th iterate of Pτ(c) has a quadratic-like restriction which is hybrid equivalent to Pc. Intuitively, the filled Julia set of Pτ(c) is obtained from that of Pa by replacing each bounded Fatou component of Pa with a copy of the filled Julia set of Pc. Douady-Hubbard’s argument involved detailed knowledge of the combinatorics of the Mandelbrot set and also continuity of the straightening map in the quadratic case, and thus breaks down for higher degree polynomials. In [9], Inou-Kiwi defined a natural analogy of the map χ = τ −1 for higher degree polynomials, which we will call the IK straightening map. Given a postcritically finite hyperbolic polynomial f0 of degree d ≥ 2, together with an internal angle system, they defined a map χ from a certain space R(f0) into a space C(T ) which consists of generalized polynomials over the reduced mapping scheme T = T (f0) of f0 with fiberwise connected Julia set. The map χ is injective and in general not continuous [8]. The map f ∈ R(f0) is the tuning of f0 with χ(f) in the sense of Douady and arXiv:2010.05199v1 [math.DS] 11 Oct 2020 Hubbard when the Julia set χ(f) is fiberwise locally connected. In the case that f0 is primitive, i.e. the bounded Fatou components have pairwise disjoint closures, the set R(f0) coincide with a combinatorially defined set C(f0) which is known to be compact. Inou-Kiwi’s argument is mostly combinatorial and they posed a conjecture that in the case that f0 is primitive, χ is surjective and with connected domain. The fact that χ is surjective means that f0 can be tuned by every g ∈C(T ). Date: October 13, 2020. 1 2 WEIXIAO SHEN AND YIMIN WANG In this paper, we shall prove the conjecture of Inou and Kiwi, using quasiconformal surgery technique. In particular, this shows that a primitive postcritically finite hyperbolic map f0 can be tuned by each g ∈C(T (f0)). Main Theorem. Let f0 be a postcritically finite hyperbolic polynomial that is primitive and fix an internal angle system for f0. Then the IK straightening map χ : C(f0) → C(T ) is bijective and C(f0) is connected. We shall recall the definition of Inou-Kiwi’s straightening map later. Let us just mention now d that if f0(z)= z + a for some a ∈ C then C(T ) is the set of all monic centered polynomials of degree d which have connected Julia sets. The main part of the proof is to show the surjectivity of the map χ. It is fairly easy to construct a quasi-regular map f which has a generalized polynomial-like restriction hybrid equivalent to a given generalized polynomial in C(T ), via qc surgery in a union of annuli domains around the critical Fatou domainse of f0. In order to show that there is a polynomial map with essentially the same dynamics as f, we run Thurston’s algorithm to f. We modify an argument of Rivera- Letelier [12] to show the convergence. In order to control distortion, we use a result of Kahn [13] on removability of certaine Julia sets. After surjectivity is proved,e we deduce connectivity of C(f0) from that of C(T ) which is a theorem of Branner-Hubbard [1] and Lavaurs [14]. In [3], qc surgery was successfully applied to construct intertwining tuning. Our case is more complicated since the surgery involves the non wandering set of the dynamics. The paper is organized as follows. In §2, we recall definition of generalized polynomial-like maps and the IK straightening map. In §3, we construct a specific Yoccoz puzzle for postcrit- ically finite hyperbolic primitive polynomials which is used to construct renormalizations. In §4, we prove a technical distortion lemma. In §5, we prove a convergence theorem for Thurston algorithm. The proof of the main theorem is given in §6 using qc surgery. Acknowledgment. We would like to thank the referee for carefully reading the paper and in particular, for pointing out an error in Section 3 of the first version. This work was supported by NSFC No. 11731003. 2. The IK straightening map We recall some basic notations and the definition of IK straightening maps. The multi-critical nature of the problem makes the definition a bit complicated. Let Polyd denote the set of all monic centered polynomials of degree d, i.e. polynomials of the d d−2 form z 7→ z + ad−2z + ··· + a0, with a0,a1, ··· ,ad−2 ∈ C. For each f ∈ Polyd, let K(f) and J(f) denote the filled Julia set and the Julia set respectively. Let P (f) denote the postcritical set of f: ∞ P (f)= {f n(c)}. ′ C n=1 f (c)=0[,c∈ [ Let Cd = {f ∈ Polyd : K(f) is connected}. 2.1. External Rays and Equipotential Curves. For f ∈ Polyd, the Green function is defined as 1 + n Gf (z) = lim log |f (z)|, n→∞ dn where log+ = max(log, 0). The Green function is continuous and subharmonic in C, satisfy- ing Gf (f(z)) = dGf (z). It is nonnegative, harmonic and takes positive values exactly in the PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 3 attracting basin of ∞ n Bf (∞) := {z ∈ C | f (z) → ∞ as n → ∞} = C \ K(f). Assume f ∈Cd. Then there exists a unique conformal map φf : Bf (∞) →{z | |z| > 1} d such that φ(z)/z → 1 as z → ∞ and such that φf ◦ f(z) = (φf (z)) on Bf (∞). The external ray of angle t ∈ R/Z is defined as −1 i2πt Rf (t)= {φf (re ) | 1 <r< ∞}, and the equipotential curve of pontential l ∈ (0, ∞) as −1 l+i2πθ Ef (l)= {φf (e ) | 0 ≤ θ< 1}. −1 i2πt We say the external ray Rf (t) lands at some point z0 if limr→1 φf (re )= z0. It is known that for each t ∈ Q/Z, Rf (t) lands at some point. On the other hand, if z0 is a repelling or parabolic periodic point, then there exists at least one but at most finitely many external rays landing at z0. See for example [16]. The rational lamination of f, denoted by λ(f), is the equivalence relation on Q/Z so that θ1 ∼ θ2 if and only if Rf (θ1) and Rf (θ2) land at the same point. 2.2. Generalized polynomials over a scheme. Now let us consider a postcritically finite hyperbolic polynomial f0. Following Milnor [17], we define the reduced mapping scheme T (= T (f0)) = (|T |,σ,δ) of f0 as follows. Let |T | denote the collection of critical bounded Fatou components of f0. For r(v) each v ∈ |T |, let r(v) be the minimal positive integer such that f0 (v) is again a critical Fatou component. Define σ : |T | → |T | and δ : |T |→{2, 3, ···} by r(v) r(v) σ(v)= f0 (v) and δ(v) = deg(f0 |v) = deg(f0|v). Definition 2.1 (Generalized Polynomials). A generalized polynomial g over T is a map g : |T |× C → |T |× C such that g(v,z) = (σ(v),gv(z)) where gv(z) is a monic centered polynomial of degree δ(v). Denote by P (T ) the set of all generalized polynomials over the scheme T . Given g ∈ P (T ), the filled Julia set K(g) of g is the set of points in |T |× C whose forward orbits are precompact. The boundary ∂K(g) of the filled Julia set is called the Julia set J(g) of g. The filled Julia set K(g) is called fiberwise connected if K(g, v) := K(g) ∩ ({v}× C) is connected for each v ∈ |T |. Let C(T ) be the set of all the generalized polynomials with fiberwise connected filled Julia set over T . We shall need external rays and Green function for g ∈C(T ) which can be defined similarly as in the case of a single polynomial. Indeed, for each v ∈ |T | there exists a unique conformal ϕv,g : C \ K(g, v) → C \ D such that ϕv,g(z)/z → 1 as z → ∞ and these maps satisfy ϕσ(v),g(gv(z)) = δ(v) −1 2πit ϕv,g(z) .