General Education and Research Center, Tokyo Polytechnic University

Shizuo Nakane

中根 静男

4KJ00004061230論文Articledepartmental bulletin pape

中根, 静男

ナカネ, シズオ

Nakane, Shizuo

1067

Tokyo Polytechnic University Repository / 東京工芸大学学術リポジトリ

Capture components for cubic polynomials withparabolic fixed pointsShizuo Nakane∗The parameter space for a family of cubic polynomials with parabolic fixed points of multiplier oneis investigated. Especially, the dynamics on the boundaries of the capture components is revealed.1 IntroductionIn this note, we will investigate the dynamics of the familyPer1(1) : Pa(z) = z3 + az2 + z, a ∈ C.of cubic polynomials with parabolic fixed point 0 of multiplier one. Especially we willstudy the paramater space of our family. One of the two critical points of the map Pabelongs to the immediate basin B∗a(0) of the parabolic fixed point 0. The dynamics of Pais completely determined by the behaviour of the orbit of another critical point. We aremuch interested in the parameters for which both critical points belong to the basin Ba(0).We call here the set of such parameters the parabolic set and its connected component aparabolic component.Our family Per1(1) in cubic polynomials has been investigated by many authors.Douady-Hubbard [DH] studied the discontinuity of the straightening map of cubic-likemaps on Per1(1). Milnor [M1] considered the family of real cubic polynomials and con-jectured the non-local connectivity of the cubic connectedness locus. Lavaurs [L] settledthis conjecture through the study of Per1(1). See also Epstein-Yampolsky [EY]. ThusPer1(1) reflects the features of the cubic dynamics much different from the quadratic one.Willumsen [W] gave necessary conditions for stretching rays to accumulate on a mapin the main parabolic component of Per1(1). Inspired by her work, the author showed,in the joint work [KN] with Y. Komori that, in a certain region of the space of real cubicpolynomials, most stretching rays have non-trivial accumulation sets on the real slice ofthe main parabolic component of Per1(1). In fact, they oscillate wildly as they approach∗Professor, General Education and Research Center, Tokyo Polytechnic University,Received Sept. 8, 200533Per1(1). In order to study stretching rays in non-real regions, we have to reveal thestructure of those parabolic components. This note is a first step toward this aim. Andwe will investigate the dynamics on the boundaries of the capture components.2 Connectedness locusNote that critical points of the map Pa :c±(a) =−a±√a2 − 33=a3(−1±√1− 3/a2).are two branches of a two-valued holomorphic function on C. It branches at a = ±√3. Orthey are holomorphic functions on the double covering space of C− {±√3}. In order toconsider them on C, we must fix their branches. Since c±(a) are holomorphic for large |a|,we fix such branches. Then they are single-valued holomorphic in C − [−√3,√3]. Notethat they are replaced by each other when we go through the slit [−√3,√3]. On this slit,both critical points belong to B∗a(0). Thus the slit is included in the connectedness locus.Since they have the following asymptotic behaviours near a =∞ :c+(a) = − 12a+O(1a3), c−(a) = −2a3+O(1a),the critical values satisfyPa(c+(a)) = O(1a), Pa(c−(a)) ≈ 4a327.Thus, for large |a|, c−(a) escapes to ∞ and c+(a) is contained in B∗a(0).Lemma 2.1. The connectedness locus M1(1) of the family Per1(1) is characterized byM1(1) = {a ∈ C; c−(a) ∈ Ka = K(Pa)}.In C−M1(1), the unique indifferent cycle 0 is persistent. Hence Pa is J-stable there.Actually we haveLemma 2.2. The complement of ∂M1(1) is the set of parameter a such that Pa is J-stable.proof. Note that Pa is J-stable if and only if the family {P ka (c±(a); k ≥ 0} forms anormal family in a neighborhood of a. If we put ma = 3max(|a|, 1), it follows Ka ⊂ Dma .Thus, if a ∈ IntM1(1), both critical points are contained in Dma and Pa is J-stable by34Montel’s theorem. If a ∈ ∂M1(1), Pa is not J-stable since {P ka (c−(a))} is not normal.This completes the proof. 2Now let c˜−(a) be the co-critical point of c−(a). It satisfies Pa(c˜−(a)) = Pa(c−(a)). Ifϕa denotes the Bo¨ttcher coordinate of Pa, ϕa(c˜−(a))3 = ϕa(Pa(c˜−(a))) ≈ 4a3/27. Thus,if we put Φ(a) = 3ϕa(c˜−(a))/3√4, we haveProposition 2.1. Φ : C −M1(1) → C − D3/ 3√4 is a conformal isomorphism satisfyinglima→∞Φ(a)/a = 1.Corollary 2.1. M1(1) is connected.Now, external rays are defined for M1(1) and we can discuss their landing properties.Note that the correspondence between the parameter space and the dynamical plane isdone through the co-critical point.3 Conformal position mapsIn this section, we will show that every parabolic components are simply connected. For aparabolic component W , the critical point c+(a) always belongs to B∗a(0) and there existsk ≥ 0 such that P ka (c−(a)) first hits B∗a(0). We call such k the preperiod ofW . W is calleda capture component if k > 0. If k = 0, that is, if both critical points are contained inB∗a(0), W is called an adjacent component. It turns out that there are only two adjacentcomponents, each containing√3 or −√3.LetW be a parabolic component of preperiod k. In order to study the global topologyof parabolic components, we use the conformal position map m after Zakeri [Z]. Letψa : D → B∗a(0) be the Riemann map satisfying ψa(0) = c+(a), ψa(1) = 0. We definem : W → D by m(a) = ψ−1a (P ka (c−(a))).Lemma 3.1. For any capture component W , ψa depends holomorphically on a ∈W .proof. Consider the map Ra = ψ−1a ◦Pa◦ψa : D→ D. Since it is a proper holomorphicmap between D, it is a Blaschke product of degree two with critical points 0 (and∞). SinceJa is locally connected, ψa can be continued to a continuous map D→ B∗a(0). Because ofthe maximum principle, it must be injective on ∂D. Thus ∂B∗a(0) is a Jordan curve andψa : D → B∗a(0) is an onto homeomorphism. Then ψa conjugates Pa to Ra also on ∂D.Hence J(Ra) = ∂D and 1 is a fixed point. Since points on ∂D are not attracted by 1, 1 isnot attracting but parabolic with multiplier one. Since Blaschke product is commutablewith the map z 7→ 1/z, C−D is also the basin of 1 and 1 has multiplicity three. That is,R′′a(1) = 0. Now it is easy to see that Ra(z) ≡ R(z) =z2 + 1/31 + z2/3, which is independent of35a. ψa maps each point on the inverse orbit by R of 1 to the point on the inverse orbit byPa of 0, which moves holomorphically on a. Thus ψa ◦ ψ−1a0 gives a holomorphic motionof the inverse orbit of 0. By the λ-lemma, it continues to to a holomorphic motion of∂B∗a(0). Hence, ψa ◦ψ−1a0 on ∂B∗a(0), and consequently ψa on ∂D depends holomorphicallyon a. Now, by the Poisson formula, ψa on D depends holomorphicall on a. 2Thus the map m : W → D is holomorphic for any capture component.Lemma 3.2. For any capture component W of preperiod k, m : W → D is proper.proof. Suppose an ∈ W tends to a0 ∈ ∂W and m(an) → m0 ∈ D. By Montel’stheorem, we may assume ψan converges to a conformal map ψ0 locally uniformly on D.Then ψ0(Dr) ⊂ B∗a0(0) for any r < 1. In fact, if there exist an r < 1 and a pointw1 ∈ ψ0(Dr)∩Ja0 , then, for any r′ > r, ψ0(Dr′) is an open neighborhood of w1 and containsa repelling periodic point w0 of Pa0 . This point is locally holomorphically continued to arepelling periodic point w(a) of Pa. Thus w(a) ∈ Ja ∩ ψ0(Dr′) for any a close to a0. Onthe other hand, since ψan(Dr′′) → ψ0(Dr′′) for any r′′ > r′, ψ0(Dr′) ⊂ ψan(Dr′′) holds forlarge n. Thus Jan does not intersect ψ0(Dr′) for large n. This is a contradiction. Thusψ0(D) ⊂ B∗a(0).Now P ka0(c−(a0)) = limn→∞ ψan(m(an)) = ψ0(m0) ∈ B∗a0(0), which implies a0 isparabolic. This contradicts the fact a0 ∈ ∂W . This completes the proof. 2Lemma 3.3. Let W be a capture component of preperiod k and a, b ∈ W . Suppose aqc-map ϕ : C→ C conjugates Pa on C−Ba(0) to Pb on C−Bb(0). If m(a) = m(b), thenthere exists a qc-conjugacy ψ on C between Pa and Pb such that ψ is conformal on Ba(0)and coincides with ϕ on C− Ba(0).proof. Since Ra ≡ R, the map ψ = ψb ◦ψ−1a gives a conformal equivalence betwee Paon B∗a(0) and Pb on B∗b (0). From the assumption m(a) = m(b), we have ψ(P ka (c−(a))) =P kb (c−(b)). Hence ψ can be holomorphically continued to Ba(0) by ψ = P−nb ◦ ψ ◦ P na .Recall that the maps ψa and ψb can be extended to the homeomorphisms from D ontoB∗a(0) and B∗b (0) respectively. Since ψ = ϕ on the inverse orbit of 0, ψ = ϕ holds alsoon ∂B∗a(0). Pulling back by Pa, the same holds on Ja = ∂Ba(0). If we extend ψ to thecomplemen of Ka by ψ = ϕ, ψ : C → C is a homeomorphism. Then ψ is a qc-map byRickman’s theorem. 2Proposition 3.1. For every capture component of preperiod k, m : W → D is a confor-mal isomorphism.proof. We have only to show the injectivity of m. Suppose m(a) = m(b). UsingBo¨ttcher coordinates, the conformal equivalence ϕa,c = ϕ−1c ◦ ϕa gives a holomorphicmotion of C −Ka. By the optimal λ-lemma of Slodkowski [Sl], this can be extended to36the holomorphic motion ϕa,c of C. We apply the previous lemma to ϕ = ϕa,b. Then thereexists a qc-conjugacy ψ between Pa and Pb on C, conformal on Ba(0) and coincides withϕ on C−Ba(0). Since ψ is a qc-map on C and conformal except on the measure zero setJa, it is conformal on C. Thus Pa is conformally conjugate to Pb on C. Then a = b. Thiscompletes the proof. 2Corollary 3.1. Every capture component is simply connected.4 Boundaries of capture componentsIn this section, we give some properties of the boundaries of capture components.Lemma 4.1. Suppose a0 6= 0. If the external ray Ra0(t) of Pa0 with angle t = p/3k ort = p/(2 ·3k) lands at z0 ∈ Ja0 and P na0(z0) is not a critical point for any n ≥ 0, then thereexists an open neighborhood U of a0 such that, for any a ∈ U , Ra(t) lands at a repellingor parabolic preperiodic point za. The landing point za depends holomorphically on a inU .Lemma 4.2. The external rays RM(t) of M1(1) with angles t = ±1/6,±1/3 land at theorigin.proof. The external rays Ra(0) and Ra(1/2) land at fixed points of Pa if they do notmeet the critical point c−(a). If Ra(0) (resp. Ra(1/2)) meets c−(a), then one of Ra(±1/3)(resp. one of Ra(±1/6)) meets the co-critical point c˜−(a), i.e. a lies on one of RM(±1/3)(resp. one of RM(±1/6)). In other words, Ra(0) (resp. Ra(1/2)) lands at a fixed point 0or −a unless a belongs to RM(±1/3) (resp. RM(±1/6)). Thus, at the accumulation pointa0 of RM(±1/3) (resp. RM(±1/6)), the landing of Ra(0) (resp. Ra(1/2)) is unstable. Onthe other hand, Lemma 4.1 implies those stabilities at a0 6= 0. Thus those rays must landat the origin. 2The four rays RM(±1/3) and RM(±1/6) and their landing point 0 separate the pa-rameter space into four parts. In the region R1 bounded by RM(−1/6) and RM(1/6),RM(0) (resp. RM(1/2)) lands at 0 (resp. −a). In the region R3 bounded by RM(1/3) andRM(−1/3), RM(1/2) (resp. RM(0)) lands at 0 (resp. −a). In the remaining two regionsR2 and R4, RM(0) and RM(1/2) land at 0.Lemma 4.3. For a point a on the boundary of a parabolic component W , Pa has neitherSiegel disks nor Cremer cycles.proof. If Pa0 has a Siegel or Cremer periodic point z0, then evidently a0 6= 0. ByLemma 3.3 of Kiwi [K], z0 and B∗a0(0) are separated by a union R of a finite collection ofclosed preperiodic external rays and R separates the orbits of critical points. Note that37preperiodic external rays must land at repelling or parabolic preperiodic points. SincePa0 has no other non-repelling cycles, the landing points of the rays in R are repellingexcept at 0. By the previous lemma, landing of those rays is stable around a0. Hence,the same holds for a ∈ W close to a0. But, in W , the orbit of c−(a) hits B∗a(0) after finiteiteration. This is a contradiction. 2We will show that, if W is capture, Pa has no parabolic cycles except 0. Suppose W iscapture and a ∈ W . Since Ka is pathwise connected, there is a path γ connecting c−(a)to c+(a). We denote za the point where γ first hits ∂B∗a(0). Since Ka is full, za is uniquelydetermined independent of the choice of γ. Since Ka is locally connected, at least twoexternal rays land at za. Among them, we take two rays Ra(t1) and Ra(t2) separatingc−(a) from B∗a(0) and consider the sector S0 bounded by these two rays and za, containingc−(a). In the following, we use the theory of orbit portraits developed in Milnor [M2].We denote by Aa(z) the set of angles of external rays of Pa landing at z.Lemma 4.4. Let W be a capture component of preperiod k and a ∈ W . Then za is aperiodic point of period m ≤ k.proof. Suppose za is not periodic. Put zj = Pja (za) and let Sj be the succesive imagesectors of S0 at zj bounded by Ra(3jt1) and Ra(3jt2). (Note that Pa(S0) covers C anddoubly covers S1.) Then, since Sj contains no critical points, it does not intersect B∗a(0)for any j ≥ 1. But its angular length 3j(t2 − t1) eventually becomes greater than one, acontradiction. 2Apparently za is repelling unless it is 0. If W ⊂ R1 orW ⊂ R3, za 6= 0 for any a ∈ W .Lemma 4.5. The point za is repelling also on ∂W unless it is 0.proof. Suppose za0 is not repelling for some a0 ∈ ∂W . It must be parabolic byLemma 4.3. Then, as a ∈ W tends to a0, za meets other repelling periodic points, sayza,j, 1 ≤ j ≤ k. Then Aa0(za0) is the union of Aa(za) and Aa(za,j). By the theory of orbitportraits, this happens only if k = 1 and the combinatorial rotation number at za0 is 0,i.e. za0 has just two angles. This contradicts the fact that za has at least two angles. 2The above proof implies that, if za 6= 0 for a ∈ W , the same holds also for a ∈ ∂W .Lemma 4.6. Suppose za 6= 0 for a ∈ W . Then Aa0(za0) = Aa(za) holds for a0 ∈ ∂Wand a ∈ W .proof. It follows za0 6= 0. Since za0 is repelling by Lemma 4.5, so is za for any a ∈Wclose to a0. By stability, it follows Aa0(za0) ⊂ Aa(za). If Aa0(za0) 6= Aa(za), there existst ∈ Aa(za) such that Ra0(t) lands at some point wa0 6= za0 . By stability, wa0 must beparabolic. Evidently wa0 6= 0. Then, for a ∈ W , the corresponding point wa 6= za isrepelling and has angle t, a contradiction. 238From the three lemmas above, it follows that, if za 6= 0 in W , the rays landing at zaseparate c−(a) and B∗a(0) for any a in a neighborhood W ′ of W . Since c−(a) 6= za, weconclude that c−(a) does not belong to an open neighborhood Ua of B∗a(0) for any a ∈ W ′.Put U ′a = Ua − ∪m−1j=0 Sj and U ′′a = Ua ∩ P−1a (Ua) − ∪m−1j=0 P−1a (Sj). Then Pa : U ′′a → U ′ais proper holomorphic. By thickening, we get a quadratic-like map Pa : Va → V ′a. Bystraightening, this map is hybrid equivalent to a quadratic polynomial p. Since Pa has aparabolic fixed point 0 of multiplier one, p(z) = z2 + 1/4.Capture component W where za = 0 sits in the region R2 or R4. In this case, tworays Ra(0) and Ra(1/2) stably lands at za = 0. Another fixed point z0 = −a is separatedby these two rays from c+(a). We take a path γ in Ka connecting z0 to 0.Lemma 4.7. There exists a sequence of points on the inverse orbit of z0 converging to 0.proof. First note that γ is not included in a Fatou component. Otherwise, thatcomponent is invariant since 0 has combinatorial rotation number 0.Suppose γ does not intersect any Fatou components. Then, since it does not containc−(a), Pa is injective on γ. Then Pa(γ) = γ since Pa fixes its endpoints. Because Pa isrepelling near both endpoints, Pa must have another fixed point in the interior of γ, acontradiction.Thus γ intersects both the Fatou set and the Julia set. The rotation number of z0is not 0 since 0, 1/2 /∈ Aa(z0). Hence, the local image of γ around z0 is another branch.Thus there exists a preimage z1 ∈ γ of z0. Let γ1 be the subpath of γ connecting 0 toz1. Suppose Pa(γ1) does not contain z1. Since the regions bounded by γ and Pa(γ1) isincluded in Ka, z1 is on the boundary of a Fatou component. Then z0 is also on theboundary of a Fatou component U . Since U is periodic, U is a periodic Fatou component,a contradiction. Thus there exists a preimage z2 ∈ γ1 of z1. Repeating this argument, weget a sequence zj ∈ γ on the inverse orbit of z0. Since its accumulation point is a fixedpoint, it must be 0. This completes the proof. 2By the same proof of Lemma 4.5, it follows that z0 is repelling for a ∈ ∂W . Moreover,at least two rays land at z0. Hence the same is true for zj. Using these rays, we get aquadratic-like map Pa : Va → V ′a, hybrid conjugate to p. We do not need thickening inthis case. Especially, since ∂B∗a(0) is homeomorphic to J(p), we get the following.Proposition 4.1. Let W be a capture component. Then ∂B∗a(0) is locally connected in aneighborhood W ′ of W .Corollary 4.1. Let W be a capture component. Then the Riemann map ψa : D→ B∗a(0)depends holomorphically on a in W ′.proof. The same proof of Lemma 3.1 works since we only use the local connectivityof ∂B∗a(0).39Lemma 4.8. Let W be a capture component of preperiod k. Then P ka (c−(a)) ∈ ∂B∗a(0) ifa ∈ ∂W .proof. Since a 7→ ∂B∗a(0) is a holomorphic motion on U , it is continuous with respectto the Hausdorff distance. For a ∈ W , we have P ka (c−(a)) ∈ B∗a(0). By continuity,P ka (c−(a)) ∈ B∗a(0) for a ∈ ∂W . Since P ka (c−(a)) /∈ B∗a(0) for a ∈ ∂W , the lemma follows.2Corollary 4.2. Let W be a capture component. Then, for a ∈ ∂W , Pa has no paraboliccycle except 0.Corollary 4.3. The map m : W → D extends to a continuous surjective map m : W →D. If, in addition, ∂W is locally connected, then m :W → D is a homeomorphism.proof. In order to prove the surjectivity of m, we define the internal ray in W as theinverse image of a ray in D by the map m. For any point w0 = e2piit ∈ ∂D, consider theinternal ray RW (t) ≡ m−1({re2piit; 0 ≤ r < 1}). We do not know whether this ray landsor not on ∂W . Take any accumulation point z0 = limn→∞m−1(rne2piit) ∈ ∂W . By thecontinuity of m, it follows m(z0) = e2piit = w0.If ∂W is locally connected, m−1 has a continuous extension to D. Then m−1 is theinverse of m also on ∂D. This completes the proof. 2References[DH] A. Douady and J. Hubbard: On the dynamics of polynomial-like mappings. Ann.Sci. Ec. Norm. Sup. 18 (1985), pp. 287–343.[EY] A. Epstein and M. Yampolsky: Geography of the cubic connectedness locus. Ann.Sci. Ec. Norm. Sup. 32 (1999), pp. 151–185.[K] J. Kiwi: Non-accessible critical points of Cremer polynomials. Stony Brook Preprint1996/2.[KN] Y. Komori and S. Nakane: Landing property of stretching rays for real cubic poly-nomials. Conformal Geometry and Dynamics 8 (2004), pp. 87–114.[L] P. Lavaurs: Syste`mes dynamiques holomorphes: explosion de points pe´riodiquesparaboliques. these Univ. Paris-Sud, 1989.[M1] J. Milnor: Remarks on iterated cubic maps. Experimental Math. 1 (1992), pp. 5–24.[M2] J. Milnor: Periodic orbits, external rays and the Mandelbrot set: An expositoryaccount. To appear in Asterisque.40[Sl] Z. Slodkowski: Extensions of holomorphic motions. Ann. Scuola Norm. Sup. Pisa Cl.Sci. 22 (1995), pp. 185–210.[W] P. Willumsen: Holomorphic dynamics : On accumulation of stretching rays. Ph.D.thesis Tech. Univ. Denmark, 1997.[Z] S. Zakeri: Dynamics of cubic Siegel polynomials. Comm. Math. Phys. 206 (1999),pp. 185–233.41

Capture Components for Cubic Polynomials with Parabolic Fixed Points

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Capture Components for Cubic Polynomials with Parabolic Fixed Points

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