Holomorphic motion of fiber Julia sets

Transcription

1 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 37 No.1 (2014) 7 Shizuo Nakane We consider Axiom A polynomial skew products on C 2 of degree d 2. The stable manifold of a hyperbolic fiber Julia set gives a holomorphic motion of the fiber Julia set. In this note, we will show that this holomorphic motion is described by the fiberwise Böttcher coordinates. 1 Introduction In this note, we consider regular polynomial skew products on C 2 of degree d 2 of the form : f(z,w) =(p(z),q(z,w)). If we set q z (w) =q(z,w), the k-th iterate of f is written by f k (z,w) =(p k (z),q k z(w)) := (p k (z),q p k 1 (z) q z (w)). Hence the dynamics on the z-plane is that of p. We call the z-plane base space. The planes {z} C are called fibers. Then f preserves the family of fibers and this enables us to investigate the dynamics. Let K p and J p be the filled Julia set and Julia set respectively of the polynomial p and A p be the set of attracting periodic points of p. Let K be the set of points with bounded orbits and put K z := {w C;(z,w) K}. The fiber Julia set J z is the boundary of K z. The second Julia set J 2, which is a right analogue of the Julia set of a one-dimensional map, is characterized by J 2 = z Jp {z} J z. If f is Axiom A, then the map z J z is continuous in J p, hence J 2 = z Jp {z} J z. See Jonsson [J]. * Professor, General Education and Research Center, Tokyo Polytechnic University, Received Sept

2 8 The stable and unstable sets of a saddle set Λ are respectively defined by W s (Λ) = {y C 2 ; f n (y) Λ}, W u (Λ) = {y C 2 ; prehistory ŷ =(y k ) Λ}. Let Λ Ap = z Ap {z} J z be the saddle set in A p C. Since the map f preserves the vertical fibers, it is easy to see that W u (Λ Ap ) A p C. Then the local stable manifold Wloc s (x) of x =(z 0,w 0 ) Λ Ap is transversal to the fiber. That is, there exist >0 and a holomorphic function ϕ(z,w 0 ) in D(z 0, ) such that Wloc(x) s ={(z,ϕ(z,w 0 )); z D(z 0, )}. This function ϕ gives a holomorphic motion of J z0 (1) ϕ(z 0, ) =id Jz0, (2) ϕ(,w) is holomorphic in D(z 0, ) for each fixed w J z0, (3) ϕ z = ϕ(z, ) is injective for each fixed z. over D(z 0, ), that is, By the λ-lemma, ϕ : D(z 0, ) J z0 C is continuous. In this note, we will show that this holomorphic motion is expressed by the fiberwise Böttcher coodrinates Φ z. They are conformal maps in a neighborhood of the point at satisfying Φ p(z) q z (w) =Φ z (w) d. Note that, if J z is connected, then Φ z extends to a conformal map Φ z : C\K z C \ D. Let φ z be the inverse of the map Φ z. 2 Continuation of the holomorphic motion The following is the main theorem of this note. Theorem 2.1. Let f be an Axiom A polynomial skew product and z 0 A p. Suppose that J z0 is connected and that the holomorphic motion ϕ z : J z0 J z exists for z U for a domain U in the immediate basin U 0 of z 0. Then φ z = ϕ z φ z0 on D for z U.

3 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 37 No.1 (2014) 9 Define a fiberwise external ray R z (θ) with angle θ by R z (θ) =φ z ({re 2πiθ ; r>1}). Then Theorem 2.1 says that, if the rays R z0 (θ j ), 1 j k, land at a same point, so do the rays R z (θ j ), 1 j k, for any z V. Recently Comerford and Woodard obtained a same result in [CW] for analytic families of bounded polynomial sequences. If f is vertically expanding over K p, we can say more : these landing properties are preserved throughout U 0. As will be seen in Example 2.1, a new landing relation may appear as z approches the bounday U 0. To prove Theorem 2.1, we need a notion in Pommerenke [P]. A family {A z ; z V } of compact sets in C is uniformly locally connected if, for any >0, there exists δ > 0 such that for any z V and for any a, b A z with a b <δ, there exists a connected subset B A z with a, b B and diam B <. Proposition 2.1. For any compact set V in U, the family {J z ; z V } is uniformly locally connected. Put ψ z = ϕ 1 z : J z J z0 for z U. Lemma 2.1. For any δ 1 > 0, there exists δ>0 such that, for any z V and a, b J z with a b <δ, we have ψ z (a) ψ z (b) <δ 1. proof. We prove the lemma by contradiction. Suppose that there exists δ 1 > 0 such that, for any n 1, there exist z n V and a n,b n J zn satisfying a n b n < 1/n, ψ zn (a n ) ψ zn (b n ) δ 1. Put ã n = ψ zn (a n ), b n = ψ zn (b n ) J z0. We may assume that z n z, a n a, b n b, ã n ã, bn b. Then a = b J z, therefore ϕ z (ã ) = lim ϕ zn (ã n ) = lim a n = lim b n = lim ϕ zn ( b n )=ϕ z ( b ).

4 10 This contradicts the injectivity of ϕ z because ã b δ 1. This completes the proof of Lemma 2.1. proof of Proposition 2.1. By the equicontinuity of the family {ϕ z ; z V }, for any >0, there exists 1 > 0 such that diam ϕ z (B) < if diam B < 1. By the local connectivity of J z0, for this 1, there exists δ 1 > 0 such that, for any ã, b J z0 with ã b <δ 1, there exists a connected subset B J z0 with diam B < 1 containing ã, b. By Lemma 2.1, for this δ 1, there exists δ>0 such that, for any z V and a, b J z with a b <δ, we have ψ z (a) ψ z (b) <δ 1. For any given >0, choose 1,δ 1 and δ as above. Then, for any z V and for any a, b J z with a b <δ, there exists a connected set B J z0 with diam B < 1 containing ψ z (a),ψ z (b). The set ϕ z (B) J z is connected, contains a, b and satisfies diam ϕ z (B) <. Thus the family {J z ; z V } is uniformly locally connected. This completes the proof of Proposition 2.1. proof of Theorem 2.1. Note that, for any w C \ D, the map z φ z (w) is continuous in U 0. From the assumption, J z0 is locally connected, hence so is J z for z U. By Proposition 2.1, the family {J z ; z V } is uniformly locally connected. By Theorem 9.11 in [P], φ z φ z0 uniformly on C \ D as z z 0. Now, take a = φ z0 (e 2πiθ ) J z0. Then, since Q n z φ z (e 2πiθ )=φ zn (e 2πidnθ ), z n = p n (z), for any n, it follows that d(q n z φ z (e 2πiθ ),Q n z 0 (a)) = d(φ zn (e 2πidnθ ),φ p n (z 0 )(e 2πidnθ )) 0. Thus (z,φ z (e 2πiθ )) W a ({z} C), hence φ z (e 2πiθ )=ϕ z (a) =ϕ z φ z0 (e 2πiθ ). This completes the proof of Theorem 2.1. If f is vertically expanding over K p, we can show a stronger result. Corollary 2.1. If f is vertically expanding over K p, both functions φ z and ϕ z extend continuously to z U 0, hence φ z = ϕ z φ z0 holds for z U 0. Example 2.1. f(z,w) =(z 2,w 2 + cz). If we set g c (w) =w 2 + c, then f n (z,w) =(z 2n,z 2n 1 g n c ( w z )). We have C p = {0} = A p. It easily follows that f is Axiom A (resp. connected) if and

5 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 37 No.1 (2014) 11 only if g c is hyperbolic (resp. J gc is connected). Thus, f is vertically expanding over K p if c lies in a hyperbolic component of the Mandelbrot set. Let φ c : C \ D C \ K gc be the inverse Böttcher coordinate of g c. Then it follows that ϕ z (w) =φ z (w) = zφ c ( w ), z D, z which depends holomorphically on z because φ c is an odd function. The internal ray R a (t) for a J 0 = D is written as R a (t) ={(re 2πit, re πit a φ c ( )); r<1}. re πit It lands at the point (e 2πit,e πit φ c (ae πit )) J 2. The fiber Julia set J z = φ z ( D) is a Jordan curve if z D, while it is a rotation of the Julia set J c if z D. Thus, pinching occurs as z approaches D. See the following figures. Figure 1: Fiber Julia sets (c = 1, from left : z =0.98, 0.999, , 1) Figure 2: Fiber Julia sets (from left : z =0.98, 0.999, , 1)

6 12 References [CW] M. Comerford & T. Woodard: Preservation of external rays in nonautonomous iteration. J. Difference Equations and Applications 19 (2013), pp [J] M. Jonsson: Dynamics of polynomial skew products on C 2. Math. Ann. 314 (1999), pp [N] S. Nakane: Postcritical sets and saddle basic sets for Axiom A polynomial skew products on C 2. Ergod. Th. & Dynam. Sys. 33 (2013), pp [P] C. Pommerenke: Univalent functions. Vandenhoeck & Ruprecht, Göttingen, Germany (1975). [R] R. Roeder: A dichotomy for Fatou components of polynomial skew products. Conformal Geometry & Dynamics 15 (2011), pp