2 次多項式の formal mating に対する canonical obstruction について

2 ^{次多項式の} formal mating ^に対する canonical obstruction ^について

Shota Kohara

For the formal mating f of two quadratic polynomials, the precise conditions for the existence of the Thurston obstruction for f is given by the theorem of Tan. When f satisfies these conditions, by the theorem of Thurston, there is no rational map which is Thurston equivalent to f. On the other hand, by using a special Thurston obstruction which is pinched under iteration of the Thurston pullback map, called canonical obstruction, we obtain some rational maps as some kind of limits of slow mating.

In this paper, for the formal mating f with Thurston obstruction, by a recursive

calculation method of Jung, we perform numerical verification of how the postcritical

set of f degenerates. By using the results, we estimate the canonical obstruction

for f and check the consistency with the theorem of Jung, and we calculate limit

rational maps. The organization of this thesis is as follows. In Section 1, we give

some definitions about Thurston map. Then, we define the Thurston obstruction,

which prevents a Thurston map from being Thurton equivalent to some rational

map, and give a statement of the theorem of Thurston. In Section 2, after explaining

polynomial dynamics, we define two types of matings, which are the methods to

obtain dynamics by pasting two polynomial dynamics. In Section 3, we define the

canonical obstruction. In Section 4, we present the theorem of Tan on the mating of

two quadratic polynomials, and the theorem of Jung on the canonical obstruction of

matings. In Section 5, we introduce Jung’s recursive calculation method. In Section 6,

we apply Jung’s method for numerical calculation of some quadratic formal matings

with Thurston obstruction. Then, we estimate the canonical obstruction and derive

rational maps that come from the pinching procedure. The program of Mathematica

used in Section 6 is written in Section 7.

2 ^{次多項式の} formal mating ^に対する canonical obstruction ^について

小原 翔太

目次

1 Thurston’s theory 4

2 Mating 7

3 Canonical obstruction 12

4 Quadratic mating 14

5 Jung’s method 17

6 Application 21

7 Appendix 29

Introduction

球面上の分岐被覆

f

P _f

Thurston map

. 2

Thurston map

, Thurston

れよりも少し弱い同値関係として

Thurston equivalence

.

,

f

Thurston equivalent

,

Thurston

([4])

, Thurston map f

Thurston equivalent

, f

Thurston obstruction

.

以上の

Thurston

. 2

P ,

Q

,

2

mating

, P , Q

Thurston map

.

, 2

2

P (z) = z ² + p, Q(z) = z ² + q (p, q ∈ C )

formal mating P ⊎ Q

Thurston obstruction

p, q

,

Tan

([11])

.

p, q

と

, Thurston

P ⊎ Q

Thurston equivalent

.

, Teichm¨ uller

Thurston pullback map

, canonical obstruction

Thurston obstruction

, P , Q

slow mating

.

そこで本論文では

, 2

formal mating

Thurston obstruction

, Jung

([12])

postcritical set

,

canonical obstruction

.

Jung

([12])

, canonical obstruction

.

以下

,

.

1

, Thurston map

, Thurston map

Thurston equivalent

, Thurston

obstruction

, Thurston

.

2

,

, 2

, mating

.

3

,

1

Thurston

obstruction

, Teichm¨ uller

Thurston pullback map σ

, canonical obstruction

.

4

, 2

formal mating

Thurston obstruction

Tan

,

formal mating

canonical obstruction

Jung

.

5

, σ

postcritical set

, Jung

.

, canonical obstruction

.

6

,

5

, canonical obstruction

.

7

,

6

Mathematica

.

Acknowledgement

本論文の執筆にあたり

,

.

,

,

.

1 Thurston’s theory

本章では

, branched covering

Thurston

.

, Thurston

Thurston obstruction

.

以下

, S ²

2

. Definition 1 (Thurston map).

f : S ² → S ²

branched map

,

a ∈ S ²

,

3

S ²

φ : U → C , ψ : V → C

n ∈ N

.

(i) a ∈ U, f (a) ∈ V (ii) φ(a) = ψ(f (a)) = 0

(iii) ψ ◦ f ◦ φ

¹ (z ) = z ⁿ on φ(U )

上の

n

, f

a

, deg(f, a)

.

Crit f := { a ∈ S ² | deg(f, a) > 1 }

f

critical point

,

P f := ∪

n

f ⁿ (Crit f )

を

f

postcritical set

. #P _f < ∞

, f

postcritically finite

, postcritically finite

branched map

Thurston map

.

Definition 2 (Thurston equivalent).

Thurston map f, g

Thurston equivalent

,

3

φ, φ

: S ² → S ²

.

(i) φ | P

= φ

| P

(ii) g = φ

◦ f ◦ φ

¹

(iii) φ, φ

P f

Definition 3 (

).

写像

f

n

f ⁿ

. a

f

, f ^m (a) = a

m ∈ N

,

m

a

.

Remark 1.

特に

, branched map f

critical point

f

, f

Thurston map

.

Definition 4 (hyperbolic orbifold).

Thurston map f

, ν f : S ² → N ∪ {∞}

.

a / ∈ P f

, ν f (a) := 1

.

, Crit f

f

b

n ∈ N

a = f ⁿ (b)

, ν _f (a) := ∞

.

a

,

ν f (a) := lcm

b

Crit

,f

(b)=x deg(f ^k , b)

とおく

. O f := (S ² , ν _f )

f

orbifold

,

χ

χ( O f ) := 2 − ∑

a

S

(

1 − 1 ν _f (a)

)

で定義する

. χ( O f ) < 0

O f

, hyperbolic orbifold

.

Thurston map

hyperbolic orbifold

, Thurston

.

,

.

Proposition 1.

#P f ≥ 5

f

hyperbolic orbifold

.

Proof. a / ∈ P f

ν f (a) = 1

, 1 − 1/ν f (a)

a ∈ P f

.

, a ∈ P f

ν f (a) ≥ 2

ν f (a) = ∞

,

χ( O f ) ≤ 2 − ∑

a

P

( 1 − 1

2 )

= 2 − #P _f 2

となるので

, #P f ≥ 5

χ( O f ) < 0

. □

Thurston obstruction

,

. S ²

, P f

S ² ∖ P f

, marked point

(S ² , P f )

.

Definition 5 (essential curve).

S ² ∖ P _f

γ

essential

, S ² ∖ γ

P f

2

.

Definition 6 (f -stable multicurve).

S ² ∖ P _f

Γ

(S ² , P _f )

multicurve

,

γ, γ

∈ Γ

, γ , γ

,

γ ∩ γ

= ∅

.

, (S ² , P f )

multicurve Γ

f -stable multicurve

,

γ ∈ Γ

, f

(γ )

essential

, Γ

P _f

.

Remark 2 ([4]).

#P f ≥ 3

, (S ² , P f )

multicurve Γ

(#P f − 3)

.

, f

Thurston map

, #Γ < ∞ .

Definition 7 (Thurston eigenvalue).

f

Thurston map

, Γ = { γ ₁ , γ ₂ , ..., γ _n }

f -stable multicurve

.

n × n

(a ij ) 1

i,j

n

, f

Thurston matrix

.

a _ij := ∑ 1

deg(f | η : η → γ j )

ただし

,

γ _i

P _f

f

(γ _j )

η

, deg(f | η )

γ j

f | η

.

, Perron-Frobenius

([4] Corollary C1.2)

λ Γ

.

λ _Γ

Γ

Thurston eigenvalue

.

Definition 8 (Thurston obstruction).

λ _Γ ≥ 1

f -stable multicurve Γ

, f

Thurston obstruction

.

, Thurston map

. Theorem 1 (Thurston, [4] Theorem 10.1.14).

hyperbolic orbifold

Thurston map f

, f

g

Thurston equivalent

, f

Thurston obstruction

.

, g

.

この定理によれば

, Thurston obstruction

Thurston map

,

.

3

canonical obstruction

と

, Thurston map

5

.

2 Mating

本章では

,

. 2

, 2

mating

.

C ˆ := C ∪ {∞} , D := { z ∈ C | | z | < 1 }

.

,

S ² = ˆ C

,

S ²

,

C ˆ

.

f : ˆ C → C ˆ

d ( ≥ 2)

. Remark 3 ([9] Theorem 2.1).

f

branched map

. Definition 9 (filled Julia set, Julia set).

K = K f := { z ∈ C | { f ⁿ (z) } n

}

f

filled Julia set

.

J = J f := ∂K f

を

f

Julia set

.

図

1 f

(z) = z

+ c

filled Julia set

Remark 4 ([5] Lemma 4.3).

f

(K _f ) = K _f = f (K _f ), f

(J _f ) = J _f = f(J _f )

.

Theorem 2 ([5] Theorem 9.5).

K

,

ϕ : C ∖ K → C ∖ D

,

w ∈ C ∖ D

.

ϕ ◦ f ◦ ϕ

¹ (w) = w ^d (1)

さらに

, ϕ

1

(d − 1)

. (

ϕ

B¨ ottcher map

.

C ∖ K ^f //

ϕ

C ∖ K

ϕ

C ∖ D ^w7→w

// C ∖ D

⟲

Outline of the proof.

r

, ϕ n (z) := (f ⁿ (z)) ^1/d

{| z | > r }

well- defined

, n → ∞

{| z | > r }

.

ϕ(z) := lim

n

ϕ n (z)

,

ϕ(f(z )) = lim

n

{ (f ⁿ⁺¹ (z )) ^1/d

} ^d = ϕ(z) ^d

となっているので

, ϕ

(1)

.

, K

Crit _f ∖ {∞} ⊂ K

, ϕ

C ∖ K

, ϕ : C ∖ K → C ∖ D

. □ Remark 5.

特に

d = 2

, B¨ ottcher map

. Definition 10 (external ray).

K

,

t ∈ R / Z

,

R t = R t (K ) := { ϕ

¹ (re ^2πit ) | 1 < r < ∞} ( ⊂ C ∖ K)

t

external ray

.

Definition 11 (landing point).

K

, Carath´ eodory

([5] Theorem 17.14)

ϕ

∂ D

.

t ∈ R / Z

,

γ(t) = γ f (t) := ϕ

¹ (e ^2πit )( ∈ J )

R t

landing point

.

Proposition 2.

K

,

t ∈ R / Z

f (R _t ) = R _d

_t , f (γ _t ) = γ _d

_t

.

Proof. t ∈ R / Z

.

1

,

1 < r < ∞

ϕ ◦ f ◦ ϕ

(re ^2πit ) = re ^2πidt

であるから

,

ϕ

¹

,

f ◦ ϕ

(re ^2πit ) = ϕ

(re ^2πidt )

が従う

.

f(R _t ) = R _d·t

. f

, f (γ _t ) = γ _d·t

. □ Remark 6 ([3] Theorem 9.1.6, [5] Theorem 18.3, 19.6, 19.7).

f

postcritically finite

, K f

.

図

2 rabbit

external ray

i = 1, 2

, f i : ˆ C → C ˆ

d ( ≥ 2)

, K f

K i

. K _i

, f _i

B¨ ottcher map ϕ _i

|

z lim

ϕ i (z) z = 1

.

Definition 12 (formal mating).

2

φ ₀ : C → D , φ

: C → C ∖ D ˆ

φ 0 (z) := z

√ | z | ² + 1 , φ

(z) := 1 φ 0 (z)

.

f i

, f 1 ⊎ f 2 : ˆ C → C ˆ

f 1 ⊎ f 2 =

 



 

φ 0 ◦ f 1 |

◦ φ 0

1 on D φ

◦ f 2 |

◦ φ

¹ on ˆ C ∖ D z 7→ z ^d on ∂ D

と定義すると

,

branched map

.

f 1 ⊎ f 2

f 1 , f 2

formal mating

図

3 basilica

rabbit

formal mating (d = 2)

Remark 7 ([6]).

各

K _i

,

t ∈ R / Z

, φ ₀ (ϕ

₁ ¹ (re ^2πit )), φ

(ϕ

₂ ¹ (re ^2πit ))

r → ∞

,

e ^2πit

.

,

K _i

, φ 0 (R t (K 1 )) ∪ φ

(R

t (K 2 ))

φ 0 (γ f

(t)), φ

(γ f

( − t))

C ˆ

.

Definition 13 (slow mating).

各

K _i

.

R > 1

.

i

,

{ ϕ

_i ¹ (Re ^2πit ) | t ∈ R / Z}

, K i

U i (R)

.

,

R

∈ (R, R ² )

, U i := U i (R

)

. z ∈ U 1 , w ∈ U 2

ϕ 1 (z) R

ϕ 2 (w)

R = 1 (

, | ϕ ₁ (z) | · | ϕ ₂ (w) | = R ^d

arg ϕ ₁ (z) = − arg ϕ ₂ (w))

,

∼

S R := U ₁ ⊔ U ₂ / ∼ ( ∼ = S ² )

を考える

.

i

,

q _i,R : U _i → S R

, { (q _i,R

¹ : q _i,R (U _i ) → C ) } i=1,2

S R

(

, S R

C ˆ

).

,

(1)

f _i (U _i (R)) = U _i (R ^d )

,

F f _R : S R → S R

.

F f R = {

q _1,R

◦ f ₁ ◦ q _1,R

on q _1,R (U ₁ ) q _2,R

◦ f ₂ ◦ q _2,R

¹ on q _2,R (U ₂ )

よって

, 2

h : S R → C ˆ , e h : S R

→ C ˆ

F R := e h ◦ F f R ◦ h

¹

, F R : ˆ C → C ˆ

(

). { F _R } R>1

f ₁ , f ₂

slow mating

.

形式的には

, R = ∞

slow mating

formal mating

, R

1

slow mating

.

5

, formal mating

slow mating

, slow mating

R → 1

formal mating

.

図

4 basilica

rabbit

slow mating (d = 2)

3 Canonical obstruction

本章では

,

1

Thurston obstruction

, Thurston pullback map

, canonical obstruction

.

f

Thurston map

. Definition 14 (Teichm¨ uller

).

2

ψ, ψ

: (S ² , P f ) → C ˆ

2

m : ˆ C → C ˆ

, ψ

ψ

.

(i) m ◦ ψ | P

= ψ

| P

(ii) m ◦ ψ

ψ

P _f

この同値関係に関する

ψ

[ψ]

,

T f := { [ψ] | ψ : (S ² , P _f ) → C ˆ

}

, f

Teichm¨ uller

.

Definition 15 (Thurston pullback map).

向きを保つ同相写像

ψ : (S ² , P _f ) → C ˆ

. ˆ C

{ (φ _i : U _i → C ) }

. ψ

, (S ² , P f )

µ := { (φ i ◦ ψ : ψ

¹ (U i ) → C ) }

で定める

.

, (S ² , P f )

µ

.

a ∈ S ² ∖ Crit _f

, f(a)

µ

φ _j ◦ ψ : ψ

(U _j ) → C

, f

(ψ

(U _j ))

a

V _j

f | V

.

, a

φ j ◦ ψ ◦ f : V j → C

.

b ∈ Crit f

, f

branched map

,

3

2

ξ _b : V _b → C , φ _k ◦ ψ : ψ

(U _k ) → C

.

(i) b ∈ V _b , f (b) ∈ ψ

¹ (U _k ) (ii) ξ b (b) = (φ k ◦ ψ)(f (b)) = 0

(iii) (φ k ◦ ψ) ◦ f ◦ ξ _b

(z) = z ^deg(f,b) on ξ b (V b )

このことから

, b

ξ b : V b → C

.

µ

S ²

S _µ ²

.

,

ψ

: (S _µ ²

, P _f ) → C ˆ

. σ f : T f → T f ; [ψ] → [ψ

]

Thurston pullback map

. (ψ

. )

(S ² , P f ) ^ψ

′

//

f

( ˆ C , ψ

(P f ))

ある有理写像

(S ² , P f ) ^ψ // ( ˆ C , ψ(P f ))

⟲

一般に

, 3

Z

, S ² ∖ Z

.

,

π : D → S ² ∖ Z

D

ρ

= 2 | dz | /(1 − | z | ² )

S ² ∖ Z

.

以下

, #P _f ≥ 3

.

Theorem 3 ([4] Proposition 3.3.8).

S ² ∖ P f

essential curve γ

, γ

S ² ∖ P f

1

.

Definition 16 (canonical obstruction).

τ = [ψ] ∈ T f , S ² ∖ P _f

essential curve γ

, Theorem 3

, ψ(γ)

C ∖ ˆ ψ(P f )

1

.

l τ (γ )

.

, τ i := σ ⁱ _f (τ ) (i ∈ N )

.

,

τ ∈ T f

l _τ

(γ ) ⁱ −→

0

S ² ∖ P _f

essential curve γ

multicurve Γ c

, f

canonical obstruction

.

Remark 8.

P _f

Γ _c

2

γ, γ

.

collaring theorem([3] Theorem 3.8.3)

, γ

0

, γ

, γ

∈ Γ c

に矛盾

.

, Γ _c

P _f

.

, f

hyperbolic orbifold

.

Theorem 4 ([7] Theorem 1.1).

(1) Γ _c = ∅

, f

Thurston equivalent

. (2) Γ c ̸ = ∅

, Γ c

f

Thurston obstruction

.

Theorem 5 ([7] Theorem 1.2).

任意の

τ ∈ T f

,

E τ

, γ / ∈ Γ _c

l _τ

(γ) ≥ E _τ

i ∈ N

.

よって特に

, τ _i

ψ _i

, i

ψ _i (P _f )

, ψ i (γ )

γ ∈ Γ c

.

5

,

i

ψ i (P f )

, Jung

.

また

, Theorem 4 (1)

Theorem 1

,

. Theorem 6.

f

obstruction

, f

canonical obstruction

.

4 Quadratic mating

本章では

, 2

f c (z) := z ² + c (c ∈ C )

, formal mating

Thurston obstruction

Tan

, Hubbard tree

formal mating

canonical obstruction

Jung

.

K _f

K _c

. Definition 17 (Mandelbrot

).

M := { c ∈ C | K c

}

Mandelbrot

. M

0

むものを

W 0

.

図

5 Mandelbrot

これを用いることで

, 2

formal mating

Thurston obstruction

, Tan

.

Theorem 7 (Tan, [11] Theorem 1.1).

f c

, f c

postcritically finite

,

c 1 , c 2

M ∖ W 0

, f c

⊎ f c

Thurston equivalent

.

以下

, 0

f c

n

. Remark 6

, K c

.

Theorem 8 ([4] Proposition 10.4.3).

K

_c

, 0

V ₀

.

,

2

.

(i) D

η _V

¹

0

◦ f _c ⁿ ◦ η _V

(z) = z ² (z ∈ D ) (

η _V

(0) = 0)

η V

: D → V 0

.

D ^z

^z

//

η

D

η

V 0 f

// V 0

⟲

(ii) K

_c

V

, f _c ^m : V → V ₀

m ∈ N

. Carath´ eodory

,

(f _c ^m | V )

¹ ◦ η _V

: D → V

∂ D

.

η V : D → V

.

Definition 18 (internal ray).

◦

K c

V

, { η V (re ^2πit ) } 0

r

1

V

t

internal ray

. Definition 19 (regulated path).

区間

[0, 1]

K c

, K

c

2

internal ray

, K _c

regulated path

.

Definition 20 (Hubbard tree).

P f

= { f _c ^k (0) } 1

k

n

, 2

K c

regulated path

T _c

, f _c

Hubbard tree

.

, (T _c ∩ K

_c ) ∖ P _f

,

Hubbard tree

edge

.

図

6

:basilica

:rabbit

Hubbard tree.

c 1 ̸ = 0

2

. (i) f c

postcritically finite

.

(ii) c ₂ := c ₁

, f _c

⊎ f _c

hyperbolic orbifold

.

このとき

Theorem 1, 6, 7

, f _c

⊎ f _c

canonical obstruction Γ

.

, c ₂ = c ₁

, T _c

edge E

, T _c

edge

.

E e

.

,

Jung

.

Theorem 9 (Jung, [12] Proposition 3.8).

任意の

γ ∈ Γ

, T _c

edge E

γ

φ ₀ (E)

φ

( E) e

1

.

Jung

,

6

canonical obstruction

.

5 Jung’s method

本章では

, formal mating

postcritical set

Thurston pullback map

, Jung

.

, canonical obstruction

, slow mating

.

p, q ∈ C

, P (z) := z ² + p, Q(z) := z ² + q

postcritically finite

. f := P ⊎ Q

.

i ∈ N ∪ { 0 }

, p i := P ⁱ (0), q i := Q ⁱ (0)

, N p := # { P ⁿ (0) } n

, N q := # { Q ⁿ (0) } n

.

Remark 9.

定義より

, 0

P

N _p

, p _N

= 0

.

, 0

Q

N q

, q N

= 0

.

R ≥ 5

, 0 ≤ t ≤ 1

R t = R ²

.

,

t

R t = R _t+1 ² ,

R t

t −→

1

, P , Q

slow mating { F R

} 0

t<

{ F _t } 0

t<

. Thurston pullback map σ _f

f

postcritical

set

,

{ ψ _t : (S ² , P _f ) → C} ˆ 0≤t<∞

換図式をみたすものを構成したい

.

S ^{2 ψ}

//

f

C ˆ

F

S ^{2 ψ}

// C ˆ

⟲

各

t ≥ 0

σ f ([ψ t ]) = [ψ t+1 ]

, σ f ([ψ 0 ]) = [ψ 1 ]

{ ψ _t } 0

t

1

.

,

F 0 (z ) = 1 + (q/R ² ) 1 + (p/R ² )

z ² + (p/R ² ) 1 + (q/R ² )z ²

.

| z | ≤ 4

,

ψ t (φ 0 (z)) := 1 + (q/R ² )(1 − t) 1 + (p/R ² )(1 − t)

z/R t

1 + (q/R ⁴ )(1 − t)(z − p) ψ t (φ

(z)) := 1 + (q/R ² )(1 − t)

1 + (p/R ² )(1 − t)

1 + (p/R ⁴ )(1 − t)(z − q) z/R t

と定めると

, R 0 = R ²

, | z | ≤ 4

F 0 ◦ ψ 1 ◦ φ 0 (z) = F 0

( z R

)

= 1 + (q/R ² ) 1 + (p/R ² )

(z ² + p)/R ²

1 + (q/R ⁴ ) { (z ² + p) − p } = ψ 0 ◦ φ 0 ◦ P (z) F ₀ ◦ ψ ₁ ◦ φ

(z) = F ₀

( R z

)

= 1 + (q/R ² ) 1 + (p/R ² )

1 + (p/R ⁴ ) { (z ² + q) − q }

(z ² + q)/R ² = ψ ₀ ◦ φ ₀ ◦ Q(z)

, φ 0 ( {| z | ≤ 4 } )

φ

( {| z | ≤ 4 } )

F 0 ◦ ψ 1 = ψ 0 ◦ f

.

ここで

Remark 6

p, q ∈ M

, M ⊂ {| z | ≤ 2 } (Hubbard 2 Proposition and Definition 10.5.4)

, | p | , | q | ≤ 2

.

R ≥ 5

, | z | ≤ 4

| ψ t (φ 0 (z)) | < 1, | ψ t (φ

(z)) | > 1

, φ 0 ( {| z | ≤ 4 } ), φ

( {| z | ≤ 4 } )

ψ t : (S ² , P f ) → C ˆ

.

, | z | ≤ 4

ψ _t (φ ₀ (z)) ≈ z/R _t , ψ _t (φ

(z)) ≈ R _t /z

, ψ _t

, | z | = 1

.

F 0 ◦ ψ 1 = ψ 0 ◦ f

ψ 0 , ψ 1

, σ f ([ψ 0 ]) = [ψ 1 ]

.

以上の

{ ψ _t } 0≤t≤1

σ _f

,

{ ψ _t } 0≤t<∞

.

t ≥ 1

, σ f ([ψ t ]) = [ψ t+1 ]

ψ 1

, ψ t+1 (0) = 0, ψ t+1 ( ∞ ) = ∞ , F t (1) = 1

.

0 ≤ t < ∞

i ∈ N

x i (t) := ψ t (φ 0 (p i )) , y i (t) := ψ t (φ

(q i ))