Joint work with Cui Guizhen Tan Lei, Universit´e de Cergy-Pontoise
July 13, 2007
F :E → L a topologicalcovering (preserving the orientation)
F :E → L a topologicalcovering (preserving the orientation) ConjugacyF ∼conj F′
F :E → L a topologicalcovering (preserving the orientation) ConjugacyF ∼conj F′
if ∃ L−→Φ
≈ L′ s.t. F′◦Φ|E = Φ◦F.
L ⊃ E −→Φ|E
≈ E′ ⊂ L′ F ↓ ↓F′
L −→Φ L′ .
F :E → L a topologicalcovering (preserving the orientation) ConjugacyF ∼conj F′
if ∃ L−→Φ
≈ L′ s.t. F′◦Φ|E = Φ◦F.
L ⊃ E −→Φ|E
≈ E′ ⊂ L′ F ↓ ↓F′
L −→Φ L′ .
Comb-equivalenceF ∼combF′
F :E → L a topologicalcovering (preserving the orientation) ConjugacyF ∼conj F′
if ∃ L−→Φ
≈ L′ s.t. F′◦Φ|E = Φ◦F.
L ⊃ E −→Φ|E
≈ E′ ⊂ L′ F ↓ ↓F′
L −→Φ L′ .
Comb-equivalenceF ∼combF′ if∃ θ isotop.∼
rel∂L id
F :E → L a topologicalcovering (preserving the orientation) ConjugacyF ∼conj F′
if ∃ L−→Φ
≈ L′ s.t. F′◦Φ|E = Φ◦F.
L ⊃ E −→Φ|E
≈ E′ ⊂ L′ F ↓ ↓F′
L −→Φ L′ .
Comb-equivalenceF ∼combF′ if∃ θ isotop.∼
rel∂L id s.t. F ◦θ−1 ∼conj F′;
F :E → L a topologicalcovering (preserving the orientation) ConjugacyF ∼conj F′
if ∃ L−→Φ
≈ L′ s.t. F′◦Φ|E = Φ◦F.
L ⊃ E −→Φ|E
≈ E′ ⊂ L′ F ↓ ↓F′
L −→Φ L′ .
Comb-equivalenceF ∼combF′ if∃ θ isotop.∼
rel∂L id s.t. F ◦θ−1 ∼conj F′;
F :E → L a topologicalcovering (preserving the orientation) ConjugacyF ∼conj F′
if ∃ L−→Φ
≈ L′ s.t. F′◦Φ|E = Φ◦F.
L ⊃ E −→Φ|E
≈ E′ ⊂ L′ F ↓ ↓F′
L −→Φ L′ .
Comb-equivalenceF ∼combF′ if∃ θ isotop.∼
rel∂L id s.t. F ◦θ−1 ∼conj F′;
i.e. if∃ L−→θ
≈ L−→Φ
≈ L′ with
F :E → L a topologicalcovering (preserving the orientation) ConjugacyF ∼conj F′
if ∃ L−→Φ
≈ L′ s.t. F′◦Φ|E = Φ◦F.
L ⊃ E −→Φ|E
≈ E′ ⊂ L′ F ↓ ↓F′
L −→Φ L′ .
Comb-equivalenceF ∼combF′ if∃ θ isotop.∼
rel∂L id s.t. F ◦θ−1 ∼conj F′;
F′◦Φ◦θ|E = Φ◦F
2. Given a repellor F, is F ∼comb a hol.-model? A criterion?
2. Given a repellor F, is F ∼comb a hol.-model? A criterion?
interests : key step in the generalization of Thurston’s theorem to non-postcritically finite maps.
2. Given a repellor F, is F ∼comb a hol.-model? A criterion?
interests : key step in the generalization of Thurston’s theorem to non-postcritically finite maps.
Applications: landing properties of parameter rays, cusp points on the boundary of hyperbolic components, hyperbolic components having a commun boundary point...
(the method is intrinsic and provides dynamical informations).
z2
−1
∗
−1
γ
γ γ0
z2
−1
∗
∗
β1 β0
β γ
γ γ0 β
−1
−1
1:1 z2
−1
∗
1:1 2:1
∗ 2:1
β1 β0
β γ
γ γ0 β
−1
−1
E1 E2
Q
L = Q, E = E1 ∪E2, with Q,E1,E2 Jordan discs.
E1 E2
F F
Q
≅ ≅
L = Q, E = E1 ∪E2, with Q,E1,E2 Jordan discs.
F :Ei →Q homeo.
E1 E2
F F
Q
≅ ≅
E 2 Q
L = Q, E = E1∪E2, with Q,E1,E2 closed annuli.
E1 E2
F F
Q
≅ ≅
F F
E1 E 2
Q
d1:1 d2 :1
L = Q, E = E1∪E2, with Q,E1,E2 closed annuli.
F :Ei →Q are di : 1 coverings
L=Q⊔A
Q a pair of pants,Aan annulus
EQQ
EAQ
L=Q⊔A
Q a pair of pants,Aan annulus F :EQQ →Q, EAQ →Q are coverings
EQQ
EQA1 EAQ
EAA2 EAA1 EQA2
0000 00 1111 11
0000 00 1111 11
L=Q⊔A
Q a pair of pants,Aan annulus F :EQQ →Q, EAQ →Q are coverings F :E∗Aj →Aare coverings
EQQ
EQA1 EAQ
EAA2 EAA1 EQA2
0000 00 1111 11
0000 00 1111 11
L=Q⊔A
Q a pair of pants,Aan annulus F :EQQ →Q, EAQ →Q are coverings F :E∗Aj →Aare coverings
F :S
E∗ → L=Q⊔Aa repellor
E1 E2
F F
Q
≅ ≅
UsuallyF 6∼conj a hol.-model
E1 E2
F F
Q
≅ ≅
UsuallyF 6∼conj a hol.-model But one has always
E1 E2
F F
Q
≅ ≅
UsuallyF 6∼conj a hol.-model But one has always
F ∼combF′ holomorphic.
E1 E2
F F
Q
≅ ≅
UsuallyF 6∼conj a hol.-model But one has always
F ∼combF′ holomorphic.
F F
Q
d1:1 d2 :1
F ∼comba hol.-model⇐⇒
E1 E2
F F
Q
≅ ≅
UsuallyF 6∼conj a hol.-model But one has always
F ∼combF′ holomorphic.
F F
E1 E 2
Q
d1:1 d2 :1
F ∼comba hol.-model⇐⇒
X
i
1 di
<1 (moduli prob.)
1 E2
F F
Q
F’ F’
E
1. Construct at first the model F′: F′ :Ei → Q
|| ||
Ei′ Q′
holomorphic .
1 E2
F F
Q
F’ F’
E
1. Construct at first the model F′: F′ :Ei → Q
|| ||
Ei′ Q′
holomorphic 2. Set Φ =id :Q →Q.
1 E2
F F
Q
F’ F’
E
1. Construct at first the model F′: F′ :Ei → Q
|| ||
Ei′ Q′
holomorphic 2. Set Φ =id :Q →Q.
3. Set
θ|Ei = (F′)−1◦F θ|∂Q = id
1 E2
F F
Q
F’ F’
E
1. Construct at first the model F′: F′ :Ei → Q
|| ||
Ei′ Q′
holomorphic 2. Set Φ =id :Q →Q.
3. Set
θ|Ei = (F′)−1◦F θ|∂Q = id
thenF′◦Φ◦θ|Ei = Φ◦F.
1 E2
F F
Q
F’ F’
E
1. Construct at first the model F′: F′ :Ei → Q
|| ||
Ei′ Q′
holomorphic 2. Set Φ =id :Q →Q.
3. Set
θ|Ei = (F′)−1◦F θ|∂Q = id
thenF′◦Φ◦θ|Ei = Φ◦F. 4. Extend θ as a homeo.
1 E2
F F
Q
F’ F’
E
1. Construct at first the model F′: F′ :Ei → Q
|| ||
Ei′ Q′
holomorphic 2. Set Φ =id :Q →Q.
3. Set
θ|Ei = (F′)−1◦F θ|∂Q = id
thenF′◦Φ◦θ|Ei = Φ◦F. 4. Extend θ as a homeo.
5. Check θ isotop.∼
rel∂Q id
F’ F’
E1 E 2
Q
Q’
d1 :1 d
2 :1
E’1 E’2
1. Construct at first a model hol. repellor F′ :Ei′ →Q′ This is possible precisely because P 1
di <1.
F’ F’
F
E1 E 2
Q
Q’
Q Φ
E’1 E’2
1. Construct at first a model hol. repellor F′ :Ei′ →Q′ This is possible precisely because P 1
di <1.
2. Set Φ :Q →Q′ a homeo.
F’ F’
E1 E 2
Q
Q’
Q Φ
E’1 E’2
1. Construct at first a model hol. repellor F′ :Ei′ →Q′ This is possible precisely because P 1
di <1.
2. Set Φ :Q →Q′ a homeo. 3. Pull-backEi′ by Φ
F’ F’
F
E1 E 2
Q
Q’
Q Φ
θ θ E’1 E’2
1. Construct at first a model hol. repellor F′ :Ei′ →Q′ This is possible precisely because P 1
di <1.
2. Set Φ :Q →Q′ a homeo. 3. Pull-backEi′ by Φ 4. Set
θ|Ei : = (Φ−1◦F′◦Φ)−1◦F :Ei−→Φ−1(Ei) θ|∂Q = id
F’ F’
F
E1 E 2
Q
Q’
Q Φ
θ θ E’1 E’2
1. Construct at first a model hol. repellor F′ :Ei′ →Q′ This is possible precisely because P 1
di <1.
2. Set Φ :Q →Q′ a homeo. 3. Pull-backEi′ by Φ 4. Set
θ|Ei : = (Φ−1◦F′◦Φ)−1◦F :Ei−→Φ−1(Ei) θ|∂Q = id
F’ F’
E1 E 2
Q’
Q Φ
θ θ
Q β
E’1 E’2
1. Construct at first a model hol. repellor F′ :Ei′ →Q′ This is possible precisely because P 1
di <1.
2. Set Φ :Q →Q′ a homeo. 3. Pull-backEi′ by Φ 4. Set
θ|Ei : = (Φ−1◦F′◦Φ)−1◦F :Ei−→Φ−1(Ei) θ|∂Q = id
5. Extend θ as a homeo. Q →Q. ThenF′◦Φ◦θ|Ei = Φ◦F.
∗
β1 β
γ β
−1
−1
γ0 β0 E
L
∗
β1 β
γ β
−1
−1
γ0 β0 E
L
Let F :E →L be a covering of degree 2. This induces a covering F :{β0, β−1, β∗, β1} → {γ0, γ−1, γ∗} of degree 2. Then
F ∼comb a hol. model ⇐⇒
∗
β1 β
γ β
−1
−1
γ0 β0 E
L
Let F :E →L be a covering of degree 2. This induces a covering F :{β , β , β , β } → {γ , γ , γ } of degree 2. Then
∗
∗
β1 β
γ
γ β
−1
−1
γ0 β0 E
L
F f
φ
Proof. Mark one point on each corner.
∗
∗
β1 β
γ
γ β
−1
−1
γ0 β0 E
L
F f
φ
Proof. Mark one point on each corner. Extend F as a branched cover with
∗
∗
β1 β
γ
γ β
−1
−1
γ0 β0 E
L
F f
φ
Proof. Mark one point on each corner. Extend F as a branched cover with PF ⊂ P, and F
∗
∗
β1 β
γ
γ β
−1
−1
γ0 β0 E
L
F f
φ
Proof. Mark one point on each corner. Extend F as a branched cover with PF ⊂ P, and F holomorphic outsideL.
Thurston
∗
∗
β1 β
γ
γ β
−1
−1
γ0 β0 E
L
F f
φ
Proof. Mark one point on each corner. Extend F as a branched cover with PF ⊂ P, and F holomorphic outsideL.
#P = 3 Thurston=⇒ there are φ=id,ψ homeomorphism, and f a rational map, such thatf ◦ψ=φ◦F, and ψisotopic toφrel P.
EQQ
EQA1 EAQ
EAA2 EAA1 EQA2
0000 00 1111 11
0000 00 1111 11
EQQ →Q, EAQ →Q E∗Aj →A
F ∼comb a hol.-model⇐⇒
EQQ
F
a renormalization
EQQ →Q, EAQ →Q E∗Aj →A
F∼comb a hol.-model⇐⇒
• (F,EQQ,Q) hasno Thurston obstructions ((F,EQQ,Q) is called arenormalizationof F).
•
EQQ
EQA1 EAQ
EAA2 EAA1 EQA2
0000 00 1111 11
0000 00 1111 11
EQQ →Q, EAQ →Q E∗Aj →A
F∼comb a hol.-model⇐⇒
• (F,EQQ,Q) hasno Thurston obstructions ((F,EQQ,Q) is called arenormalizationof F).
γ
γ 1
2
γ4 γ3
Let Γ ={homotop. cl. (withinL) of∂L}
= {γ1, γ2, γ3, γ4} be the boundary multic- urve. Its transition matrix D = (aij) is de- fined by:
γ
γ 1
2
γ4 γ3
Let Γ ={homotop. cl. (withinL) of∂L}
= {γ1, γ2, γ3, γ4} be the boundary multic- urve. Its transition matrix D = (aij) is de- fined by: aij =X
α
1
deg(F :α→γj),
γ
γ 1
2
γ4 γ3
Let Γ ={homotop. cl. (withinL) of∂L}
= {γ1, γ2, γ3, γ4} be the boundary multic- urve. Its transition matrix D = (aij) is de- fined by: aij =X
α
1
deg(F :α→γj), withα running through the curves in
F−1(γj) homotopic (withinL=Q⊔A) toγi.
γ4
α24
α44
0000 00 1111 11
0000 00 1111 11
Let Γ ={homotop. cl. (withinL) of∂L}
= {γ1, γ2, γ3, γ4} be the boundary multic- urve. Its transition matrix D = (aij) is de- fined by: aij =X
α
1
deg(F :α→γj), withα running through the curves in
F−1(γj) homotopic (withinL=Q⊔A) toγi.
γ4
α24
α44
0000 00 1111 11
0000 00 1111 11
Let Γ ={homotop. cl. (withinL) of∂L}
= {γ1, γ2, γ3, γ4} be the boundary multic- urve. Its transition matrix D = (aij) is de- fined by: aij =X
α
1
deg(F :α→γj), withα running through the curves in
F−1(γj) homotopic (withinL=Q⊔A) toγi. λ(D)<1 corresponds toP 1
<1, deals with moduli problems.
γ4
α24
α44
0000 00 1111 11
0000 00 1111 11
Let Γ ={homotop. cl. (withinL) of∂L}
= {γ1, γ2, γ3, γ4} be the boundary multic- urve. Its transition matrix D = (aij) is de- fined by: aij =X
α
1
deg(F :α→γj), withα running through the curves in
F−1(γj) homotopic (withinL=Q⊔A) toγi.
Theorem 1. Let (F,E,L) be a repellor, of constant complexity (see below). If
Theorem 1. Let (F,E,L) be a repellor, of constant complexity (see below). If
every renormalization has no Thurston obstructions;
andF has no boundary obstructions,
Theorem 1. Let (F,E,L) be a repellor, of constant complexity (see below). If
every renormalization has no Thurston obstructions;
andF has no boundary obstructions, thenF ∼comb a hol.-model.
Theorem 1. Let (F,E,L) be a repellor, of constant complexity (see below). If
every renormalization has no Thurston obstructions;
andF has no boundary obstructions, thenF ∼comb a hol.-model.
(Combining this theorem with quasi-conformal surgery, we get a generalization of Thurston’s theorem for hyperbolic maps.)
Theorem 1. Let (F,E,L) be a repellor, of constant complexity (see below). If
every renormalization has no Thurston obstructions;
andF has no boundary obstructions, thenF ∼comb a hol.-model.
(Combining this theorem with quasi-conformal surgery, we get a generalization of Thurston’s theorem for hyperbolic maps.) What isconstant complexity?
Theorem 1. Let (F,E,L) be a repellor, of constant complexity (see below). If
every renormalization has no Thurston obstructions;
andF has no boundary obstructions, thenF ∼comb a hol.-model.
(Combining this theorem with quasi-conformal surgery, we get a generalization of Thurston’s theorem for hyperbolic maps.) What isconstant complexity?
renormalization?
Theorem 1. Let (F,E,L) be a repellor, of constant complexity (see below). If
every renormalization has no Thurston obstructions;
andF has no boundary obstructions, thenF ∼comb a hol.-model.
(Combining this theorem with quasi-conformal surgery, we get a generalization of Thurston’s theorem for hyperbolic maps.) What isconstant complexity?
renormalization?
Thurston obstructions? Boundary obstructions?
Q a hole
Qand EQ E
A repellor (F,E,L) is ofconstant complexity, if,
∀ non-(disc-annular)L-piece Q,
∃! E-piece EQ ⊂Q s.t each curve in ∂Q is
Q a hole
Qand EQ
Q E
FORBIDDEN:
new homotopic cl.
A repellor (F,E,L) is ofconstant complexity, if,
∀ non-(disc-annular)L-piece Q,
F∗-periodic Q induces arenormalization (F ,EQ,Q).
Extension. Let P ={one point in each component ofCrQ}. Let h be a postcritically finitebranched cover extension of (Fp,EQp,Q) withPh⊂ P andh(P)⊂ P.
Fp
F∗-periodic Q induces arenormalization (F ,EQ,Q).
Extension. Let P ={one point in each component ofCrQ}. Let h be a postcritically finitebranched cover extension of (Fp,EQp,Q) withPh⊂ P andh(P)⊂ P.
A Jordan curveγis saidnon-peripheralif
γ∩ P=∅and each component ofCrγcontains≥2 points ofP.
F∗-periodic Q induces arenormalization (F ,EQ,Q).
Extension. Let P ={one point in each component ofCrQ}. Let h be a postcritically finitebranched cover extension of (Fp,EQp,Q) withPh⊂ P andh(P)⊂ P.
A Jordan curveγis saidnon-peripheralif
γ∩ P=∅and each component ofCrγcontains≥2 points ofP. Γ ={γ1,· · · , γk} is said amulticurve, if theγi’s are
non-peripheral, mutually disjoint and mutually non-homotopic.
F∗-periodic Q induces arenormalization (F ,EQ,Q).
Extension. Let P ={one point in each component ofCrQ}. Let h be a postcritically finitebranched cover extension of (Fp,EQp,Q) withPh⊂ P andh(P)⊂ P.
A Jordan curveγis saidnon-peripheralif
γ∩ P=∅and each component ofCrγcontains≥2 points ofP. Γ ={γ1,· · · , γk} is said amulticurve, if theγi’s are
non-peripheral, mutually disjoint and mutually non-homotopic.
Γ induces atransition matrixDΓ= (bij) with bij =X
δ
1
deg(h:δ→γj), where δruns through the curves in h−1(γj) homotopic (relP) toγi.
F∗-periodic Q induces arenormalization (F ,EQ,Q).
Extension. Let P ={one point in each component ofCrQ}. Let h be a postcritically finitebranched cover extension of (Fp,EQp,Q) withPh⊂ P andh(P)⊂ P.
A Jordan curveγis saidnon-peripheralif
γ∩ P=∅and each component ofCrγcontains≥2 points ofP. Γ ={γ1,· · · , γk} is said amulticurve, if theγi’s are
non-peripheral, mutually disjoint and mutually non-homotopic.
Γ induces atransition matrixDΓ= (bij) with bij =X
δ
1
deg(h:δ→γj), where δruns through the curves in h−1(γj) homotopic (relP) toγi.
(Fp,Ep,Q) hasa Thurston obstructionif∃Γ such thatλ(DΓ)≥1.
Let (F,E,L) be a repellor of constant complexity. Itsboundary multicurve is:
Let (F,E,L) be a repellor of constant complexity. Itsboundary multicurve is:
Y =
{curves is∂L, non null-homotopic withinL}/homotopy withinL.
Let (F,E,L) be a repellor of constant complexity. Itsboundary multicurve is:
Y =
{curves is∂L, non null-homotopic withinL}/homotopy withinL.
The corresponding transition matrix D= (aij) is defined by:
Let (F,E,L) be a repellor of constant complexity. Itsboundary multicurve is:
Y =
{curves is∂L, non null-homotopic withinL}/homotopy withinL.
The corresponding transition matrix D= (aij) is defined by:
aij =X
α
1
deg(F :α→γj),
Let (F,E,L) be a repellor of constant complexity. Itsboundary multicurve is:
Y =
{curves is∂L, non null-homotopic withinL}/homotopy withinL.
The corresponding transition matrix D= (aij) is defined by:
aij =X
α
1
deg(F :α→γj),withα running through the curves in F−1(γj) homotopic (withinL=Q⊔A) toγi.
Let (F,E,L) be a repellor of constant complexity. Itsboundary multicurve is:
Y =
{curves is∂L, non null-homotopic withinL}/homotopy withinL.
The corresponding transition matrix D= (aij) is defined by:
aij =X
α
1
deg(F :α→γj),withα running through the curves in F−1(γj) homotopic (withinL=Q⊔A) toγi.
We say thatF hasa boundary obstruction if λ(D)≥1.
For renormalizations. If a renormalization (Fp,EQp,Q) has no boundary obstructions, and
has no Thurston obstructions,
then it is ∼comb a hol.-model (with any prescribed equipotentials).
(Proof: Thurston+Shishikura.)
For renormalizations. If a renormalization (Fp,EQp,Q) has no boundary obstructions, and
has no Thurston obstructions,
then it is ∼comb a hol.-model (with any prescribed equipotentials).
(Proof: Thurston+Shishikura.)
For a periodic cycle of renormalization. If (F,Sp
i=1Ei,Sp
i=1Qi) is a periodic cycle of renormalization satisfying
F has no boundary obstructions, and has no Thurston obstructions,
then it is ∼comb a hol.-model (with any prescribed equipotentials).
For renormalizations. If a renormalization (Fp,EQp,Q) has no boundary obstructions, and
has no Thurston obstructions,
then it is ∼comb a hol.-model (with any prescribed equipotentials).
(Proof: Thurston+Shishikura.)
For a periodic cycle of renormalization. If (F,Sp
i=1Ei,Sp
i=1Qi) is a periodic cycle of renormalization satisfying
F has no boundary obstructions, and has no Thurston obstructions,
then it is ∼comb a hol.-model (with any prescribed equipotentials).
This,+ Gr¨otzsch ‘≤’, proves:
For repellors. Let (F,E,L) be a repellor, of constant complexity. If it has no boundary obstructions, and
branched coverings, holomorphic on U(acc(PG)), without
Thurston obstructions. Then G is comb-equivalent to a hyperbolic rational mapR, i.e. ∃θ, φ such thatG ◦θ−1∼φ−conjR, with θ∼id relU∪ PG, andφ|U holomorphic. Moreover R is unique.
branched coverings, holomorphic on U(acc(PG)), without
Thurston obstructions. Then G is comb-equivalent to a hyperbolic rational mapR, i.e. ∃θ, φ such thatG ◦θ−1∼φ−conjR, with θ∼id relU∪ PG, andφ|U holomorphic. Moreover R is unique.
+ distortion controls, plumbing and pinching surgeries:
For geometrically finite maps Let G :C→C be a geometrically type branched coverings, holomorphic on U∗(acc(PG)), without Thurston obstructions. Then G is comb-equivalent to a unique geometrically finite rational mapR, i.e. ∃θ, φ such that
G ◦θ−1∼φ−conjR, with θ∼id relU ∪ PG, and φ|U holomorphic.
branched coverings, holomorphic on U(acc(PG)), without
Thurston obstructions. Then G is comb-equivalent to a hyperbolic rational mapR, i.e. ∃θ, φ such thatG ◦θ−1∼φ−conjR, with θ∼id relU∪ PG, andφ|U holomorphic. Moreover R is unique.
+ distortion controls, plumbing and pinching surgeries:
For geometrically finite maps Let G :C→C be a geometrically type branched coverings, holomorphic on U∗(acc(PG)), without Thurston obstructions. Then G is comb-equivalent to a unique geometrically finite rational mapR, i.e. ∃θ, φ such that
G ◦θ−1∼φ−conjR, with θ∼id relU ∪ PG, and φ|U holomorphic.
with the same price:
Application for parabolic perturbations A parabolic map is stably
branched coverings, holomorphic on U(acc(PG)), without
Thurston obstructions. Then G is comb-equivalent to a hyperbolic rational mapR, i.e. ∃θ, φ such thatG ◦θ−1∼φ−conjR, with θ∼id relU∪ PG, andφ|U holomorphic. Moreover R is unique.
+ distortion controls, plumbing and pinching surgeries:
For geometrically finite maps Let G :C→C be a geometrically type branched coverings, holomorphic on U∗(acc(PG)), without Thurston obstructions. Then G is comb-equivalent to a unique geometrically finite rational mapR, i.e. ∃θ, φ such that
G ◦θ−1∼φ−conjR, with θ∼id relU ∪ PG, and φ|U holomorphic.
with the same price:
Application for parabolic perturbations A parabolic map is stably accessible from hyperbolic components with any given stable
E U={|z|>r>1}
W
{|z|=r }2
Q
E
hol. attractor
repellor z 2
wandering cr. pt.
cv=
ρe2iπθ
G :C→Ca branched cover,
E U={|z|>r>1}
W
{|z|=r }2
Q
E
hol. attractor
repellor z 2
wandering cr. pt.
cv=
ρe2iπθ
G :C→Ca branched cover, with (G,U) a hol. attractor,
E U={|z|>r>1}
W
{|z|=r }2
Q
E
hol. attractor
repellor z 2
wandering cr. pt.
cv=
ρe2iπθ G :C→Ca branched cover, with (G,U) a hol. attractor, (G,E,Q) a repellor,
E U={|z|>r>1}
W
{|z|=r }2
Q
E
hol. attractor
repellor z 2
wandering cr. pt.
cv=
ρe2iπθ
G :C→Ca branched cover, with (G,U) a hol. attractor, (G,E,Q) a repellor, G(W)⊂U,
E U={|z|>r>1}
W
{|z|=r }2
Q
E
hol. attractor
repellor z 2
wandering cr. pt.
cv=
ρe2iπθ
G :C→Ca branched cover, with (G,U) a hol. attractor, (G,E,Q) a repellor,
G(W)⊂U, and cv(G) =ρe2πiθ.
E U={|z|>r>1}
W
{|z|=r }2
Q
E
hol. attractor z 2
wandering cr. pt.
cv=
ρe2iπθ
hol.
G :C→Ca branched cover, with (G,U) a hol. attractor, (G,E,Q) a repellor,
G(W)⊂U, and cv(G) =ρe2πiθ. 1. Replace G by G◦θ−1 withθ(Q) =Q and θisotop.∼
rel∂Q id, we may assume G| holomorphic, andG| quasi-regular.
E U={|z|>r>1}
W
{|z|=r }2
Q
E
hol. attractor z 2
wandering cr. pt.
cv=
ρe2iπθ
hol.
G :C→Ca branched cover, with (G,U) a hol. attractor, (G,E,Q) a repellor,
G(W)⊂U, and cv(G) =ρe2πiθ. 1. Replace G by G◦θ−1 withθ(Q) =Q and θisotop.∼
rel∂Q id, we may assume G|U∪E holomorphic, andG|W quasi-regular.
2. ∃φqc., and hol. on U such thatφ◦G ◦φ−1 is z2+c
E U={|z|>r>1}
W
{|z|=r }2
Q
E
hol. attractor z 2
wandering cr. pt.
cv=
ρe2iπθ
hol.
G :C→Ca branched cover, with (G,U) a hol. attractor, (G,E,Q) a repellor,
G(W)⊂U, and cv(G) =ρe2πiθ. 1. Replace G by G◦θ−1 withθ(Q) =Q and θisotop.∼
rel∂Q id, we may assume G| holomorphic, andG| quasi-regular.
obstructions. Then G is comb-equivalent to a hyperbolic rational map R, i.e. ∃θ, φsuch that G ◦θ−1 ∼φR.
Sketch of the proof.
repellor P
repellor S~id S=id on U S=id on PG
P P
repellor q.r.
repellor S~id S=id on U S=id on PG
P q.r. P
repellor
constant L E
P
repellor q.r.
choice careful
of L
complexity
repellor S~id S=id on U S=id on PG
P q.r. P
repellor
constant L E
Thm. 1 T~id
rel Lc P
repellor q.r.
choice careful
of L
U
P
µ 0−hol.
q.r.
complexity
GS T−1−1
µ 0−hol.
repellor S~id S=id on U S=id on PG
P q.r. P
repellor
constant L E
Thm. 1 T~id
rel Lc P
repellor q.r.
choice careful
of L
Φµ qc−conj.
U
P
µ 0−hol.
q.r. q.r.
complexity
GS T−1
H
−1
ν=ν H*
µ 0−hol.
repellor S~id S=id on U S=id on PG
P q.r. P
repellor
constant L E
Thm. 1 T~id
rel Lc P
repellor q.r.
choice careful
of L
Φµ qc−conj.
U
P
µ 0−hol.
Φν qc−conj.
q.r. R
hol.
globally q.r.
complexity
GS T−1
H
−1
−PG′ := cluster(PG) := cluster(critical orbits) is finite and 6=∅
−G is hol. on neighborhood(PG′ ), per. points = attrac. / parabolic
−PG′ := cluster(PG) := cluster(critical orbits) is finite and 6=∅
−G is hol. on neighborhood(PG′ ), per. points = attrac. / parabolic
• with no obstructions(generalized Thurston obstructions)
−PG′ := cluster(PG) := cluster(critical orbits) is finite and 6=∅
−G is hol. on neighborhood(PG′ ), per. points = attrac. / parabolic
• with no obstructions(generalized Thurston obstructions)
=⇒ ∃!g rational map∼combG, i.e. for
U =⊔(round attract. discs)⊔(attract. petals) =basic set
−PG′ := cluster(PG) := cluster(critical orbits) is finite and 6=∅
−G is hol. on neighborhood(PG′ ), per. points = attrac. / parabolic
• with no obstructions(generalized Thurston obstructions)
=⇒ ∃!g rational map∼combG, i.e. for
U =⊔(round attract. discs)⊔(attract. petals) =basic set
∃ θ, φ homeo., s.t. G◦θ−1 ∼φ−conjg:
−PG′ := cluster(PG) := cluster(critical orbits) is finite and 6=∅
−G is hol. on neighborhood(PG′ ), per. points = attrac. / parabolic
• with no obstructions(generalized Thurston obstructions)
=⇒ ∃!g rational map∼combG, i.e. for
U =⊔(round attract. discs)⊔(attract. petals) =basic set
∃ θ, φ homeo., s.t. G◦θ−1 ∼φ−conjg: U,C −→φ,≈ V,C θ|U =id
θ,≈↑↑G g θ∼PG id
U,C φ|U hol.
−PG′ := cluster(PG) := cluster(critical orbits) is finite and 6=∅
−G is hol. on neighborhood(PG′ ), per. points = attrac. / parabolic
• with no obstructions(generalized Thurston obstructions)
=⇒ ∃!g rational map∼combG, i.e. for
U =⊔(round attract. discs)⊔(attract. petals) =basic set
∃ θ, φ homeo., s.t. G◦θ−1 ∼φ−conjg: U,C −→φ,≈ V,C θ|U =id
θ,≈↑↑G g θ∼PG id
U,C φ|U hol.
Moreover g = limn→∞gn(Thurston algorithm of successive liftings converges).
cycle,
cycle,
Vx =discaround x ∈X, Vx(r): bounded by equi-potentialr.
cycle,
Vx =discaround x ∈X, Vx(r): bounded by equi-potentialr.
Then ∃ C(r)ց0 asr ց0 such that
cycle,
Vx =discaround x ∈X, Vx(r): bounded by equi-potentialr.
Then ∃ C(r)ց0 asr ց0 such that
∀ N,∀ φ:C r [
x∈g−N(X)
Vx(r)→C univalent
cycle,
Vx =discaround x ∈X, Vx(r): bounded by equi-potentialr.
Then ∃ C(r)ց0 asr ց0 such that
∀ N,∀ φ:C r [
x∈g−N(X)
Vx(r)→C univalent
distortion(φ)≤C(r)ց0.
cycle,
Vx =discaround x ∈X, Vx(r): bounded by equi-potentialr.
Then ∃ C(r)ց0 asr ց0 such that
∀ N,∀ φ:C r [
x∈g−N(X)
Vx(r)→C univalent
distortion(φ)≤C(r)ց0.
Applications of Theorem A + Theorem B + easy surgeries: I. (Douady-Hubbard) convergence of preperiodic periodic rays of M.
II. Parabolic maps are accessible from various hyperbolic
with parabolics Control of distortions *
with parabolics Control of distortions * F
star-like plumb. surgery ↑ G
with parabolics Control of distortions * F Hyperbolic version
∼ f
star-like plumb. surgery↑ G
with parabolics Control of distortions * F Hyperbolic version
∼ f
star-like plumb. surgery ↑ ցstar-like pinching
G ft
with parabolics Control of distortions * F Hyperbolic version
∼ f
star-like plumb. surgery ↑ ցstar-like pinching
G ⇇
∗ ft
with parabolics Control of distortions * F Hyperbolic version
∼ f
star-like plumb. surgery ↑ ցstar-like pinching
G g ⇇
∗ ft
with parabolics Control of distortions * F Hyperbolic version
∼ f
star-like plumb. surgery ↑ ցstar-like pinching
G conclusion∼ g ⇇
∗ ft
(Douady-Hubbard)
(Douady-Hubbard)
Qc:z 7→z+c,θ= 2qp,c(t) such thatϕc(t)(et+2πiθ) =c(t) (t>0).
(Douady-Hubbard)
Qc:z 7→z+c,θ= 2qp,c(t) such thatϕc(t)(et+2πiθ) =c(t) (t>0).Then c(0) exists and c(t)→ c(0) as t →0.
(Douady-Hubbard)
Qc:z 7→z+c,θ= 2qp,c(t) such thatϕc(t)(et+2πiθ) =c(t) (t>0).Then c(0) exists and c(t)→ c(0) as t →0.
Existence of c(0): Qc(t0) cut-glue−→ F Thurston−→ Qc(0)
(Douady-Hubbard)
Qc:z 7→z+c,θ= 2qp,c(t) such thatϕc(t)(et+2πiθ) =c(t) (t>0).Then c(0) exists and c(t)→ c(0) as t →0.
Existence of c(0): Qc(t0) cut-glue−→ F Thurston−→ Qc(0)
Perturbation of c(0): ∀1>>t>0,
(Douady-Hubbard)
Qc:z 7→z+c,θ= 2qp,c(t) such thatϕc(t)(et+2πiθ) =c(t) (t>0).Then c(0) exists and c(t)→ c(0) as t →0.
Existence of c(0): Qc(t0) cut-glue−→ F Thurston−→ Qc(0)
Perturbation of c(0): ∀1>>t>0, Qc(0)cut-glue−→ FtThm A
−→
Qc(t)
↑n→∞
Qc(t,n)
