Breaking parabolic points, Pisa, 10-2013

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Breaking parabolic points along stable directions

CUI Guizhen and TAN Lei

Academy of Mathematics and Systems Science, CAS Universite d’Angers

Pisa, October 2013

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Motivations

We want to generalize the following results to rational maps:

In the quadratic family{z7→z²+c},

1. the fat rabbit parameter is accessible from both the main cardioid and the rabbit hyperbolic component. In the main cardioid direction, the

parabolic point ’breaks’ into an attracting fixed point and a period-3 repelling cycle, whereas in the rabbit direction, the parabolic point ’breaks’ into a repelling fixed point and a period-3 attracting cycle.

These two types of nearby dynamical systems are kind of ’stable’ (as opposite to ’implosive’) perturbations of the parabolic rabbit dynamical system.

2. a Misiurewicz parameter is accessible from the escape hyperbolic component.

We want to construct such accesses for more general maps, using intrinsic dynamical perturbations, rather than extrinsic parameter parametrizations.

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§1. Breaking parabolic points along stable directions

A parabolic point takes the formz(1 +zⁿ+o(zⁿ)),n≥1 (after passing to some iterates) and has a flowerP of 2nsepals such thatf(P) =P. A nearby map will break the fixed point 0 into a set of nearby fixed points with total multiplicitypfollowing some compatible combinatorics.

P1 P2

P3 P4

P1

P4

b

In this case,P2 andP3merge after the breaking. You can choose to merge any pair of neighboring sepals, or choose to merge simultaneously two pairs:

τ τ b b b

or

τ

bbbb

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In general, a compatible stable breaking combinatoricsis a choice of pairs of sepals to be merged, so that:

• distinct pairs do not cross

• each group of consecutive un-paired sepals has an even number of sepals

• compatible with the dynamics (rotation, period).

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Theorem (

breaking parabolic points )

Letgbe a rational map with parabolic cyclesY and with at most finite orbited critical points on the Julia set. Givenany compatible stable breaking combinatoricsofY,∃ a continuous path of rational maps{f_r}0<r≤r0 realizing the combinatoricsandconverging dynamicallytog.

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Theorem (

breaking parabolic pointsa more precise statement)

Letgbe a rational map with parabolic cyclesY and with at most finite orbited critical points on the Julia set. Givenany compatible stable breaking combinatoricsofY,∃ a continuous path of rational maps{fr}0<r≤r0 realizing the combinatoricstogether with qc-conjugaciesφ_rfrom f_r₀ to f_rsuch that:

f_r φr

uniform

−→r→0

g

φ and (fr0,J_f_r₀) ^φsemi-conjugate

reassembling the breaking−→ (g,J_g).

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Theorem (

breaking parabolic pointsa more precise statement)

Letgbe a rational map with parabolic cyclesY and with at most finite orbited critical points on the Julia set. Givenany compatible stable breaking combinatoricsofY,∃ a continuous path of rational maps{f_r}0<r≤r0 realizing the combinatoricstogether with qc-conjugaciesφ_rfrom f_r₀ to f_rsuch that:

fr

φr

uniform

−→r→0

g

φ and (fr0,Jfr0) ^φsemi-conjugate

reassembling the breaking−→ (g,Jg).

A particular case concerns Milnor’s conjecture in the geometrically finite setting: that parabolic cycles of a rational map can always be converted to attracting cycles without changing the topology of the Julia set (see also P.

Haissinsky and T. Kawahira).

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How many compatible combinatorics?

The following is computed by J. Tomasini :

For a parabolic fixed point with 2n sepals and with multiplier 1, i.e. in the form z(1 +zⁿ+o(zⁿ)),

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How many compatible combinatorics?

n 2 3 4 5 6 7 · · ·

#{compa. comb.}=

3n+ 1 n

n+ 1 7 30 143 728 3846 21318 · · ·

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How many compatible combinatorics?

n 2 3 4 5 6 7 · · ·

#{compa. comb.}=

3n+ 1 n

n+ 1 7 30 143 728 3846 21318 · · ·

#{generic comb.}=

2n

n

n+ 1 2 5 14 42 132 429

Therefore a map with a parabolic point of 7 pairs of sepals is stably accessible from 21318 distinct ’hyperbolic components’ on various slices with persistent parabolic points...

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How many compatible combinatorics?

n 2 3 4 5 6 7 · · ·

#{compa. comb.}=

3n+ 1 n

n+ 1 7 30 143 728 3846 21318 · · ·

#{generic comb.}=

2n

n

n+ 1 2 5 14 42 132 429

Therefore a map with a parabolic point of 7 pairs of sepals is stably accessible from 21318 distinct ’hyperbolic components’ on various slices with persistent parabolic points...

The case with rotationsseems more complicated ...

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§2. Misiurewicz polynomials are accessible by external rays

Douady-Hubbard (generalized by Kiwi to higher degree polynom.)

Forg(z) =z²+c such that 0 is strictly preperiodic, andθsuch that the external rayRc(θ) lands atc, there is a pathc(r)r>0 such thatc(r)∈R_c(r)(θ) with potentiale^r andc(r)→c asr→0.

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§2. Misiurewicz polynomials are accessible by external rays

Douady-Hubbard (generalized by Kiwi to higher degree polynom.)

Forg(z) =z²+c such that 0 is strictly preperiodic, andθsuch that the external rayRc(θ) lands atc, there is a pathc(r)r>0 such thatc(r)∈R_c(r)(θ) with potentiale^r andc(r)→c asr→0.

We will see a new proof of this theorem that works for more general settings.

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§3. A preliminary convergence criterion

Let: g be a rational map without critical points¹on the Julia set;

X0be a finite collection ofparabolic and repelling cycles;

X be thegrand orbit ofX0;

1or with only finite orbited critical points _{7 / 21}

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§3. A preliminary convergence criterion

{Ux(r)}0<r<1,x∈X0 be families small concentric disc-neighborhoods² generating by pullback a similar family{Uy(r)} for everyy∈XrX0.

1or with only finite orbited critical points

2after uniformizingU

x(1) _{7 / 21}

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§3. A preliminary convergence criterion

Theorem

(a convergence criterion)

a) Letpr,nnormalizedunivalent maps outside [

y∈g⁻ⁿ(X0)

Uy(r), and V ⊂⊂ Xc

. Thenpr,n|V −→

r→0iduniformly onn.

x(1) _{7 / 21}

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§3. A preliminary convergence criterion

Theorem

y∈g⁻ⁿ(X0)

. Thenpr,n|V −→

b) Let

f_r,n rational map of V ^p−→^r,n+1 the same degree,g(V)⊂V g↓ ↓fr,n

andg⁻¹(z)⊂V for some z, s.t. V −→

pr,n

Thenf_r,n−→

r→0guniformly onn.

The part a)=⇒b) is easy. So the main point is a).

x(1) _{7 / 21}

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§4. Application to g(z) = z

²

+ c with 0 preperiodic

1 Replacegby a (quasi-regular) mapF_r which is equal tog outside a small neighborhoodWrof 0 butFr(0) =z(r) on a rayRc(θ) landing at c(of potentiale^r). This new map isnot holomorphicin Wr.

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§4. Application to g(z) = z

²

+ c with 0 preperiodic

1 Replacegby a (quasi-regular) mapF_r which is equal tog outside a small neighborhoodW_rof 0 butF_r(0) =z(r) on a rayR_c(θ) landing at c(of potentiale^r). This new map isnot holomorphicin W_r.

2 (this does not work)Pullback the standard complex structure infinitely many times to get anF_r-invariant structure and integrate using Measurable Riemann Mapping Theorem.

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§4. Application to g(z) = z

²

+ c with 0 preperiodic

2 Instead pullback step by stepthe standard complex structure and

integrate, to get

p−→r,n+1

withpr,n+1univalent outside Fr↓ ↓fr,n

[n j=0

F_r^−jWr= [n j=0

g^−jWr.

pr,n

−→

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§4. Application to g(z) = z

²

+ c with 0 preperiodic

integrate, to get

p−→r,n+1

[n j=0

F_r^−jWr= [n j=0

g^−jWr.

pr,n

−→

3

So forV = the outside V ^p^r,n+1−→^,univalent of a large equipotential, g=Fr↓ ↓fr,n

V −→

pr,n,univalent

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§4. Application to g(z) = z

²

+ c with 0 preperiodic

integrate, to get

p−→r,n+1

[n j=0

F_r^−jWr= [n j=0

g^−jWr.

pr,n

−→

3

V −→

pr,n,univalent

4

∞ f_r

↑ ↑ n f_r,n −→

r→0 g unif. onn,

withfr:z7→z²+c(r) and c(r)∈Rc(r)(θ) (of potentiale^r).

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§4. Application to g(z) = z

²

+ c with 0 preperiodic

integrate, to get

p−→r,n+1

[n j=0

F_r^−jWr= [n j=0

g^−jWr.

pr,n

−→

3

V −→

pr,n,univalent

4

∞ f_r

↑ ↑ soց n f_r,n −→

r→0 gunif.onn,

withfr:z7→z²+c(r) and c(r)∈Rc(r)(θ) (of potentiale^r).

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§5 Application to parabolic points: plumbing surgery

This surgery makes a new Riemann surface, and a new local holomorphic dynamics with the desired combinatorices.

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two ways to get a global map

Given a rational mapg with a parabolic sepal flower of sizer,

1 use a quasi-regular modification in a neighborhood of the strict first preimage of the flower, to get a branched coveringFr realizing holomorphically (locally) the prescribed combinatorics, withF_r holomorphic except on [

y∈g⁻¹(X0)rX0

U_y(r).

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two ways to get a global map

y∈g⁻¹(X0)rX0

U_y(r).

2 use a consecutive sequence of plumbing surgery to get a pair of sequences (fr,n, pr,n)n realizing holomorphically (locally) the prescribed

combinatorics (see next slice).

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two ways to get a global map

y∈g⁻¹(X0)rX0

U_y(r).

2 use a consecutive sequence of plumbing surgery to get a pair of sequences (fr,n, pr,n)n realizing holomorphically (locally) the prescribed

combinatorics (see next slice).

3 prove a generalized Thurston theorem for parabolic maps,use it to conclude that eachF_ris c-equivalent to a rational map f_r, and its Thurston’s algorithm gives the same sequencef_r,n, so

f_r ∞

↑ as ↑ fr,n n

.

4 Applythe convergence criterionto (fr,n, p_r,n) to getf_r,n→guniformly onn. Therefore

∞ fr

↑ ↑ soց n fr,n −→

r→0 g unif.onn.

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Consecutive plumbing surgery

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Consecutive plumbing surgery

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Consecutive plumbing surgery

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Hope to get a limit like this

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Hope to get a limit like this

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§6. Proof of the convergence criterion (sketch)

Theorem

y∈g⁻ⁿ(X0)

. Thenpr,n|V −→

b) Let

fr,n rational map of V ^p−→^r,n+1 the same degree,g(V)⊂V g↓ ↓fr,n

pr,n

Thenf_r,n−→

Proof of a)=⇒b). p_r,n|_V −→

r→0iduniformly onnimpliesf_r,n|_V −→

r→0g uniformly onn.

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§6. Proof of the convergence criterion (sketch)

Theorem

y∈g⁻ⁿ(X0)

. Thenpr,n|V −→

b) Let

fr,n rational map of V ^p−→^r,n+1 the same degree,g(V)⊂V g↓ ↓fr,n

pr,n

Thenf_r,n−→

Proof of a)=⇒b). p_r,n|_V −→

r→0iduniformly onnimpliesf_r,n|_V −→

By control of degree,fr,n|C_b −→

In order to prove a), we first forget the normalization and control the distortion ofp_r,n from being globally conformal (i.e. M¨obius transformation).

We then use the normalization to control their spherical distance to the identity.

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§7. Univalent maps and M¨obius transformations

Given η:V ^univalent֒→ Cb on an open setV ⊂Cb, How far isη being globally conformal?

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§7. Univalent maps and M¨obius transformations

Given η:V ^univalent֒→ Cb on an open setV ⊂Cb, How far isη being globally conformal?

E^′

E η(E)

η(E^′) V

η

E^′

E η(E)

η(E^′)

A newA

η(V)

We choose to define

D(η, V) = sup{|modA−modnewA|, E, E^′ disjoint full continua inV}.

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Let:

g be a rational map without critical points³ on the Julia set;

X be thegrand orbit ofX0; ∆ = the unit disc;

{Ux(r)}0<r<1,x∈X be families small concentric disc-neighborhoods.

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Let:

g be a rational map without critical points³ on the Julia set;

X be thegrand orbit ofX0; ∆ = the unit disc;

{Ux(r)}0<r<1,x∈X be families small concentric disc-neighborhoods.

Theorem

(conformal distortion ofpr,n)

LetV ⊂⊂(X)^c. Then∃C(r) for smallrwithC(r)−→

r→00 such that

∀n≥0, ∀pr,n univalent outside [

y∈g⁻ⁿ(X0)

Uy(r), D(pr,n, V)≤C(r).

Clearly only the overlapping properties of theU_y(r)’s will play a role, but not the dynamical relations between them.

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§8. Univalent maps outside overlapped disks

LetX ⊂Cbe a finite set together with a family {Dx(r)}of concentric disc-neighborhoods.

Letλ∈[0,1) be a constant and d(r) : (0, r0]→(0,1] is a function with d(r)−→

r→00. We will say that the discs{Dx}x∈X are (λ, d(r))-nested if : (1) Any two are either disjoint or one contains the other.

(2) IfD_y∩D_x(r)6=∅, thenD_y ⊂D_x(d(r)) (near the center they are not too long)

(3) area(S

y∈Dxr{x}Dy)≤λareaDx.

Theorem (a priori bound, no dynamics involved)

There exists a functionC(r)−→

r→00 depending only onλandd(r), such that for any set{Dx}x∈X of (λ, d(r))-nested discs and any disjointV,

∀univalent mapφoutside [

Dx(r), D(φ, V)≤C(r)·M(∪Dx⊂(V)^c), whereM(D⊂W) is a constant depending only onD andW.

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§9. Proof of the apriori bound

We usean extremal length argument.

LetAbe an annulus with A^c⊂V. Let ρ0(z) be the extremal metric on A so that

ℓ_width(ρ0, A) = 1, ℓheight(ρ0, A) = areaρ0(A) = mod(A).

Key lemma

LetK⊂A be compact andϕbe univalent onA−K. The for anylength increasingconformal metricρ(z)|dz|supported onArK,

|modA−modnewA|< areaρ(A)−areaρ0(A).

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§9. Proof of the apriori bound

Fork≥1 letI_k,I_k(r) be the union ofD_x andD_x(r) of nesting depthk respectively.

Fixr >0 small enough. A disk D_xof depth kisoff centeredif Dx∩Il(r) =∅ forl < k.

LetI_k^′ ,I_k^′(r) be the union of off centeredDx andDx(r) of depthk respectively.

Define ρ_k(z) inductively onk by

ρ_k(z) =







ρk−1(z) onA−I_k^′, ρ0(z)(1−p

d(r))^−k onI_k^′ −I_k^′(d(r)),

0 onI_k^′(d(r)).

Then after finitely many depths, the metric stabilize to a metricρ, which increases length

ℓ_height(ρ, A)≥ℓ_hight(ρ0, A) = modA, ℓ_width(ρ, A)≥ℓ_width(ρ0, A) = 1 and0≤areaρ(A)−areaρ0(A)≤C(r)M(D⊂W).

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§9. Proof of the apriori bound

Let nowφa univalent map outside∪Dx(r),Aan annulus whose complement is in the domain of definition ofφ. LetA^′⊂Cb be the annulus whose complement isφ(Crb A) andρthe conformal metric defined above.

|mod(A^′)−mod(A)|^{Key lemma}≤ areaρ(A)−areaρ0(A)≤C(r)M(D ⊂W).

So

D0(φ, V)≤C(r)M(D ⊂W).

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§10. Cui’s program (around 1998)

0. Characterization of postcritically finite rational maps, by W. Thurston.

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§10. Cui’s program (around 1998)

1. Characterization of hyperbolic and sub hyperbolic rational maps (two published proofs by Jiang-Zhang and Cui-T. respectively, see also the sub sequel thesis of Godillon and Wang).

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§10. Cui’s program (around 1998)

2. Breaking parabolic points to star-like attracting cycles

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§10. Cui’s program (around 1998)

2. Breaking parabolic points to star-like attracting cycles 3. Characterization of geometrically finite rational maps

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§10. Cui’s program (around 1998)

4. Breaking parabolic points along arbitrary stable combinatorics(today)

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§10. Cui’s program (around 1998)

4. Breaking parabolic points along arbitrary stable combinatorics(today) 5. Convergence of pinching paths to boundary parabolic maps.

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§10. Cui’s program (around 1998)

(2-5 are almost written).

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§10. Cui’s program (around 1998)

Next step : go to the implosive direction, for this one needs to control critical orbits...

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§10. Cui’s program (around 1998)

Next step : go to the implosive direction, for this one needs to control critical orbits...

——– Other issues:

To check (use the work of Buff-T.) the convergence of the dimension along the paths...

Study the strata structure of these nearby ’hyperbolic components’.

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Grazie ! Thank you !

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