Folding surgery and wandering continuatanlei/papers/copenhagen.pdf · Holomorphic Dynamics, Around Thurston’s Theorem Roskilde , September 27 - October 1 2010 1 / 21 §1. Statements

Folding surgery and wandering continua

CUI Guizhen & TAN Lei

Holomorphic Dynamics, Around Thurston’s Theorem

Roskilde , September 27 - October 1 2010

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§1. Statements

Let f be a rational map.

A wandering continuum is a non-degenerate continuum K ⊂ Jf such

that fn(K) ∩ fm(K) = ∅ for 0 ≤ n < m.

Theorem (from Thurston, Kiwi, Levin)

f polynomialJf connected and locally connected =⇒ No wandering continuum .

Renormalization

f polynomial =⇒ use equipotentials and external raysto build renormalization pieces

Why interesting?

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§1. Statements






Renormalization


Why interesting?

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§1. Statements






Renormalization


Why interesting?2 / 21

§1. 1 Examples.

McMullen Pilgrim-T.

Godillon’s example

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§1.2 Statements

Theorem 1. ∃ rational maps f having wandering continua

Jf disconnected Jordan curve,McMullen, Pilgrim-T. // Cui

Lattes example segment(irrational slope, easy)

postcr. finite (sub)-hyperbolic Jordan curve,Cui-T. (Thurston thm.)

Theorem 2 (Cui-Peng-T.) Combinatorial criterion, #Pf < ∞

∃ Jordan curve wandering continua ⇐⇒ ∃ Cantor multicurve.

renormalizable ⇐= ∃ Cantor multicurve.

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§1.2 Statements








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§1.2 Statements








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§2. Cantor multicurve Γ = {γ1, · · · , γn}Assume Pf , the postcritical set of f , is finite.

Γ is an irreducible (ergodic) multicurve in CrPf , if

every γi is a non-peripheral Jordan curve in CrPf ,

they are disjoint and non-homotopic in CrPf ,The following unweighted transition matrix is irreducible.

f#,Γ = (bij) bij = #{curves in f−1(γj) homotopic to γi}.

In addition, Γ is a Cantor multicurve if one of the following occurs (theyare all equivalent)

• f#,Γ is not a permutation matrix,• The leading eigenvalue λ(f#,Γ) > 1,• #{curves in f−1(Γ) homotopic to curves in Γ} > #Γ,• #{curves in f−n(Γ) homotopic to curves in Γ}−→∞ (exponentially).

The equator of a mating is irreducible, but NOT a cantor multicurve.

Folding provides a cantor multicurve with one curve.

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Folding provides a cantor multicurve with one curve.9 / 21

§3. Mating and Folding of polynomials

An application of Thurston’s theorem usually takes the following steps:

1. Create a branched covering model F (not always easy).

2. Rule out eventual obstructions (can be very hard).3. Apply Thurston’s theorem to get a rational map R (only one line).4. Study the topology/geometry of JR (not always easy).

Let’s practice this to find a postcritically finite R with

a Cantor multicurve,a wandering continuum,a renormalization.

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1. Create a branched covering model F (not always easy).2. Rule out eventual obstructions (can be very hard).

3. Apply Thurston’s theorem to get a rational map R (only one line).4. Study the topology/geometry of JR (not always easy).



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1. Create a branched covering model F (not always easy).2. Rule out eventual obstructions (can be very hard).3. Apply Thurston’s theorem to get a rational map R (only one line).

4. Study the topology/geometry of JR (not always easy).



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1. Create a branched covering model F (not always easy).2. Rule out eventual obstructions (can be very hard).3. Apply Thurston’s theorem to get a rational map R (only one line).4. Study the topology/geometry of JR (not always easy).



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1. Create a branched covering model F (not always easy).2. Rule out eventual obstructions (can be very hard).3. Apply Thurston’s theorem to get a rational map R (only one line).4. Study the topology/geometry of JR (not always easy).



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§3.1. Topological folding

W = C ∪ {∞ · e2iπθ, θ ∈ R}, W ′ = C′ ∪ {(∞ · e2iπθ)′, θ ∈ R},A = [−1, 1]× S1,S = W tA tW ′/ ∼,with ∞ · e2πiθ ∼ (−1, e2πiθ) and (+1, e2πiθ) ∼ (∞ · e−2πiθ)′,π = id : W ′ → W .

f, g polynomials, degree=d, d′, postcritically finite,

d = d′, f, g monic folding(f, fold) : F :

mating(f, g) : M : F 2(CA) ⊂ Kf , (pre)periodic

f

W W'A

W W'A

gcover

W W'

ff°π fold

A

W

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§3.2. Foldings are obstructed or not?

Partial answers :

Proposition.

A folding F can not have Levy-type obstructions.

Theorem. A folding map F

(easy) is obstructed if d = 2, or equivalently 1d + 1

d ≥ 1.

(harder) is unobstructed if{

d ≥ 3F (CF ∩A) ⊂ a star-like of Bottcher rays.

(not trivial) ∃ obstructed examples with d = 4.

We will postpone the proof of this.

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Partial answers :

Proposition.




d ≥ 1.





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Partial answers :

Proposition.




d ≥ 1.





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Partial answers :

Proposition.




d ≥ 1.





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§3.3 Applying Thurston’s theorem

An unobstructed folding F is

combinatorially equivalent to a rational map R :

Sh1

≈//

F

��

C

R

��S

h0

≈ // C,

h1|PF= h0|PF

, h1 ∼PFh0

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§3.4. Properties of R

∂W is a cantor multicurve for F =⇒ h0(∂W ) is one for R.

Rees-Shishikura

Sh1

≈//

F

��

C

R

��S

h0

≈ // C,

promotion−→

Sh //

F

��

C

R

��S

h// C,

h is continuous and surjective (and

reduces the space between Kf and Kg to vacuum in the mating case).

Proposition. For an unobstructed folding F

h(A) ⊃ non empty annulus A s.t.A separates h(W ) and h(W ′),A ⊃ two essential annuli A1, A2 with R : A1 → A and R : A2 → Acoverings, and ∂A ⊂ ∂(A1 ∪A2).

=⇒ R has a polynomial renormalization and wandering continua, and they are

all Jordan curves due to uniform expansion (as in Pilgrim-T.).

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Rees-Shishikura

Sh1

≈//

F

��

C

R

��S

h0

≈ // C,

promotion−→

Sh //

F

��

C

R

��S

h// C,






all Jordan curves due to uniform expansion (as in Pilgrim-T.).

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Rees-Shishikura

Sh1

≈//

F

��

C

R

��S

h0

≈ // C,

promotion−→

Sh //

F

��

C

R

��S

h// C,






all Jordan curves due to uniform expansion (as in Pilgrim-T.).14 / 21

§3.4 Survival of the central annulus

Proposition.


Proof. Assume T := h(W ) ∩ h(W ′) 6= ∅.

Set T ∗ = ∩k>0Rk(T ). Then R(T ) ⊂ T =⇒ R(T ∗) = T ∗.

(1) ∃ an backword orbit {wk}k≥0 ⊂ T ∗ s.t. R(wk+1) = wk.(2) We may assume that F : h−1(w1) → h−1(w0) is a homeomorphism.(3) If #h−1(w0) ∩ ∂W = 1, then

#h−1(w1) ∩ F−1(∂W ) = #h−1(w1) ∩ (∂W ∩ ∂W ′) ={

1 (homeo)2 (homotopy) .

A contradiction.(4) If not, iterate backward a few times...

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§3.5 No obstructions

The idea is similar.

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§4. Decomposition along a Cantor multicurve

Theorem.For a postcritically finite rational map g,

∃ a Cantor multicurve Γ = {γ1, · · · ,γn} =⇒{∃ renormalization∃ wandering continua.

Key proposition (homotopy inv. −→ invariance)

Each γi can be promoted to an annulus Ai ⊂ CrPg such thatcore-curve(Ai)∼ γi,Ai ∩Aj = ∅ (i 6= j),components A′ of g−1(Aj) homotopic to γi are actually contained in Ai,∂

⋃Ai ⊂ ∂

⋃A′.

Proof.

Deform g into a branched covering G s.t.

G is combinatorially equivalent to g,the annuli survive under Rees-Shishikura’s semi-conjugacy (same proof).

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§4. Decomposition along a Cantor multicurve

Theorem.For a postcritically finite rational map g,

∃ a Cantor multicurve Γ = {γ1, · · · ,γn} =⇒{∃ renormalization∃ wandering continua.

Key proposition (homotopy inv. −→ invariance)

Each γi can be promoted to an annulus Ai ⊂ CrPg such thatcore-curve(Ai)∼ γi,Ai ∩Aj = ∅ (i 6= j),components A′ of g−1(Aj) homotopic to γi are actually contained in Ai,∂

⋃Ai ⊂ ∂

⋃A′.

Proof. Deform g into a branched covering G s.t.

G is combinatorially equivalent to g,the annuli survive under Rees-Shishikura’s semi-conjugacy (same proof).

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§5. An obstructed folding

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§6. Decomposition of hyperbolic rational maps

JR disconn. =⇒a canonical multicurve

renormalizations with connected J ⇒

cantor multicurve

renormalizations ⇒ ...

(For non-hyperbolic ones, see next talk)

Thanks for your attention !

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§7. Questions.

If f is a polynomial with connected but non-locally connected Julia set,

does f have a wandering continuum in J ? Parital answers: Yes by

Inou-Shishikura if: deg(f) = 2, with a periodic indifferent point, with

high-type rotation number and no critical point on the boundary of the Siegel

disc (if exists);

Kiwi: f poly. no irrationnally neutral periodic point. Then ’no wandering

continuum’ is equivalent to local connectivity.

Yes if deg(f)=2 and infinitely renormalizable and Jn 6→ point.

Otherwise ?

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Is there a

postcritically finite

and non-Lattesrational map having a full wandering

continuum?

How to characterize the dynamics of f having no multicurves? Or having

no Cantor multicurves?

Is there a wandering Julia component K such that fn(K) is complex type

for all n ≥ 0?

Are there infinitely many complex type Julia components?

Let f be a subhyperbolic rational map. Is the number of cycles of

complex Julia components bounded by C(deg(f))?

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Folding surgery and wandering continuatanlei/papers/copenhagen.pdf · Holomorphic Dynamics, Around Thurston’s Theorem Roskilde , September 27 - October 1 2010 1 / 21 §1. Statements

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