folding surgery and wandering continuatanlei/papers/copenhagen.pdf · holomorphic dynamics, around...
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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?
2 / 21
§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?
2 / 21
§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?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
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
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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§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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§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.
9 / 21
§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.
9 / 21
§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.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.
10 / 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.
10 / 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.
10 / 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.
10 / 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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§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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§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.
12 / 21
§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.
12 / 21
§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.
12 / 21
§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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§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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§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.).14 / 21
§3.4 Survival of the central annulus
Proposition.
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).
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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