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Combinatorial rigidity of infinitely renormalization
unicritical polynomials
Davoud Cheraghi
University of Warwick
Roskilde, Denmark, Sept 27- Oct 1, 2010
Let fc(z) = zd + c , c ∈ C and d ≥ 2, with Julia set denoted byJ(fc ).
Let fc(z) = zd + c , c ∈ C and d ≥ 2, with Julia set denoted byJ(fc ).The problem is to show that the dynamics of fc on J(fc ) iscombinatorially rigid! In particular, the geometry of orbits on J(fc )are uniquely determined by some combinatorial data!
Let fc(z) = zd + c , c ∈ C and d ≥ 2, with Julia set denoted byJ(fc ).The problem is to show that the dynamics of fc on J(fc ) iscombinatorially rigid! In particular, the geometry of orbits on J(fc )are uniquely determined by some combinatorial data!
M(d) := {c ∈ C : J(fc) is connected}
We only consider the third case in
1. fc has an attracting periodic point
2. fc has a neutral periodic point
3. all periodic points of fc are repelling
We only consider the third case in
1. fc has an attracting periodic point
2. fc has a neutral periodic point
3. all periodic points of fc are repelling
fcBc
z 7→ z3
Rθ
E r
The puzzle pieces in the cubic case
The puzzle pieces in the cubic case
The puzzle pieces in the cubic case
By Yoccoz in 1990 ford = 2 and byKahn-Lyubich in 2005 ford ≥ 2, the nests of puzzlepieces shrink to pointsunless the map isrenormalizable
renormalization and straightening, repeating the process
b
×
f 3c
×
f tc
renormalization and straightening, repeating the process
b
×
f 3c
×
f tc
q.c. mapping
fc′
q.c. mapping
fc′
The combinatorics of an infinitely renormalizable map
The combinatorics of an infinitely renormalizable map
Md1
The combinatorics of an infinitely renormalizable map
Md1
S1
The combinatorics of an infinitely renormalizable map
Md1
S1
The combinatorics of an infinitely renormalizable map
Md1
S1
Md2
The combinatorics of an infinitely renormalizable map
Md1
S1
Md2
S2 . . .
Md3
The combinatorics of an infinitely renormalizable map
Md1
S1
Md2
S2 . . .
Md3
Com(c):=〈Md
i〉i=1,2,3,...
The combinatorics of an infinitely renormalizable map
Md1
S1
Md2
S2 . . .
Md3
Com(c):=〈Md
i〉i=1,2,3,...
Conjecture (combinatorial rigidity)
If Com(c)=Com(c’), then c = λc ′, for some λ with λd−1 = 1.
Secondary limbs
Secondary limbs
• Secondary limbs condition: all Md
ibelong to a finite number
of secondary limbs.
Secondary limbs
• Secondary limbs condition: all Md
ibelong to a finite number
of secondary limbs.
• A priori bounds: the moduli of the fundamental annuli areuniformly away from zero.
Theorem (Rigidity)
For every d ≥ 2, if fc and fc′ satisfy a priori bounds and secondarylimbs conditions, then Com (c)=Com(c’) implies c = λc ′ for some(d − 1)-th root of unity λ.
Theorem (Rigidity)
For every d ≥ 2, if fc and fc′ satisfy a priori bounds and secondarylimbs conditions, then Com (c)=Com(c’) implies c = λc ′ for some(d − 1)-th root of unity λ.
Earlier results
• for d = 2 by Lyubich; His proof uses linear growth of moduliand does not work for arbitrary degree.
• for d = 2 and real maps by Graczyk-Swiatek; This also useslinear growth of moduli and, because of symmetry, nohomotopy argument is needed.
• for real polynomials by Kozlovski-Shen-van Strien; Arbitrarynumber of critical points are involved, but the symmetry ofthe map is used.
Combinatorial equivalence
Topological conjugacy
Quasi-conformal conjugacy
Conformal conjugacy
Combinatorial equivalence
Topological conjugacy
Quasi-conformal conjugacy
Conformal conjugacy
local connectivity of Julia sets by A. Douady and Y. Jiang
Open closed argument, or no invariant line fields by McMullen
Combinatorial equivalence
Topological conjugacy
Quasi-conformal conjugacy
Conformal conjugacy
local connectivity of Julia sets by A. Douady and Y. Jiang
Open closed argument, or no invariant line fields by McMullen
Thurston Equivalence
fc is Thurston equivalentto fc′ if there exists a q.c.mapping H : C → C whichis homotopic to atopological conjugacy Ψbetween fc and fc′ , relativethe post-critical set of fc .
Lemma (Thurston-Sullivan?)
Thurston equivalence implies q.c. equivalence
Lemma (Thurston-Sullivan?)
Thurston equivalence implies q.c. equivalence
Proof.
C CH ∼ Ψ
fc fc′
C C
fc fc′
H1
C C
f n−2c f n−2
c′
H2
C CHn
b
b
b
Lemma (Thurston-Sullivan?)
Thurston equivalence implies q.c. equivalence
Proof.
C CH ∼ Ψ
fc fc′
C C
fc fc′
H1
C C
f n−2c f n−2
c′
H2
C CHn
b
b
b
Hn −→ G
Lemma (Thurston-Sullivan?)
Thurston equivalence implies q.c. equivalence
Proof.
C CH ∼ Ψ
fc fc′
C C
fc fc′
H1
C C
f n−2c f n−2
c′
H2
C CHn
b
b
b
Hn −→ G
z ∼ Jc Hn(z) ∼ Jc′
The multiply connected regions and buffers for gluing:
gl
hl
gl+1
hl+1
b
b
bb
b
b
Building a Thurston equivalence;Depending on the type of renromalization on level n and maybe onlevel n+ 1 as in
A: on level n primitive type;
B: on level n satellite type, and on level n + 1 primitive type;
C: on level n satellite type, and on level n + 1 also satellite type.
we define the q.c. mappings between multiply connected regions.
Building a Thurston equivalence;Depending on the type of renromalization on level n and maybe onlevel n+ 1 as in
A: on level n primitive type;
B: on level n satellite type, and on level n + 1 primitive type;
C: on level n satellite type, and on level n + 1 also satellite type.
we define the q.c. mappings between multiply connected regions.
The second case (naturally) imposes the SL condition on us, asthere may be bad scenarios.
The right number of twists for gluings
bb
b b
The right number of twists for gluings
bb
b b
The right number of twists for gluings
bb
b b