combinatorial rigidity of infinitely renormalization …combinatorial rigidity of infinitely...

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

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


Md1


Md1

S1


Md1

S1

Md2


Md1

S1

Md2

S2 . . .

Md3


Md1

S1

Md2

S2 . . .

Md3

Com(c):=〈Md

i〉i=1,2,3,...


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




Conformal conjugacy

local connectivity of Julia sets by A. Douady and Y. Jiang

Open closed argument, or no invariant line fields by McMullen




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



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



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



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

combinatorial rigidity of infinitely renormalization …combinatorial rigidity of infinitely...

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