Topology of the Lyubich-Minsky
Laminations
for Quadratic Maps:
Deformation and Rigidity(3rd lecture)

10-12 decembre 2008
Jussieu, Paris
Nagoya Univ. / LATP CMI
Tomoki KAWAHIRA

Abstruct of Today's Talk
Rational map:
Kleinian Groupholo. group action
Poincare Extension

holo. dynam.

-Lamination
cyclic group action

Extension in the "universal setting"-Laminationisometric gr. action
properly discontinuous
Quotient by the Action

leafwise isometric gr. action
properly discontinuous
Quotient by the Action

Quotient Lamination
hyperbolic 3-manifold/orbifold

hyp. 3-orbifold lamination1

Abstruct of Today's Talk

Fuchsian group

Symmetric quadratic map

Deformation

Quasi-Fuchsian group(on a Bers slice)

Deformation

Quadratic map(on the Mandelbrot set)

We will compare them in terms of the quotient manfold
and the quotient lamination.
2

LM Hyperbolic 3-Laminations:
Construction(in the universal setting)3

Recall: Natural Extension
Rational map:

backward orbits

Inverse limit:
natural extension

Natural lifted action:right shift
left shiftprojection

semiconj.

4Recall: Regular Part
Definition: A backward orbit

there exists a nbd.

ofstis regular if

eventually univalent

Definition: The set of regular backward orbits in

is called the regular part.

Fact 1: The regular partis a "rough" Riem. surf. lamin.

Fact 2: The leaves are

In particular, any leaf

or annuli (only Herman rings).
is dense in.

Fact 3: The action

is a leafwise conformal homeo.

5Recall: Affine Part (C-lamination)
Definition: The affine part
Ex: Whenis the union of leaves, we have".

-lamination"

leaf
uniformization

Semiconjugacy
univ. covering = exp.
:meromorphic6

Embedding to the Univsersal SettingNote: This part is a brief summary of what I explained witha black board.

7

3D Extention and Taking Quotient
Note: This part is a brief summary of what I explained witha black board.

8

An Analogy
Bers slice vs. Mandelbrot set

2a9

Example of Kleinian Lamination

Julia setquotient

Sullivan's Solenoid

mirror image
3

H -lamination inside

Hyp. 3- lamin. with product str.

Analogy to Fuchsian group:
quotient

hyperbolic surface

Limit set3

H insidemirror image
Hyp. 3-mfd. with product str.

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Quasi-Fuchsian deformationLet us look the action ofthough a special qc-lens:conformal on the upper half

qc-LENS

Bers' Theorem:"Bers slice"

11

The Mandelbrot Set
The deformation ofin the Mandelbrot set

is similar:

Fact: For any
with, the dynamics on thebasin at infitnity is conformally conjugate to that of.
conformal on the upper half

Sullivan's Solenoid
Deformation!
mirror image????

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The Rabbits
f3
superattracting

f2
attracting

f0

superattracting

f1g

attracting

parabolic13

Suparattracting -- Attracting
Dynamics near the Julia sets are stable.
f0

superattracting

f1attracting

f3

superattracting

f2

attracting

Stable dynamics implies stable topology:
Theorem(K+LM): For small enough perturbation of ,the affine partsandare quasiconformally
homeomorphic.
The qc homeo is lifted to the hyperbolic 3-laminations andthe Kleinian laminations.

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Attracting--Parabolic--Attracting
f1
attractinggparabolicf2
attracting

Topology of the C-lamination changes!

15

Pinching Semiconj. (downstairs)
g
f1

f2

parabolicattracting

attracting

We can construct pinching semiconjugacies by using "tessellation".

16

Pinching Semiconj. (upstairs)
g
f1

f2

parabolicattracting

lift to C-lamin

attracting

lift to hyp 3-lamin

We can show that their the C and hyp. 3-laminations havedifferent topologies.

17

Quotient / Kleinian Laminationsg

f1

attracting

f2parabolicattracting"Fact" (K): For this continuous motion, the leafwise
topology of the quotient lamination is preserved.
However, the Kleinian laminations have different
topologies each other. The difference can be described by
the combinatorial Dehn-twist of the lower ends.
A caricature of the lower ends:
pinchingplumpingtwisting

glued

glued

glued

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Future Program- Quotient laminations
for infinitely renormalizable maps
- Refining rigidity theorems- Applying the strategy to other dynamics- In particular for cpx 2-dim. maps
- etc.
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