Topology of the Lyubich-Minsky
Laminations
for Quadratic Maps:
Deformation and Rigidity(2nd lecture)

10-12 decembre 2008
Jussieu, Paris
Nagoya Univ. / LATP CMI
Tomoki KAWAHIRAThe Riemann Surface
Laminations
Constructions/Examples
II:
Lyubich-Minsky C-laminations

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Abstruct of Today's Talk
Holo. Dynamics

Riemann Surf. Lamin.
Topology/Geometryholo. dynam.

natural extension?
Problem of Rigidity
regular partaffine part= C-lamination

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Natural Extension
Rational map:

backward orbits

Inverse limit:
natural extension

Natural lifted action:right shift
left shiftprojection

semiconj.

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"Regular" Backward Orbits
Ex.

backward univalent orbitsorEx.

singular backward orbit

Definition: A backward orbit

there exists a nbd.

ofst

is regular if

eventually univalent4

Regular Part
Definition: The set of regular backward orbits in

is called the regular part.

Ex: Backward orbits in a repelling cycle:

in an attracting or parabolic cycle:

regular pt.
irregular pt.

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Affine Part: C-lamination
Definition: For future 3D extension ("the union of leavesby") we denote, and call it the affine part.

Ex: Backward orbits in a repelling cycle are in
Ex: When.

, we have

Proposition(Lyubich-Minsky):
If is critically non-recurrent, thencyclic backward orbits in
attracting/parabolic cycles

6Deformation and Rigidity ofL-M Affine Parts (C-laminations)7

Deformation of Hyperbolic Maps
A rational map is called Hyperbolic if all critical points areattracted to attracting cycles.
Consider a perturbation of Hyperbolic in the space ofrational functions of the same degree >1.
Fact: For small enough perturbation of , the dynamics
near the Julia sets are quasiconformally the same. (NOT
globally conjugate, because of superattracting cycles!)Stable dynamics implies stable topology:
Theorem(K): For small enough perturbation of , theaffine partsandare quasiconformally homeomorphic.
(No matter how many superattracting cycles are!)8Quadratic Maps: Cabrera's Theorem
Letandbe hyperbolic quadraticmaps. If and are in the same hyperbolic component of
the Mandelbrot set, then we have
(qc homeo) by
previous Theorem (K). As we expect, the converse is true:
Topology determines the holomorphic dynamics!
Theorem(Cabrera): If there exists an orientation
preserving homeo. betweenand, and must bein the same hyperbolic component.
Cor 1: These laminations are actually qc homeo.
Cor 2: If they have superattracting cycles,.
Cf. Mostow's Rigidity: If two complete finite volume hyperbolic
3-manifolds are homeomorphic, then they are isometrically homeo.
(hence are quotients of conformally the same Kleinian groups).

9Topology and Rigidity ofLM's C-lamination for
infinitely renormalizable
quadratic maps

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Quadratic-like maps / Straightening
A Quadratic-like map
covering of degree 2, like this:is a proper holo. branchedqc-LENS

conformal on the filled Julia

A technical assumption: The critical orbit never goes out by

iteration (This implies the connected filled Julia set.).

By Douady-Hubbard's Straightening Map, we may regard
the Q-like map
as a deformed image of aquadratic map
with uniquely determined. 11Renormalization / Combinatorics
A Q-like map"sub-Q-like map"

is renormalizable if there exists
like this:
DH-Straightening

Another technical assumption:
Renormalization "non-crossing"

By Douady-Hubbard's Tuning
Theorem, we can represent thecombinatorics by a uniquelydetermined superattracting dynamics

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Infinite Renormalization
A quad. map
is infinitely renormalizable if
there exist "nested-Q-like maps"
like this::renormaliazation with
The sequence of superattracting
parameters
given byis called the combinaroticsofor.
Question (Combinatorial Rigidity):(Comb. Rigidity

MLCHyperbolic Density)

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Rigidity of Combinatorics
An infinitely renormalizable
has a priori
bounds when each level of the renormalization is separated
by a definite size (modulus) of annulus:

Proposition (Kaimanovich-Lyubich):
has a priori bounds(C-lamination)

Topology of lamin. determines the combinatorics:
Theorem(Cabrera-K): Ifthen:
Cor: Plus, MLC atand

have a priori bounds(homeo.)13Structure Theorem
For the proof, we use the following theorem:
Structure Theorem(C-K): For infinitely renormalizablewith combinatorics, its natural extension
supports a decomposion by blocks
andas
follows:

for any

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