Conjugacy between polynomial basins

0.12 MB·English

by  Xiaoguang Wang

CONJUGACY BETWEEN POLYNOMIAL BASINS
XIAOGUANGWANG
3
1
Abstract. In this article, we study the properties of conjugacies be-
0
tween polynomial basins. For any conjugacy, there is a quasiconformal
2
conjugacy in the same homotopy class minimizing the dilatation. We
n
computetheprecisevalueoftheminimaldilatation. Thequasiconformal
a
conjugacy minimizingthedilatation isnotuniqueingeneral. Wegivea
J
necessary and sufficient condition when the extremal map is unique.
5

]
S
D
1. Introduction
.
h
Let f be a polynomial of degree d ≥ 2. The complex plane can be
t
a
decomposed into the union of its open and connected basin of infinity
m
[ X(f):= {z ∈ C : fn(z) → ∞ as n → ∞}

1 and its complement, the compact filled Julia set K(f). The Green function
v G :C → [0,∞) of f is given by
1 f
2 G (z) = lim d−nlog+|fn(z)|,
9 f
n→∞
0
. where log+(t) = max{0,logt} for t ∈ [0,∞). The function G vanishes ex-
1 f
0 actlyonK(f),harmoniconX(f),continuousonC,andsatisfiesGf(f(z)) =
3 dG (z) for all z ∈ C.
f
1
We denote by C(f) the critical set of f in C, and P(f) := ∪ fk(C(f))
: k≥1
v
the postcritical set. Consider the dynamic of f restricted in X(f). We
i
X say that φ is a conjugacy between (f,X(f)) and (g,X(g)) if φ : X(f) →
r X(g) is a homeomorphism such that φ◦f = g ◦φ in X(f). It determines
a
a homotopy class [φ] consisting of all conjugacies between (f,X(f)) and
(g,X(g)), homotopic to φ rel the set P(f) ∩ X(f). It’s known that the
homotopy class [φ] always contains a quasiconformal (or smooth) map.
Let φ bea quasiconformal conjugacy between (f,X(f)) and(g,X(g)), let
∂φ φ dz
z
µ = =
φ
∂φ φ dz
z
be its Beltrami form. The esssup norm kµ k of µ measures the dilatation
φ φ
of φ. Let δ = inf{kµ k :ψ ∈ [φ] is a quasiconformal map}. Note that the
[φ] ψ
Date: December11, 2013.
2010 Mathematics Subject Classification. Primary 37F10; Secondary 37F30.
Key words and phrases. basin of infinity,conjugacy.
1
2 Xiaoguang Wang
quasiconformal maps in [φ] form a normal family, there is a quasiconformal
map ϕ∈ [φ] whose dilatation equals δ . The natural questions are:
[φ]
Suppose φ is a conjugacy between (f,X(f)) and (g,X(g)), what is δ ?
[φ]
Further, let ϕ ∈ [φ] with kµ k= δ , when is ϕ unique?
ϕ [φ]
This note is devoted to answer these questions. We compute the precise
value δ and give a necessary and sufficient condition when the extremal
[φ]
map is unique. Since some basic properties of conjugacies are needed first,
we state the result in the next section. The main result is Theorem 2.6.
This work is inspired by DeMarco and Pilgrim’s sequel works [DP1,DP2,
DP3]. The notations we use here are borrowed from their’s. The original
idea concerning the deformation of polynomial basins comes from the wring
motion (also called the Branner-Hubbard motion) introduced by Branner
and Hubbard [BH], and the Teichmu¨ller theory of rational maps developed
by McMullen and Sullivan [MS]. The uniqueness of the extremal quasi-
conformal conjugacy follows from a result of Reich and Strebel [RS]. The
theory of extremal quasiconformal mappings is used by Cui [C] to construct
the extremal conjugacy between rational maps.
WedenotebyDtheunitdisk,H = {ℜz > 0}therighthalfplaneendowed
the metric |dz|/ℜz.
2. Basic properties of conjugacy
Let P be the space of monic and centered polynomials. That is each
d
f ∈ P takes the form
d
f(z) = zd+a zd−2+···+a z+a ,
d−2 1 0
where a ,··· ,a ∈ C. Any polynomial of degree d is conjugate by an
d−2 0
automorphism of C to some element of P .
d
Set M(f)= max{G (c) : c∈ C(f)}. The B¨ottcher map φ of f is defined
f f
near ∞ with φ′ (∞) = 1 andit can beextended to an isomorphism(c.f. [M])
f
φ : {z ∈C : G (z) > M(f)} → {w ∈ C :|w| > eM(f)}.
f f
2.1. Conjugacy preserves escape order.
Lemma 2.1. Let φ be a conjugacy between (f,X(f)) and (g,X(g)), then
for any z ,z ∈ X(f),
1 2
1. G (z ) > G (z ) if and only if G (φ(z )) > G (φ(z )).
f 1 f 2 g 1 g 2
2. If φ is quasiconformal, then
G (z )−G (z )
f 1 f 2
≤ G (φ(z ))−G (φ(z )) ≤ K(φ)(G (z )−G (z )).
g 1 g 2 f 1 f 2
K(φ)
Proof. We assume G (z ) > G (z ). Choose k ≥ 0 such that G (fk(z )) ≥
f 1 f 2 f 2
M(f), then the set A = {z ∈ C : G (fk(z )) < G (z) < G (fk(z ))} is an
f 2 f f 1
Conjugacy between polynomial basins 3
annulus. Since mod(φ gnφ(A)) = mod(gnφ(A)) = dnmod(φ(A)) → ∞ as
g
n→ ∞, we conclude by ( [Mc], Theorem 2.1) that when n is large enough,
max|p| < min|q|,
p∈αn q∈βn
where α and β are inner and outer boundaries of φ gnφ(A), respec-
n n g
tively. Since φ gn+k(φ(z )) ∈ β and φ gn+k(φ(z )) ∈ α , we have that
g 1 n g 2 n
|φ gn+k(φ(z ))| > |φ gn+k(φ(z ))|. Thus G (φ(z )) > G (φ(z )). The ‘if’
g 1 g 2 g 1 g 2
part follows from the same argument, by considering φ−1.
To prove the bi-Lipschitz inequality when φ is quasiconformal, we first
observe that G (z ) = G (z ) if and only if G (φ(z )) = G (φ(z )) by the
f 1 f 2 g 1 g 2
above conclusion. We may still assume G (z ) > G (z ), then
f 1 f 2
φ(A) = {w ∈ C : G (gk(φ(z ))) < G (w) < G (gk(φ(z )))}.
g 2 g g 1
Note that 2πmod(φ(A)) = dk(G (φ(z ))−G (φ(z ))) and 2πmod(A) =
g 1 g 2
dk(G (z )−G (z )). By the modular distortion
f 1 f 2
K(φ)−1mod(A) ≤ mod(φ(A)) ≤ K(φ)mod(A),
we get the required inequality. (cid:3)
Given a polynomial f ∈ P , the DeMarco-McMullen tree T(f) of f is the
d
quotientofX(f)obtainedbycollapsingeach connectedcomponentofalevel
set of G to a single point, see [DM]. Let π : X(f) → T(f) be the quotient
f f
map, then f induces a tree map σ :T(f)→ T(f) by σ ◦π = π ◦f. The
f f f f
function G descends to a height function h : T(f)→ R by G = h ◦π ,
f f + f f f
which induces a metric d on T(f). A consequence of Lemma 2.1 is
f
Corollary 2.2. A conjugacy φ between (f,X(f)) and (g,X(g)) induces an
isomorphism ι : (T(f),σ ,d ) → (T(g),σ ,d ) preserving the tree dynam-
φ f f g g
ics. If φ is quasiconformal, then ι is K(φ)-isometric:
φ
d (x,y)
f
≤ d (ι (x),ι (y)) ≤ K(φ)d (x,y), ∀x,y ∈ T(f).
g φ φ f
K(φ)
2.2. Conjugacypreservesangulardifference. The1-formω = 2∂G =
f f
2∂Gfdz satisfies f∗ω = dω and it provides a dynamically determined con-
∂z f f
formal metric |ω | on X(f), with singularities at the escaping critical points
f
and their inverse preimages (c.f. [DM,DP2,DP3]). In the metric |ω |, for
f
any L > 0, we have
|ω | = 2π.
f
ZGf=L
Lemma 2.3. Let φ be a conjugacy between (f,X(f)) and (g,X(g)). Then
for any L > 0 and any connected arc γ on the level set {G = L}, we have
f
|ω |= |ω |.
f g
Zγ Zφ(γ)
4 Xiaoguang Wang
Proof. For k ≥ 0, let I (f,γ) (resp. I (g,φ(γ))) be the integer part of
k k
1 |(fk)∗ω | (resp. 1 |(gk)∗ω |). Since φ is a conjugacy, we have
2π γ f 2π φ(γ) g
I (f,γ) = I (g,φ(γ)) for all k ≥ 0. Thus
k R k R
2πI (f,γ) 2πI (g,φ(γ))
k k
|ω | = lim = lim = |ω |.
Zγ f k→∞ dk k→∞ dk Zφ(γ) g
(cid:3)
2.3. Branner-Hubbard motion. In the rest of the paper, we assume
C(f) ∩ X(f) 6= ∅. In that case M(f) > 0. The fundamental annulus
is A(f) = {z ∈ C : M(f) < G (z) ≤ dM(f)}. Set ℓ = M(f). The
f 0
critical orbits meet A(f) at the level sets {G = ℓ },1 ≤ j ≤ N with
f j
ℓ < ℓ < ··· < ℓ = dℓ . These level sets decompose A(f) into N suban-
0 1 N 0
nuli A = {z ∈ C : ℓ < G (z) < ℓ },1 ≤ j ≤ N. Note that some level set
j i−1 f j
{G = ℓ } may meet at least two critical orbits.
f k
By Lemma 2.1, any conjugacy φ between (f,X(f)) and (g,X(g)) can
induce a map S : R → R such that S ◦G = G ◦φ on X(f). It is
φ + + φ f g
monotone increasing and satisfies S (dt) = dS (t) for all t > 0.
φ φ
We remark that by identifying t and dt in R , the map S descends to a
+ φ
circle homeomorphism τ :S → S.
φ
By Lemma 2.3, when t > M(f), the angular difference between φ (z)
f
and φ (φ(z)) is a constant for all z ∈ {G = t}. Thus φ induces a map
g f
T : (M(f),+∞) → R such that T (G (z)) = argφ (φ(z))−argφ (z) when
φ φ f g f
G (z) > M(f). The map T satisfies T (dt) = dT (t) for all t > M(f). We
f φ φ φ
can extend this map to a map from R to R, still denoted by T , such that
+ φ
T (dt) = dT (t) for all t > 0. Following Branner and Hubbard [BH], we call
φ φ
S the stretching part of φ and T the turning part of φ.
φ φ
Note that for any two conjugacies φ,ϕ between (f,X(f)) and (g,X(g)),
one has S (ℓ ) = S (ℓ ), ∀1 ≤ j ≤ N. So the homotopy classes of conjuga-
φ j ϕ j
cies are determined by their turning parts. The following fact is immediate:
Lemma 2.4. Let φ,ϕ be two conjugacies between (f,X(f)) and (g,X(g)).
Then [φ]= [ϕ] if and only if
T (ℓ ) = T (ℓ ), ∀1 ≤j ≤ N.
φ j ϕ j
In fact T (ℓ )−T (ℓ ) ∈ 2πZ for some integer k ≥ 1, which is the fold
φ j ϕ j kj j
of symmetry of the level set {G = ℓ }.
f j
In the following, we assume φ is a quasiconformal conjugacy between
(f,X(f)) and (g,X(g)), then φ is differentiable almost everywhere in X(f).
This implies S′(t) and T′(t) exist for a.e t ∈ R . By Lemma 2.1, we have
φ φ +
K(φ)−1 ≤ S′(t)≤ K(φ) a.e t ∈ R .
φ +
Lemma 2.5. Let φ be a quasiconformal conjugacy between (f,X(f)) and
(g,X(g)). Then we have
∂φ Sφ′ ◦Gf +iTφ′ ◦Gf −1∂Gf
= .
∂φ S′ ◦G +iT′ ◦G +1∂G
φ f φ f f
Conjugacy between polynomial basins 5
Proof. It suffices to verify the equation in {z ∈ C : G (z) > M(f)} since
f
both sides are f-invariant.
When G (z) > M(f), we have
f
φ (z)
φ (φ(z)) = eSφ(Gf(z))+iTφ(Gf(z)) f .
g
|φ (z)|
f
Set E(z) = eSφ(Gf(z))+iTφ(Gf(z)), then
∂(φ ◦φ) φ (z) ∂G ∂ φ (z)
g = E(z) (S′(G (z))+iT′(G (z))) f f + f ,
∂z φ f φ f |φ (z)| ∂z ∂z |φ (z)|
f f
h (cid:16) (cid:17)i
∂(φ ◦φ) φ (z) ∂G ∂ φ (z)
g = E(z) (S′(G (z))+iT′(G (z))) f f + f .
∂z φ f φ f |φ (z)| ∂z ∂z |φ (z)|
f f
h (cid:16) (cid:17)i
By calculation,
∂ φ (z) φ (z) ∂G (z) ∂ φ (z) φ (z) ∂G (z)
f f f f f f
= − , = .
∂z |φ (z)| |φ (z)| ∂z ∂z |φ (z)| |φ (z)| ∂z
f f f f
(cid:16) (cid:17) (cid:16) (cid:17)
We have
∂φ ∂(φg ◦φ) Sφ′(Gf(z))+iTφ′(Gf(z))−1∂Gf
= = .
∂φ ∂(φ ◦φ) S′(G (z))+iT′(G (z))+1∂G
g φ f φ f f
(cid:3)
2.4. Minimal dilatation and uniqueness. Let φ be the conjugacy be-
tween (f,X(f)) and (g,X(g)). Set
S (ℓ )−S (ℓ ) T (ℓ )−T (ℓ ) τ∗−1
τ∗ = φ j φ j−1 +i φ j φ j−1 , δj = j , j = 1,··· ,N.
j ℓ −ℓ ℓ −ℓ [φ] τ∗+1
j j−1 j j−1 j
(cid:12) (cid:12)
(cid:12) (cid:12)
Note that τ∗ depends only on the homotopy class(cid:12)[φ]. (cid:12)
j
Theorem 2.6. Let φ be a conjugacy between (f,X(f)) and (g,X(g)). Then
j
δ = max δ .
[φ] 1≤j≤N [φ]
The extremal quasiconformal conjugacy is unique if and only if
δ1 = ··· = δN .
[φ] [φ]
To prove Theorem 2.6, we need the following result:
Theorem 2.7 (Reich-Strebel). Let φ : R → R be a quasiconformal map
1 2
between hyperbolic Riemann surfaces. If there exist a holomorphic quadratic
differential q on R with |q| < +∞ and a number 0 ≤ k < 1, such that
1 R1
µ = kq/|q|, then every quasiconformal map ψ :R → R ,ψ 6= φ homotopic
φ R 1 2
to φ modulo the boundary satisfies kµ k > kµ k.
ψ φ
6 Xiaoguang Wang
The proof is as in [RS], p.380.
Proof of Theorem 2.6. Note that every map ψ ∈ [φ] is determined by the
restrictions of S ,T in [ℓ ,ℓ ].
ψ ψ 0 N
We construct a quasiconformal conjugacy ψ ∈ [φ] such that
(S (ℓ ),T (ℓ )) = (S (ℓ ),T (ℓ )),j = 0,··· ,N,
ψ j ψ j φ j φ j
and S ,T are linear in each interval (ℓ ,ℓ ). Then by Lemma 2.5,
ψ ψ j−1 j
∂ψ N τ∗−1 ∂G
j f
= χ ,
∂ψ τ∗+1 fk(Aj)∂G
k∈Zj=1 j f
XX
where χ is the characteristic function, defined so that χ (x) = 1 if x ∈ E
E E
and χ (x) = 0 if x ∈/ E.
E
j
The map ψ satisfies: µ | = k q /|q |, where k = δ and
ψ Aj j j j j [φ]
τ∗−1 ∂G 2
j f
q = dz
j τ∗+1 ∂z
j
(cid:16) (cid:17)
is a holomorphic and integrable quadratic differential on A . By Theorem
j
2.7, the map ψ| is the unique extremal map on A modulo the boundary
Aj j
∂A . This implies δ = max δj . Further, if δ1 = ··· = δN , then ψ
j [φ] 1≤j≤N [φ] [φ] [φ]
is the unique extremal map.
If kµ | k< δ for some k, then one can deform ψ| to another map ϕ
ψ Ak [φ] Ak
modulothe boundarywith kµ |k≤ δ . This yields another quasiconformal
ϕ [φ]
conjugacy homotopic to ψ and with the same dilatation as ψ. (cid:3)
2.5. Appendix: The Teichmu¨ller space of (f,X(f)). The general Te-
ichmu¨ller theory of rational maps is developed in [MS]. It’s shown that the
Teichmu¨ller space of (f,X(f)) is isomorphic to HN. Here we precise this
statement by Theorem 2.6.
We begin with the definition of the Teichmu¨ller space of (f,X(f)) fol-
lowing [MS]. Let QC(f,X(f)) be the set of quasiconformal self-conjugacies
of (f,X(f)). QC (f,X(f)) ⊂ QC(f,X(f)) consists of those conjugacies h
0
isotopic to the identity in the following sense: there is a family (h ) ⊂
t t∈[0,1]
QC(f,X(f)) with h = id,h = h such that (t,z) → (t,h (z)) is a homeo-
0 1 t
morphismfrom[0,1]×Contoitself. TheTeichmu¨llerspaceTeich(f,X(f))is
thesetofequivalenceclasses[((g,X(g)),φ)] ofpairs((g,X(g)),φ), whereg ∈
Pd,φisaquasiconformbalconjugacybetween(f,X(f))and(g,X(g)), andwe
say ((g ,X(g )),φ ) and ((g ,X(g )),φ ) are equivalent if there is a confor-
1 1 1 2 2 2
malconjugacy ψ between(g ,X(g ))and(g ,X(g ))suchthatφ−1◦ψ◦φ ∈
1 1 2 2 2 1
QC (f,X(f)). The Teichmu¨ller metric between ζ = [((g ,X(g )),φ )],j =
0 i j j j
1,2 is defined by
d(ζ ,ζ ) = inflogK(φ),
1 2
Conjugacy between polynomial basins 7
wheretheinfimumisoverallquasiconformalconjugaciesbetween(g ,X(g ))
1 1
and(g ,X(g )), homotopic to φ ◦φ−1. We remark thatthis metric (instead
2 2 2 1
of its half) can make the map in Theorem 2.8 isometric.
Give any pair of points a = (a ,··· ,a ), b = (b ,··· ,b ) ∈ HN, we
1 N 1 N
define the distance d (a,b) by
HN
dHN(a,b) = max dH(aj,bj).
1≤j≤N
For τ = (τ ,··· ,τ ) ∈ HN, let φ : C → C solve the Beltrami equation
1 N τ
N
∂φ τ −1 ∂G
τ j f
= χ .
∂φ τ +1 fk(Aj)∂G
τ k∈Zj=1 j f
XX
Theholomorphicfamilyofquasiconformalmapsisnormalizedsothatφ =
(1,···,1)
id and f = φ ◦f ◦φ−1 ∈ P . By the proof of Theorem 2.6, we have (com-
τ τ τ d
pare [DP1], Lemma 5.2)
Theorem 2.8. The map
HN → Teich(f,X(f)),
Φ :
(τ 7→ [((fτ,X(fτ)),φτ)].
is biholomorphic and isometric.
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DepartmentofMathematics,ZhejiangUniversity,Hangzhou,310027,P.R.China
E-mail address: [email protected]