A dichotomy for Fatou components of polynomial skew products

0.2 MB·

by  Roland K. W. Roeder

A DICHOTOMY FOR FATOU COMPONENTS
OF POLYNOMIAL SKEW PRODUCTS
ROLANDK. W. ROEDER1
1
Abstract. Weconsider polynomial mapsof theform f(z,w)=(p(z),q(z,w)) that extend
1 as holomorphic maps of CP2. Mattias Jonsson introduces in [12] a notion of connectedness
0
for such polynomial skew products that is analogous to connectivity for the Julia set of a
2
polynomial map in one-variable. We prove the following dichotomy: if f is an Axiom-A
n polynomial skew product, and f is connected, then every Fatou component of f is home-
a
omorphic to an open ball; otherwise, some Fatou component of F has infinitely generated
J
first homology.
6

]
S
D 1. Introduction
.
h We consider the dynamics of mappings f :C2 C2 of the form
t →
a
(1) f(z,w) = (p(z),q(z,w))
m
[ wherep and q are polynomials. It will beconvenient to assumethat deg(p) = deg(q) =d and

p(z) = zd+O(zd−1)andq(z,w) = wd+O (wd−1)sothatf extendsasaholomorphicmapping
2 z
v f : CP2 CP2. (Throughout this paper we will assume that d 2.) Since f preserves the
→ ≥
2 family of vertical lines z C, onecan analyze f via thecollection of one variable fibermaps
5 { }×
q (w) = q(z,w), for each z C. In particular, one can define fiber-wise filled Julia sets K
2 z ∈ z
2 and Julia sets Jz that are analogous to their one-dimensional counterparts. For this reason,
. polynomial skew products provide an accessible generalization of one variable dynamics to
5
0 two variables. They been previously studied by many authors, includingHeinemann in [6, 7],
0 Jonsson in [12], DeMarco-Hruska in [2], and Hruska together with the author of this note
1
in [8].
:
v A more general situation in which the base is allowed to be an arbitrary compact topolog-
i
X ical space, while the vertical fibers are copies of C, has been considered by Sester in [17, 18].
r Meanwhile, generalization to semigroups of polynomial (and rational) mappings of the Rie-
a
mann sphere has been studied by extensively by Hinkkanen, Martin, Ren, Stankewitz, Sumi,
Urban´ski, and many others—we refer the reader to the excellent bibliography from [21] for
further references.
As an analogy with polynomial maps of one variable,
Definition 1.1. (Jonsson) A polynomial skew product f is connected if J is connected
p
and J is connected for all z J .
z p
∈
Here, J is the Julia set for the “base map” z p(z). Jonsson proves in [12, Sec. 6] that if
p
7→
f is a connected polynomial skew product, then the support J := suppµ of its measure of
2
Date: January 7, 2011.
2010 Mathematics Subject Classification. Primary 32H50; Secondary 37F20, 57R19.
Key words and phrases. Fatou components, linking numbers,closed currents,holomorphic motions.
1Research was supported in part bystartup fundsfrom theDepartment of Mathematics at IUPUI.
1
2 R.K.W. Roeder
maximal entropy µ (see 2, below) is connected, the Hausdorff dimension of µ is 2, and the
§
associated Lyapunov exponents are precisely logd. (Note that J can be connected even if f
2
is a disconnected polynomial skew product; see [2, Lemma 5.5].)
The Fatou set U(f) of a holomorphic map f : CP2 CP2 is the maximal open set
in CP2 on which the sequence of iterates is normal and→the Julia set is its complement:
J(f):= CP2 U(f). In this note, we prove
\
Theorem 1.2. Suppose that f is an Axiom-A polynomial skew product. If f is connected,
then every Fatou component of f is homeomorphic to an open ball. Otherwise, some Fatou
component of f has infinitely generated first homology.
The Axiom-A assumption ensures hyperbolicity of f on the non-wandering set. See, for
example, [12, Sec. 8] and [2]. Using the equivalence between (1) and (2) from Theorem
2.1 (below), connectivity of f can be expressed in terms of “escape of the critical locus”, so
Theorem 1.2 is an analog of the fundamental dichotomy from polynomial dynamics in one
variable (see, for example [15, Theorem 9.5]).
Corollary 1.3. If f is an Axiom-A polynomial skew product for which J = suppµ is dis-
2
connected, then some Fatou component of f has infinitely generated first homology.
We will discuss in Remark 4.4 the natural question of whether Theorem 1.2 holds if f is
not Axiom-A. We first show that some hypothesis of hyperbolicity is needed, by providing
an example of a disconnected product mapping that is not Axiom-A, whose Fatou set has a
single component that is homeomorphic to an open ball. We will then show that the proof
of Theorem 1.2 that is presented in Section 4 continues to hold if we replace Axiom-A with
slightly weaker conditions.
Acknowledgments. Many of the techniques and ideas from this paper were originally in-
spired by John H. Hubbard during discussions that we had when I was his student (5 years
ago). I have also benefited greatly from discussions with Laura DeMarco, Mattias Jonsson,
Suzanne Hruska, Lex Oversteegen, and Rodrigo Perez.
The anonymous referee has provided many helpful comments leading to substantial im-
provements in the paper.
2. Background on polynomial skew products
We present some background on polynomial endomorphisms of CP2 from [3, 9, 22, 19, 1]
and on polynomial skew products from [12].
Let f : C2 C2 be given by
→
f(z,w) = (p(z,w),q(z,w)),
with p and q polynomials of degree d 2, and let p and q be the homogeneous parts of
d d
≥
p and q, respectively, of the maximal degree d. Such a mapping extends as a holomorphic
mapping f : CP2 CP2 if and only if p (z,w) = 0 = q (z,w) implies that (z,w) = (0,0).
d d
Any holomorphic m→apping f :CP2 CP2 that is obtained in this way is called a polynomial
endomorphism of CP2. →
Suppose that f : C2 C2 extends as a polynomial endomorphism of CP2. The (affine)
→
Green’s function
1
(2) G(z,w) = lim log fn(z,w) , where log := max log,0 ,
dn +|| || + { }
Dichotomy for polynomial skew products 3
is a plurisubharmonic (PSH) function having the property that (z,w) C2 is in U(f) if and
∈
only if G(u,v) is pluriharmonic (PH) in a neighborhood of (z,w). The Green’s current T :=
1 ddcG is a closed positive (1,1) current on C2 with J(f):= suppT and U(f):= C2 suppT.
2π \
If f is a polynomial skew product, i.e. p(z,w) p(z), there is a refinement of this
≡
description. The base map p(z) has a Julia set J C and, similarly, a Green’s function
p
⊂
G (z) := lim 1 log pn(z) . Furthermore, one can define a fiber-wise Green’s function
p n→∞ dn +|| ||
by:
G (w) := G(z,w) G (z).
z p
−
For each fixed z, G (w) is a subharmonic function of w and one defines the fiber-wise Julia
z
sets by K := G (w) = 0 and J := ∂K .
z z z z
The extensio{n of f to C}P2 is given by
(3) f([Z :W : T])= [P(Z,T) :Q(Z,W,T) :Td],
where P(Z,T) and Q(Z,W,T) are the homogeneous versions of p and q. The point [0 :1 :0]
thatis“vertically atinfinity”withrespecttotheaffinecoordinates(z,w)isatotally invariant
super-attracting fixed point and (z,w) Ws([0 :1 :0]) if and only if w C K .
z
∈ ∈ \
The action of f on the line at infinity Π := T = 0 is given by the polynomial map
{ }
f (u) = q (1,u), where u = w/z and q (z,w) is the homogeneous part of q of maximal
Π d d
degree d. As usual, one can consider the associated Julia sets K and J .
Π Π
One can extend the Green’s current T as a closed positive (1,1) current on all of CP2
satisfyingf∗T = d T andhavingthepropertythatJ(f)= suppT andU(f) = CP2 suppT.
· \
Moreover, the wedge product µ := T T is a measure satisfying f∗µ = d2 µ, which happens
∧ ·
to be the measure of maximal entropy for f. It is customary to define a second Julia set
J J (f):= suppµ J(f).
2 2
≡ ⊂
If f = (p,q) is a polynomial skew product, then [12, Cor. 4.4] gives
J = z J .
2 z
{ }×
z[∈Jp
The following is (an excerpt from) Theorem 6.5 from [12]:
Theorem 2.1. (Jonsson) The following are equivalent:
(1) f is connected.
(2) C K and C K , for all z J .
p p z z p
⊂ ⊂ ∈
(3) J is connected, J is connected, and J is connected, for all z C.
p Π z
∈
(4) C K , C K , and C K , for all z C.
p p Π Π z z
⊂ ⊂ ⊂ ∈
Here C and C are the critical points of p and f , while C := (z,w) :∂q/∂w = 0 .
p Π Π z
{ }
The following appears in [8]:
Theorem 2.2. (Hruska-R) If J is connected for every z J , then Ws([0 : 1 : 0]) is
z p
∈
homeomorphic to an open ball. Otherwise, the first homology H (Ws([0 :1 : 0])) is infinitely
1
generated.
The proof that Ws([0 :1 :0]) is homeomorphic to a ball is an application of G (w) as a type
z
of Morse function, while the latter uses non-trivial linking numbers between closed loops in
Ws([0 :1 :0]) and the Green’s current T.
4 R.K.W. Roeder
3. Background on linking with a closed positive (1,1) current in CP2.
We present a brief summary (without proofs) of material from [8, 3]. See also [16].
§
Any closed positive (1,1) current S on a complex manifold N can be described using an
open cover U of N together with PSH functions v : U [ , ) that are chosen so
i i i
{ } → −∞ ∞
that S = ddcv in each U . The functions v are called local potentials for S and they are
i i i
requiredtosatisfythecompatibilityconditionthatv v isPHonanynon-emptyintersection
i j
−
U U = . The support of S and polar locus of S are defined by:
i j
∩ 6 ∅
suppS := z N : if z U then v is not PH at z , and
j j
{ ∈ ∈ }
polS := z N : if z U then v (z) = .
j j
{ ∈ ∈ −∞}
The compatibility condition assures that that above sets are well-defined. Moreover, since
PH functions are never , we have polS suppS.
−∞ ⊂
Let M be another complex manifold, possibly of dimension different from that of N. If
f : M N is a holomorphic map with f(M) polS, then the pull-back f∗S is a closed
→ 6⊂
positive (1,1) current defined on M by pulling back the system of local potentials for S to
form a system of local potentials on M that define f∗S. See [19, Appendix A.7] and [9, p.
330-331] for further details.
Given any closed positive (1,1) current S on N and any piecewise smooth two chain σ in
N with ∂σ disjoint from supp S, we can define
σ,S = η ,
S
h i
Zσ
where η is a smooth approximation of S within it’s cohomology class in N ∂σ, see [4,
S
−
pages 382-385]. The resulting number σ,S will depend only on the cohomology class of
h i
S and the homology class of σ within H (N,∂σ). If σ is a holomorphic chain, this pairing
2
simplifies to be the integral of the measure1 σ∗S over σ.
The following invariance property is useful:
Proposition 3.1. Suppose that S is a closed positive (1,1) current on N and f : M N is
→
holomorphic, with f(M) polS. If σ is a piecewise smooth two chain in M with ∂σ disjoint
6⊂
from suppf∗S, then f σ,S = σ,f∗S .
∗
h i h i
Notice that H (CP2) is generated by the class of any complex projective line L CP2.
2
⊂
Since S is non-trivial, L,S = 0, so that after an appropriate rescaling we can assume that
h i 6
L,S = 1. This normalization is satisfied by the Green’s Current T that was defined in 2.
h i §
Definition 3.2. Let S be a normalized closed positive (1,1) current on CP2 and let γ be a
piecewise smooth closed curve in CP2 supp(S). We define the linking number lk(γ,S) by
\
lk(γ,S) := Γ,S (mod 1)
h i
where Γ is any piecewise smooth two chain with ∂Γ = γ.
Unlike linkingnumbersbetween closed loops inS3, itis often thecase that that Γ,S Z,
h i 6∈
resulting in non-zero linking numbers (mod 1).
Proposition 3.3. If γ and γ are homologous in H (CP2 supp S), then lk(γ ,S) =
1 2 1 1
\
lk(γ ,S).
2
1This measure is well-defined,since ∂σ is disjoint from suppS ⊃polS.
Dichotomy for polynomial skew products 5
Moreover, since the pairing ,S is linear in the space of chains σ (having ∂σ disjoint from
h· i
supp S), the linking number descends to a homomorphism:
lk(,S) :H (CP2 supp S) R/Z.
1
· \ →
Similarly lk(,S) : H (Ω) R/Z for any open Ω CP2 supp S.
1
· → ⊂ \
Theorem 3.4. Suppose that f :CP2 CP2 is a holomorphic endomorphism and Ω U(f)
→ ⊂
is contained in a union of basins of attraction of attracting periodic points for f. If there
are c H (Ω) with linking number lk(c,T) = 0 arbitrarily close to 0 in Q/Z, then H (Ω) is
1 1
∈ 6
infinitely generated.
4. Proof of the Main result
We recall the characterization of Axiom-A polynomial skew products from [12, 8]. (For
§
the actual definition of Axiom-A, see, for example, [12, Def. 8.1].) Let A be the set of
p
attracting periodic points of p and consider the following postcritical sets:
D := pnC , where C = z : p′(z) = 0 ,
p p p
{ }
n≥1
[
D := fnC , where C := (z,w) : z J ,q′(w) = 0 ,
Jp Jp Jp { ∈ p z }
n≥1
[
D := fnC , where C := (z,w) : z A ,q′(w) = 0 , and
Ap Ap Ap { ∈ p z }
n≥1
[
D := fnC , where C := λ :f′ (λ) =0 .
Π Π Π Π { Π }
n≥1
[
Let
J = z J .
Ap { }× z
z[∈Ap
The following appears as [12, Cor. 8.3]:
Proposition 4.1. A polynomial skew product f = (p,q) is Axiom-A on CP2 if and only if:
(1) D J = ,
p p
∩ ∅
(2) D J = ,
Jp ∩ 2 ∅
(3) D J = , and
Ap ∩ Ap ∅
(4) D J = .
Π Π
∩ ∅
In particular, if f is Axiom-A, then
everyFatoucomponentofp(respectivelyf )isthebasinofattractionofanattracting
Π
•
cycle whose immediate basin contains a critical point from C (respectively C ), and
p Π
C J = over every z A J .
z z p p
• ∩ ∅ ∈ ∪
We can describe the fiberwise dynamics as follows: Let L := W = 0 be the horizontal
projective line. The vertical projection π(z,w) = (z,0) induces a{rationa}l map π : CP2 L
→
whose only point of indeterminacy is [0 :1 :0] (which blows up under π to the entire line L).
Although L is not invariant under f, we can consider the action of p on the line L and we
let µ be the harmonic measure on J , i.e. µ := ddcG (z). Within CP2 we have the vertical
p p p p
current T := π∗µ .
p p
6 R.K.W. Roeder
Throughout the remainder of the paper we will denote by C the closure in CP2 of the
“vertical critical points” (z,w) : q′(w) = 0 . It is an algebraic curve of degree d 1 with
{ z } −
C = C (z C) and C Π = C . Since q (w) = wd +O (w), we have that [0 : 1 : 0] C.
z Π z z
∩ × ∩ 6∈
Therefore, π :C L is a branched covering of degree degC = d 1.
→ −
Proposition 4.2. If f is a polynomial skew product with J connected for every z J , then:
z p
∈
(1) For any piecewise-smooth two-chain Γ C, we have Γ,T = Γ,T .
p
⊂ h i h i
(2) LetU beany Fatou component U of pand C := π−1(U). The sequenceof restrictions
U |C
fn is a normal family2.
{ |CU}
Proof. It suffices to consider any irreducible component of C, which we parameterize by its
C
normalization ρ : ; see [5]. Since has dimension 1, is a compact Riemann surface.
C → C C C
The following diagram summarizes the maps ρ and π:
b ρ b
>
C C
π
b
π ρ
◦ >∨
L.
Since [0 : 1 : 0] , we need only consider within two systems of affine coordinates:
6∈ C C
the original affine coordinates (z,w) and the coordinates t = T/Z and u = W/Z, defined
in a neighborhood Ω of [1 : 0 : 0]. In the (z,w) coordinates the Green’s current is given by
T = 1 ddcG(z,w) with
2π
(4) G(z,w) = G (w)+G (z).
z p
Meanwhile, in the (t,u) coordinates it is given by T = 1 ddcG (t,u) with
2π Ω
(5) G (t,u) = G (u)+G#(t).
Ω t p
#
Here, G (t) is obtained by extending G (1/t) log(1/t) continuously through t = 0 and
p p
−
G (u) is the PSH extension of G (w) described in [12, Lemma 6.3]. One has that (0,u) K
t z Π
∈
if and only if G (u) = 0 and, for t = 0, (t,u) K if and only if G (u) = 0.
0 1/t t
6 ∈
Because J is connected for every z J , Proposition 6.4 from [12] gives that C K
z p z z
∈ ⊂
for every z C and C K . Therefore, if (z,w) C2 we have that G (w) = 0 and
Π Π z
(4) gives tha∈t the restrict⊂ion of G to C2 coincide∈s wCi∩th G (z). Similarly, using (5), the
p
C ∩
#
restriction of G to Ω is just G (t).
Ω p
C ∩
The above calculations give
(6) ρ∗ T = ρ∗ T = ρ∗π∗µ .
p p
Let Γ be any piecewise smooth two chain and Γ its lift to the normalization. Then,
⊂ C ⊂ C
Γ,T = Γ,ρ∗T = Γ,ρ∗T = Γ,T .
h i bp bh pi
D E D E
There is a general principle that for any complex manifold M and holomorphic map φ :
b b
M CP2, the family fn φ is normal in a neighborhood of x M if and only if x
→ { ◦ } ∈ ∈
M suppφ∗T; See [3] and also [12, p. 409].
\
In our situation, it follows that
(7) fn ρ: CP2
◦ C →
2In other words, {fn◦ρ} is a normal family for abny parameterization ρ of CU (see below).
Dichotomy for polynomial skew products 7
is normal at x if and only if x supp(ρ∗ T ). Since G is harmonic outside of J ,
p p p
∈ C ∈ C \
we have that is disjoint from supp T and, hence, that ˆ is disjoint from supp(ρ∗ T ).
U p U p
C C
Therefore, fn ρ ibs normal on ˆ . b (cid:3)
U
◦ C
Corollary 4.3. Suppose that f is Axiom-A and satisfies the hypotheses of Proposition 4.2,
z is an attracting periodic point for p (possibly z = ), and U is any component of the
0 0
∞
immediate basin for z . Then, C is in the immediate basins for some attracting periodic
0 U
points of f within z = z .
0
Proof. Since f is Axiom-A, is in the immediate basins of attraction for some
Cz0 ⊂ CU
attracting periodic points of f . Because the line z = z is transversely attracting,
|{z=z0} 0
these periodic points are also attracting for f in CP2, giving an open subset of is in these
U
C
immediate basins. Since fn ρ is normal on ρ−1( ), all of is in the union of these
U U
immediate basins. { ◦ } C C (cid:3)
Proof of Theorem 1.2:
Case 1: f is connected:
We begin with Fatou components for f that are bounded in C2, i.e. those on which
G(z,w) 0.
≡
Since J is connected and p is hyperbolic, the Fatou set U(p) consists of the basins of
p
attraction of finitely many attracting periodic orbits together with the basin of attraction
for the superattracting fixed point at z = . Moreover, each component of these basins is
∞
conformally equivalent to the open unit disc D. For simplicity of exposition, we supposethat
each attractingperiodicorbitsforpisafixedpoint(otherwise, onecanpasstoanappropriate
iterate of f).
Let z C be an attracting fixed point of p and let U be the component of its basin of
0 0
∈
attraction that contains z .
0
Let x and y denote points in C2.
Sincef isAxiom-A,J isahyperbolicset,havingalocalstablemanifoldWs (J )formed
z0 loc z0
as the union of stable curves Ws (x) of points x J with each Ws (x) being holomorphic.
loc ∈ z0 loc
These stable curves satisfy the invariance
(8) f(Ws (x)) Ws (f(x)).
loc loc
⊂
They are transverse to the vertical line z = z since it is the unstable direction of J .
{ 0} z0
Hence, there is a sufficiently small ǫ > 0 so that each Ws (x) is the graph of a holomorphic
loc
function over z D (z ) := z z < ǫ . In other words, this describes Ws (J ) as a
∈ ǫ 0 {| − 0| } loc z0
holomorphic motion (see, for example, [10, 5.2]) over the open disc z D (z ).
ǫ 0
§ ∈
Because f : z C z C is hyperbolic, its Fatou set consists of the basins of
|z=z0 { 0}× → { 0}×
attraction of finitely many attracting periodic points (including w = ). We will use some
∞
hyperbolic theory to show that if z D (z ), then either (z,w) Ws (J ) or (z,w) is in
∈ ǫ 0 ∈ loc z0
the basin of one of these finitely many attracting periodic points for f . Because most
|z=z0
classical treatments of hyperbolictheory arefor diffeomorphisms, we appealto the discussion
for endomorphisms that appears in [11].
The natural extension of J is
z0
J = (x ) : x J andf(x ) = x .
z0 { i i≤0 i ∈ z0 i i+1}
Associated to any prehistory xˆ J is a local unstable manifold Wu (xˆ), which, in this case,
b ∈ z0 loc
is just an open disc in the vertical line z = z .
0
b
8 R.K.W. Roeder
We will now verify that J has a local product structure; see Definition 2.2 from [11]. If
z0
Ws (x) Wu (yˆ) is non-empty for some x J and yˆ J , then Ws (x) Wu (yˆ) = x,
loc ∩ loc ∈ z0 ∈ z0 loc ∩ loc
since Wu (yˆ) is contained inbthe vertical line z = z and Ws (x) is the graph of a function
loc { 0} loc
of z. In particular, the intersection is a unique point. Morebover, the unique prehistory xˆ of
x satisfying that x Wu (fi(yˆ)) for all i 0 is in J , since every local unstable manifold
i ∈ loc ≤ z0
lies in z = z .
0
{ }
Because J has a local product structure, Corollbary 2.6 from [11] gives that there is a
z0
neighborhood V of J in C2 so that if fn(z,w) V for all n 0, then (z,w) Ws (x)
z0 ∈ ≥ ∈ loc
for some x b∈ Jz0. If z ∈ Dǫ(z0), then fn(z,w) must converge to the line {z = z0} since
D (z ) U . Existence of the neighborhood V implies that (z,w) is either in Ws (J )
ǫ 0 ⊂ 0 loc z0
or is in the basin of attraction for one of the finitely many attracting periodic points for
f : z C z C.
|z=z0 { 0}× → { 0}×
We can now see that if z D (z ), then J = Ws(J ) ( z C). From the preceding
∈ ǫ 0 z z0 ∩ { }×
paragraph we see that if (z,w) Ws (J ), then w J . Meanwhile, if (z,w) Ws (J ),
6∈ loc z0 6∈ z ∈ loc z0
then the family of iterates fn cannot be normal at w, giving that w J .
{z}×C z
| ∈
Repeatedly taking the preimages of Ws (J ) under f that intersect the vertical line z =
loc z0 {
z , we obtain a stable set Ws(J ) over all of U . For any z U , we have that (z,w) is
0} z0 0 ∈ 0
either in Ws(J ) or is in the basin attraction for one of the finitely many attracting periodic
z0
points of f . We also have J = Ws(J ) ( z C).
|z=z0 z z0 ∩ { }×
ByCorollary4.3,C U(f),sothatC isdisjointfromWs(J ). Arepeatedapplication
U0 ⊂ U0 z0
of the Inverse Function Theorem, together with (8), allows us to extend (in the parameter
z) the holomorphic motion of J from z D (z ) to a holomorphic motion of J over all of
z0 ∈ ǫ 0 z0
U . (For details, see the proof of Theorem 6.3 from [8].)
0
TheimageofthisholomorphicmotionispreciselyWs(J ). BySlodkowski’sTheorem[20],
z0
it will extend (in the fiber w) as a holomorphic motion of the entire Riemann Sphere CP1
that is parameterized by U . Therefore, every bounded Fatou component of f lying above
0
U is given as the holomorphic motion of a Fatou component of f . Each of these is a
0 |z=z0
disc(since f is hyperbolicwith connected Julia set), so every boundedFatou component
|z=z0
lying above U is homeomorphic to an open ball.
0
Letz beann-thpreimageunderpofz andU bethecomponentofWs(z )containingz .
n 0 n 0 n
We will use induction on n to show that there is a holomorphic motion of J parameterized
zn
by z U so that
n
∈
(1) for every z U , J is obtained as the motion of J , and
∈ n z zn
(2) the motion of any w J is precisely stable curve of w.
∈ zn
Then, as above, Slodkowski’s Theorem will give that every bounded Fatou component of
f lying over U is obtained as the holomorphic motion of some component of K J ,
n zn \ zn
parameterized by z U . In particular, every such Fatou component will be homeomorphic
n
∈
to an open ball.
The desired holomorphic motion already exists for n = 0, so we suppose that it exists for
n = k in order to prove it for n= k+1.
It is sufficient to show that the part C of the horizontal critical locus that lies above
Uk+1
U is intheFatou set U(f). In thatcase, theImplicitFunction Theorem can beusedtolift
k+1
theentireholomorphicmotionofJ underf toaholomorphicmotionofJ parameterized
zk zk+1
by U . The result will automatically satisfy (1) and (2).
k+1
Dichotomy for polynomial skew products 9
Let be some irreducible component of C . By Proposition 4.2, the family of
CUk+1 Uk+1
restrictions of iterates fn is normal. Therefore, fk+1 is either in the Fatou set
{ |CUk+1} CUk+1
U(f) or is within the stable curve of a single x J .
Suppose the latter happens. Then, every p∈ointz0of fn+1 lying over any z U is
CUk+1 ∈ 0
in J . Let be portion of the horizontal critical locus lying over ∂U J that is
z C∂Uk+1 k+1 ⊂ p
in the closure of . Lower semi-continuity of the mapping z J (see [12, Prop. 2.1]),
CUk+1 7→ z
combined with continuity of fk+1, gives that
fk+1 z J z J = J .
C∂Uk+1 ⊂ { }× z ⊂ { }× z 2
z∈[∂U0 z[∈Jp
However, since f is Axiom-A, Proposition 4.1 implies that fk+1 D is disjoint from
C∂Uk+1 ⊂ Jp
J . Therefore, we conclude that every component of C is in the Fatou set.
2 Uk+1
We now consider the Fatou components for f on which G(z,w) > 0. Recall that (z,w)
∈
Ws([0 : 1 : 0]) if and only if G (w) > 0. Since J is connected for every z J , it follows
z z p
∈
from Theorem 2.2 that Ws([0 : 1: 0]) is homeomorphic to an open ball.
It remains to consider Fatou components on which G (z) > 0 and G (w) = 0. Let U
p z ∞
⊂
CP1 be the basin of attraction of under p. Since J is connected, U is simply connected.
p ∞
∞
For simplicity of exposition, we suppose that 0 U (otherwise, one can conjugate f by an
∞
6∈
appropriate translation in the z-coordinate).
Let us work in the system of local coordinates t = T/Z and u = W/Z in which
td Q(1,u,t)
f(t,u) = , (r(t),s (u)),
t
P(1,t) P(1,t) ≡
(cid:18) (cid:19)
which is a rational skew product. The first coordinate has 0 as a totally-invariant super-
attracting fixed point, whose basin of attraction U is entirely contained in the copy of
∞
C parameterized by t. The Fatou components of f that remain to be studied each have
projection under (t,u) t lying entirely in U .
∞
7→
The line Π is given by t = 0. Since f is Axiom-A, J is hyperbolic, having a local stable
Π
manifold Ws (J ) that is formed as the union of stable curves of points x J . Each of
loc Π ∈ Π
these stable curves is transverse to Π, since Π is the unstable direction of J . Thus, we can
Π
choose ǫ > 0 sufficiently small so that Ws (J ) is described as a holomorphic motion of J
loc Π Π
that is parameterized by t D (0).
ǫ
∈
Using the same proof as for J , one can check that J has a local product structure, so
z0 Π
that if (t,u) satisfies t < ǫ, then (t,u) is in the basin of attraction for one of the finitely
| |
many attracting periodic pointsbof fΠ (including u = b, which corresponds to [0 : 1 : 0]).
∞
Similarly, if t < ǫ, then J(f) (C t ) = Ws(J ) (C t ).
Π
| | ∩ ×{ } ∩ ×{ }
By Corollary 4.3, the horizontal critical locus
∂s (u)
t
C = (t,u) : = 0 and t U
U∞ ∂u ∈ ∞
(cid:26) (cid:27)
lies in the Fatou set for f. As before, the Implicit Function Theorem can be used repeatedly
to extend the holomorphic motion of J to one that is parameterized by all of U . Then,
Π ∞
Slodkowski’s Theorem can beused to extend this holomorphicmotion in the fiberw. Each of
theFatoucomponentsoff thatliesoverU (otherthanWs([0 : 1: 0]))isthenaholomorphic
∞
motion of a bounded Fatou component of the polynomial map f , parameterized by the
Π
10 R.K.W. Roeder
simply connected domain U . This gives that each such component is homeomorphic to an
∞
open bidisc.
Case 2: f is disconnected:
If J is disconnected for any z J , then Theorem 2.2 gives that H (Ws([0 : 1 : 0])) is
z p 1
∈
infinitely generated. It remains to consider the case that J is connected for every z J
z p
∈
and J is disconnected.
p
Let U be the basin of attraction of z = for p(z). Corollary 4.3 gives that each of the
∞
irreducible components of C = π−1(U ) i∞s in the immediate basin of attraction Ws(ζ ) of
U∞ |C ∞ 0 i
one of the finitely many attracting periodic points
ζ := ζ1,...,ζn1 ,...,ζ := ζ1,...,ζnm
1 { 1 1 } m { m m }
of f .
Π
The proof of Proposition 4.2 from [8] shows how to generate a sequence of piecewise
smooth one-cycles υ U bounding regions Υ C with the property that Υ ,µ 0.
n ∞ n n p
⊂ ⊂ h i →
Perturbing the υ slightly (if needed), we can suppose that none of them lie on the finitely
i
many critical values of π : C L. For each i we let
→
γ := π−1(υ ) and Γ := π−1(Υ ),
i i i i
so that each γ is a finite union of closed loops in C bounded by a piecewise smooth chain
i U∞
Γ . Moreover, π :Γ Υ is a ramified cover of degree d 1.
i i i
→ −
Proposition 4.2 gives Γ ,T = Γ ,T and the invariance properties of , from Propo-
i i p
h i h i h· ·i
sition 3.1 give
Γ ,T = Γ ,π∗µ = π Γ ,µ = (d 1) Υ ,µ .
i p i p ∗ i p i p
h i h i h i − h i
Since Υ ,µ 0, we have that
i p
h i →
lk(γ ,T) = Γ ,T (mod 1) = (d 1) Υ ,µ (mod 1) 0 (mod 1).
i i i p
h i − h i →
Proposition 3.4 gives that the first homology of Ws(ζ ) is infinitely generated. Hence at
i 0 i
least one of the finitely many Fatou components from this union has infinitely generated
homology. (cid:3) S
Remark 4.4. We will now discuss the question of whether Theorem 1.2 holds without the
hypothesisthatf beAxiom-A.Toseethatsomeconditiononhyperbolicityisneeded,consider
the disconnected product mapping
f(z,w) = (z2 6,w2 +i),
−
which has that J is aCantor set and that J is a (connected) dendritefor every z C. Since
p z
∈
J is connected for every z J , Theorem 2.2 gives that Ws([0 : 1 : 0]) is homeomorphic to
z p
∈
an open ball. Moreover, the closure of Ws([0 : 1 : 0]) is all of CP2, giving that it is the only
Fatou component for f.
Meanwhile, the author does not presently know of any example of a connected polynomial
skew product having any Fatou component that is not homeomorphic to an open ball.
One can somewhat weaken the hypothesis that f be Axiom-A and still have the proof of
Theorem 1.2 that is presented in this section hold. (We leave the Axiom-A condition in the
statement of Theorem 1.2, in order to keep it simple.) More specifically:
If J is disconnected for some z J , then no hypothesis is needed in order to determine
z p
∈
that Ws([0 :1 :0]) has infinitely generated first homology. See Example 5.5.