752 O. KOZLOVSKI, W. SHEN, AND S. VAN STRIEN
Like the previous successful approach in the quadratic case, we exploit
the powerful tool, Yoccoz puzzle. Also we require a “complex bounds” theorem
to treat infinitely renormalizable maps. The main difference is as follows. In
the proof of [10], [20], a crucial point was that quadratic polynomials display
decay of geometry: the moduli of certain dynamically defined annuli grow at
least linearly fast, which is a special property of quadratic maps. The proof in
[38] does not use this property explicitly, but instead a combinatorial bound
was adopted, which is also not satisfied by higher degree polynomials. So
all these proofs break down even for unimodal polynomials with degenerate
critical points. Our approach was inspired by a recent observation of Smania
[40], which was motivated by the works of Heinonen and Koskela [13], and
Kallunki and Koskela [15]. The key estimate (stated in the Key Lemma) is the
control of geometry for appropriately chosen puzzle pieces. For example, if c
is a nonperiodic recurrent critical point of f with a minimal ω-limit set, and
if f is not renormalizable at c, our result shows that given any Yoccoz puzzle
piece P c, there exist a constant δ > 0 and a sequence of combinatorially
defined puzzle pieces Q
n
, n = 1, 2, . . . , which contain c and are pullbacks of P
with the following properties:
• diam(Q
n
) → 0;
• Q
n
contains a Euclidean ball of radius δ · diam(Q
n
);
• there is a topological disk Q
n
⊃ Q
n
such that Q
n
− Q
n
is disjoint from
the orbit of c and has modulus at least δ.
In [40], Smania proved that in the nonrenormalizable unicritical case this
kind of control implies rigidity. To deduce rigidity from puzzle geometry con-
trol, we are not going to use this result of Smania directly - even in the
nonrenormalizable case - but instead we shall use a combination of the well-
known spreading principle (see Section 5.3) and the QC-criterion stated in
Appendix 1. This spreading principle states that if we have a K-qc homeo-
morphism h: P →
˜
P between corresponding puzzle neighbourhoods P,
˜
P of
the critical sets (of the two maps f,
˜
f) which respects the standard bound-
ary marking (i.e. agrees on the boundary of these puzzle pieces with what is
given by the B¨ottcher coordinates at infinity), then we can spread this to the
whole plane to get a K-qc partial conjugacy. Moreover, together with the
QC-criterion this also gives a method of constructing such K-qc homeomor-
phisms h, which relies on good control of the shape of puzzle pieces Q
i
⊂ P ,
˜
Q
i
⊂
˜
P with deeper depth. This different argument enables us to treat in-
finitely renormalizable maps as well. In fact, in that case, we have uniform
geometric control for a terminating puzzle piece, which implies that we have
a partial conjugacy up to the first renormalization level with uniform regular-
ity. Together with the “complex bounds” theorem proved in [37], this implies
rigidity for infinitely renormalizable maps, in a similar way as in [10], [20].
RIGIDITY FOR REAL POLYNOMIALS
753
In other words, everything boils down to proving the Key Lemma. It
is certainly not possible to obtain control of the shape of all critical puzzle
pieces in the principal nest. For this reason we introduce a new nest which
we will call the enhanced nest. In this enhanced nest, bounded geometry and
decay in geometry alternate in a more regular way. The successor construction
we use is more efficient than first return domains in transporting information
about geometry between different scales. In addition we use an ‘empty space’
construction enabling us to control the nonlinearity of the system.
1.2. Organization of this work. The strategy of the proof is to reduce it in
steps. In Section 2 we reduce the Density of Axiom A Theorem to the Rigidity
Theorem stated above. Then, in Section 3, we reduce it to the Reduced Rigidity
Theorem. These two sections can be read independently from the rest of this
paper, which is occupied by the proof of the Reduced Rigidity Theorem.
The idea of the proof of the Reduced Rigidity Theorem is to reduce all
difficulties to the Key Lemma.
In Section 4, we give the precise statement of the Key Lemma on control
of puzzle geometry for a polynomial-like box mapping which naturally appears
as the first return map to a certain open set. In Section 5, we review a few
facts on Yoccoz puzzles. These facts will be necessary to derive our Reduced
Rigidity Theorem from the Key Lemma, which is done in the next two sections,
Section 6 and Section 7.
The remaining sections are occupied by the proof of the Key Lemma. In
Section 8 we construct the enhanced nest, and show how to derive the Key
Lemma from lower and upper control of the geometry of the puzzle pieces
in this nest. In Section 9, we analyze the geometry of the real trace of the
enhanced nest. These analysis will be crucial in proving the lower and up-
per geometric control for the puzzle pieces, which will be done in Section 10
and Section 11 respectively. The statement and proof of a QC-criterion are
given in Appendix 1 and some general facts about Poincar´e discs are given in
Appendix 2.
We organized the paper in this way to emphasize that our proof shows
that if the properties asserted in the conclusion of the Key Lemma hold, then
Rigidity and Density of Hyperbolicity follow. If the reader is happy to assume
the Key Lemma and only interested in the nonrenormalizable case then he/she
only needs to read Sections 2-6. To deal with the infinitely renormalizable case
in addition, he/she also needs to read Sections 7. The later sections only deal
with the proof of the Key Lemma and therefore could be skipped if one could
prove the Key Lemma in a different way. But again, if he/she only wants to
see how the Key Lemma follows from the upper and lower bounds, then it is
sufficient to read Section 8. The proof of the lower and upper bounds is the
most technical part of this paper, and these are proved in Sections 10 and 11.
754 O. KOZLOVSKI, W. SHEN, AND S. VAN STRIEN
Real Bounds §9
⇓
Construction and Proper-
ties of the Enhanced Nest,
see §8.2 and §8.1
=⇒
Lower Bounds §10 &
Upper Bounds §11
⇓ 8.3
Key Lemma (Stated in §4)
⇓ §7
Spreading Principle §5.3
and QC-Criterion
§7
=⇒
Reduced Rigidity Theorem in
the infinitely renormalizable case,
stated in Prop osition 6.1
⇓ §6
Spreading Principle §5.3
and QC-Criterion
§6
=⇒
Reduced Rigidity Theo-
rem, stated in §1.1
⇓ §3
Rigidity Theorem, stated in §1.1
⇓ §2
Density of Hyperbolicity, stated in §1.1
1.3. General terminologies and notation. Given a topological space X and
a connected subset X
0
, we use Comp
X
0
(X) to denote the connected component
of X which contains X
0
. Moreover, for x ∈ X, Comp
x
(X) = Comp
{x}
(X).
For a bounded open interval I = (a, b) ⊂ R, C
I
= C − (R − I). For
any θ ∈ (0, π) we use D
θ
(I) to denote the set of points z ∈ C
I
such that the
angle (measured in the range [0, π]) between the two segments [a, z] and [z, b]
is greater than θ.
We usually consider a real-symmetric proper map f : U → V , where each
of U and V is a disjoint union of finitely many simply connected domains in C,
and U ⊂ V . Here “real-symmetric” means that U and V are symmetric with
respect to the real axis, and that f commutes with the complex conjugate. A
point at which the first derivative f
vanishes is called a critical point. We use
Crit(f) to denote the set of critical points of f. We shall always assume that
f
n
(c) is well defined for all c ∈ Crit(f) and all n ≥ 0, and use PC(f) to denote
the union of the forward orbit of all critical points:
PC(f) =
c∈Crit(f)
n≥0
{f
n
(c)}.
As usual ω(x) is the omega-limit set of x.
An interval I is a properly periodic interval of f if there exists s ≥ 1
such that I, f(I), . . . , f
s−1
(I) have pairwise disjoint interiors and such that
RIGIDITY FOR REAL POLYNOMIALS
755
f
s
(I) ⊂ I, f
s
(∂I) ⊂ ∂I. The integer s is the period of I. We say that f is
infinitely renormalizable at a point x ∈ U ∩R if there exists a properly periodic
interval containing x with an arbitrarily large period.
A nice open set P (with respect to f) is a finite union of topological disks
in V such that for any z ∈ ∂P and any n ∈ N, f
n
(z) ∈ P as long as f
n
(z) is
defined. The set P is strictly nice if we have f
n
(z) ∈ P .
Given a nice open set P , let D(P ) = {z ∈ V : ∃k ≥ 1, f
k
(z) ∈ P }. The
first entry map
E
P
: D(P ) → P
is defined as z → f
k(z)
(z), where k(z) is the minimal positive integer with
f
k(z)
(z) ∈ P . The restriction of E
P
to P is called the first return map to P ,
and is denoted by R
P
. The first landing map
ˆ
L
P
: D(P ) ∪ P → P
is defined as follows: for z ∈ P ,
ˆ
L
P
(z) = z, and for z ∈ D(P ) \ P ,
ˆ
L
P
(z) =
R
P
(z) (we use the roof notation in
ˆ
L
P
, to indicate that if a point z is already
‘home’, i.e. in P , then
ˆ
L
P
(z) = z). A component of the domain of the first
entry map to P is called an entry domain. Similar terminology applies to
return, landing domain. For x ∈ D(P ), L
x
(P ) denotes the entry domain
which contains x. For x ∈ D(P ) ∪P,
ˆ
L
x
(P ) denotes the landing domain which
contains x. So if x ∈ D(P ) \P , L
x
(P ) =
ˆ
L
x
(P ). We also define inductively
L
k
x
(P ) = L
x
(L
k−1
x
(P )).
We shall also frequently consider a nice interval, which means an open
interval I ⊂ V ∩ R such that for any x ∈ ∂I and any n ≥ 1, f
n
(x) ∈ I. The
terminology strictly nice interval, the first entry (return, landing) map to I as
well as the notation L
x
(I),
ˆ
L
x
(I) are defined in a similar way as above.
By a pullback of a topological disk P ⊂ V , we mean a component of f
−n
(P )
for some n ≥ 1, and a pullback of an interval I ⊂ V ∩R will mean a component
of f
−n
(I) ∩ R (rather than f
−n
(I)) for some n ≥ 1.
See Section 4 for the definition of a polynomial-like box mapping, child,
persistently recurrent, a set with bounded geometry and related objects.
See Section 9 for the definition of a chain and its intersection multiplic-
ity and order. Also the notions of scaled neighbourhood and δ-well-inside are
defined in that section.
For definitions of quasi-symmetric (qs) and quasi-conformal (qc) maps,
see Ahlfors [1].
At the end of the paper we put a list for notation we have used.
756 O. KOZLOVSKI, W. SHEN, AND S. VAN STRIEN
2. Density of Axiom A follows from the Rigidity Theorem
One of the main reason for us to look for rigidity is that it implies density
of Axiom A among certain dynamical systems. Our rigidity theorem implies
the following, sometimes called the real Fatou conjecture.
Theorem 2.1. Let f be a real polynomial of degree d ≥ 2. Assume that
all critical points of f are real and that f has a connected Julia set. Then f
can be approximated by hyperbolic real polynomials with real critical points and
connected Julia sets.
The rigidity theorem implies the instability of nonhyperbolic maps. As is
well-known, in the unicritical case the above theorem then follows easily: If
a map f is not stable, then the critical point of some nearby maps g will be
periodic, and so g will be hyperbolic. In the multimodal case, the fact that
the kneading sequence of nearby maps is different from that of f, does not
directly imply that one can find hyperbolic maps close to f. The proof in the
multimodal case, given below, is therefore more indirect.
By means of conjugacy by a real affine map, we may assume that the
intersection of the filled Julia set with R is equal to [0, 1]. Let Pol
d
denote
the family of all complex polynomials g of degree d such that g(0) = f(0) and
g(1) = f (1). Note that this family is parametrized by an open set in C
d−1
. Let
Pol
R
d
denote the subfamily of Pol
d
consisting of maps with real coefficients and
let X denote the subfamily of Pol
R
d
consisting of maps g which have only real
critical points and connected Julia set (so there is no escaping critical points).
Moreover, let Y denote the subset of X consisting of maps g satisfying the
following properties:
• Every critical point of g is nondegenerate;
• Every critical point and every critical value of g are contained in the open
interval (0, 1).
Note that Y is an open set in Pol
R
d
.
Lemma 2.1. X =
Y .
Proof. This statement follows from Theorem 3.3 of [33]. In fact X is the
family of boundary-anchored polynomial maps g : [0, 1] → [0, 1] with a fixed
degree and a specified shape which are determined by the degree and the sign
of the leading coefficient of f . Recall that given a real polynomial g ∈ X, its
critical value vector is the sequence (g(c
1
), g(c
2
), ··· , g(c
m
)), where c
1
≤ c
2
≤
··· ≤ c
m
are all critical points of g. That theorem claims that the critical
value vector determines the polynomial, and any vector v = (v
1
, v
2
, . . . , v
m
) ∈
R
m
, such that these v
i
lie in the correct order, is the critical value vector
RIGIDITY FOR REAL POLYNOMIALS
757
of some map in X. In any small neighborhood of the critical value vector
of f , we can choose a vector v = (v
1
, v
2
, ··· , v
m
) so that v satisfies the strict
admissible condition, i.e., these v
i
are pairwise distinct. The polynomial map
corresponding to this v is contained in Y .
Therefore by a perturbation, if necessary, we may assume that f ∈ Y . For
every g ∈ Y , let τ (g) be the number of critical points which are contained in
the basin of a (hyperbolic) attracting cycle. Note that the map τ : Y → N∪{0}
is lower semicontinuous. Let
Y
= {g ∈ Y : τ(g) is lo cally maximal at g}.
As τ is uniformly bounded from above, Y
is dense in Y . Moreover, from the
lower semicontinuity of τ , it is easy to see that τ is constant in a neighborhood
of any g ∈ Y
. Thus Y
is open and dense in Y . Note also that every map in Y
does not have a neutral cycle (this is well-known, because otherwise one can
perturb the map so that the neutral cycle becomes hyperbolic attracting; see
for example the pro of of Theorem VI.1.2 in [8]). Doing a further perturbation
if necessary, we assume that f ∈ Y
. Let r = τ (f).
Let c
1
< c
2
< ··· < c
d−1
be the critical points of f, and let Λ denote the
set of i such that c
i
∈ AB(f ), where AB(f) is the union of basins of attracting
cycles. Let U be a small ball in Pol
d
centered at f. (Recall that Pol
d
is
identified with an open set C
d−1
.) Then there exist holomorphic functions
c
i
: U → C, 1 ≤ i ≤ d −1,
such that c
i
(g) are all the critical points of g. By shrinking U if necessary, we
may assume that for any g ∈ U ∩ X, c
1
(g) < c
2
(g) < ··· < c
d−1
(g) and for
any g ∈ U and for any i ∈ Λ, c
i
(g) ∈ AB(g).
For a map g ∈ U, by a critical relation we mean a triple (n, i, j) of positive
integers such that g
n
(c
i
(g)) = c
j
(g). Given any submanifold S of U which
contains g, we say that the critical relation is persistent within S if for any h ∈
S, we have h
n
(c
i
(h)) = c
j
(h). Each critical relation corresponds to an algebraic
subvariety of Pol
d
of codimension one. Therefore, by a further perturbation if
necessary, we may assume that there is no critical relation ( n, i, j) for f with
i ∈ Λ. By shrinking U if necessary, we find that this statement remains true
for any g ∈ U.
Let
QC(f) = {g ∈ Pol
d
: g is quasiconformally conjugate to f}.
By Theorem 1 in [35], f does not support an invariant line field in its Julia set,
and thus by Theorem 6.9 of [29], the (complex) dimension of the Teichm¨uller
space of f is at most r (since we assumed there are no periodic critical points, it
is not an orbifold; see Theorem 6.2 of [29]). Consequently, QC(f) is covered by
758 O. KOZLOVSKI, W. SHEN, AND S. VAN STRIEN
countably many embedded complex submanifolds of Pol
d
which have (complex)
dimension at most r, and hence
QC
R
(f) = QC(f) ∩Pol
R
d
is covered by countably many embedded real analytic submanifolds M
i
of X
which have (real) dimension at most r (and so of codimension at least one).
The same argument applies to any map in Y
.
Completion of proof of Theorem 2.1. Let us keep the notation and
assumption on f as above. We shall prove that U ∩Pol
R
d
contains a hyperbolic
map. Arguing by contradiction, assume that every map g in U ∩ Pol
R
d
is not
hyperbolic. Then r = τ (f) < d − 1.
For positive integers n, 1 ≤ i, j ≤ d − 1, let
M
n,i,j
= {g ∈ U ∩ X : g
n
(c
i
(g)) = c
j
(g)}.
Each of these M
n,i,j
is a subvariety of U ∩ X with dimension at most d − 2.
By assumption M
n,i,j
= ∅ for i ∈ Λ. We claim that there exists some (n, i, j)
such that the dimension of M
n,i,j
is d − 2.
To see this we use the following fact, whose proof is easy and left to the
reader.
Fact 2.1. Let m be a positive integer, and let B be a Euclidean ball in
R
m
. Let M
i
, i = 1, 2, . . . be embedded real analytic submanifolds of B such
that dim(M
i
) ≤ m − 2. Then B −
∞
i=1
M
i
is arc-connected.
If all the M
n,i,j
’s have dimension less than d −2, then U ∩X −
M
n,i,j
is
arc-connected. By the standard kneading theory, [32], [25], it follows that any
g ∈ U ∩ X −
M
n,i,j
is topologically conjugate to f on the real line. By our
Rigidity Theorem, g ∈ QC
R
(f). Therefore, U ∩ X ⊂
M
n,i,j
∪ QC
R
(f). So
U ∩X is a countable union of manifolds of codimension at least one, which is
impossible.
Therefore, we obtain a real analytic codimension-one embedded subman-
ifold V
1
of U ∩ X which has a persistent critical relation (n, i, j) with i ∈ Λ.
Let us now apply the same arguments to the new (d − 2)-dimensional family
V
1
. More precisely, if r = d − 2, then this implies that every map in V
1
is
hyperbolic, which is a contradiction. So r < d − 2. Take any f
1
∈ V
1
. As the
Teichm¨uller space of f
1
also has (complex) dimension r, QC(f
1
) ∩X is covered
by a countable union of codimension one submanifolds of V
1
. Proceeding as
above, we will find a real analytic embedded submanifold V
2
of V
1
which has
dimension d − 3 and has two distinct persistent critical relations. Repeating
this argument we complete the proof.
RIGIDITY FOR REAL POLYNOMIALS
759
3. Derivation of the Rigidity Theorem
from the Reduced Rigidity Theorem
Definition 3.1. Let f and
˜
f be two polynomials of degree d, d ≥ 2. We
say that they are Thurston combinatorially equivalent if there exist homeomor-
phisms H
i
: C → C, i = 0, 1, such that
˜
f ◦ H
1
= H
0
◦ f, and H
0
∼ H
1
rel
PC(f) (i.e., H
0
and H
1
are homotopic rel PC(f)). The homeomorphism H
0
is called a Thurston combinatorial equivalence between these two polynomials,
and H
1
is called a lift of H
0
(with respect to f and
˜
f).
Proposition 3.1. Let f and
˜
f be real polynomials of degree d ≥ 2 with
only nondegenerate real critical points. Assume that they are topologically con-
jugate on the real axis, and let h : R → R be a conjugacy. Let H : C → C be a
real-symmetric homeomorphism which coincides with h on PC(f ). Then H is
a Thurston combinatorial equivalence between f and
˜
f.
Remark 3.1. Let H, H
be two real-symmetric homeomorphisms of the
complex plane which coincide on a set E ⊂ R. Then it is clear that H ∼ H
rel E.
Proof. Without loss of generality, we may assume that h is orientation-
preserving. Let c
1
< c
2
< ··· < c
d−1
and ˜c
1
< ˜c
2
< ··· < ˜c
d−1
be the
critical points of f and
˜
f respectively. It suffices to prove that there exists a
real-symmetric homeomorphism H
1
: C → C such that
˜
f ◦ H
1
= H ◦ f and
H
1
|R preserves the orientation. Indeed, we will then have H
1
= H on PC(f)
automatically, which implies that H
1
∼ H rel PC(f ).
Let us add a circle X = {∞e
i2πt
: t ∈ R/Z} to the complex plane. Then
C ∪ X is naturally identified with the closed unit disk, and f extends to a
continuous map from C ∪ X to itself, which acts on X by the formula t → dt
if the coefficient of the highest term of f is positive, or t → dt + 1/2 otherwise.
Let T = f
−1
(R), and T
0
= T − Crit(f ). Note that T
0
is a (disconnected)
one-dimensional manifold.
Let x
i
= ∞e
(d−i)π/d
for each 0 ≤ i ≤ 2d−1. Since each component of C−T
is a univalent preimage of one of the half planes, it is obviously unbounded.
Therefore there cannot be a closed curve in T
0
, and thus each component of
T
0
is diffeomorphic to the real line. The ends of these components can only
be a critical point or a point x
i
. By local behaviour of the critical points, for
each critical c
i
, there is a component γ
i
of T
0
which is contained in the upper
half plane and has c
i
as one end. Note that the other end of γ
i
must be in X,
for otherwise, C − T would have a bounded component. As these curves γ
i
,
1 ≤ i ≤ d − 1, are pairwise disjoint, the end of γ
i
at infinity must be x
i
. We
have proved that the intersection of T with the upper half plane consists of
d − 1 curves γ
i
, which connects x
i
and c
i
. By symmetry, the intersection of T
760 O. KOZLOVSKI, W. SHEN, AND S. VAN STRIEN
with the lower half plane consists of d −1 curves γ
i
, d + 1 ≤ i ≤ 2d −1, which
connects x
i
and c
2d−i
.
Similarly,
˜
T =
˜
f
−1
(R) has the same structure as T . Thus we can define a
real-symmetric homeomorphism H
1
: T →
˜
T as a lift of the map H : R → R.
Since each component of C −T is a univalent preimage of the upper or lower
half plane, H
1
extends to a homeomorphism of C, as a lift of H : C → C.
Derivation of the Rigidity Theorem from the Reduced Rigidity Theorem.
Let f and
˜
f b e two real polynomials as in the Rigidity Theorem, and let
h : R → R be a homeomorphism such that
˜
f ◦h = h ◦f. The Reduced Rigidity
Theorem implies that we can find a real-symmetric qc map Φ : C → C such
that Φ = h on PC(f ), and such that
˜
f ◦ Φ = Φ ◦ f holds on a neighborhood
of infinity and also on a neighborhood of each periodic attractor of f. By
Proposition 3.1, Φ is a Thurston combinatorial equivalence between f and
˜
f. Let Φ
0
= Φ and let Φ
n
, n ≥ 1, be the successive lifts. Then all these
homeomorphisms Φ
n
are quasiconformal with the same maximal dilatation as
that of Φ. Note that Φ
n
is eventually constant out of the Julia set J(f) of f.
Since J(f ) is nowhere dense, Φ
n
converges to a qc map which is a conjugacy
between f and
˜
f.
Although our main interest is in real polynomials with real critical points,
we shall frequently need to consider a slightly larger class of maps: real poly-
nomials with real critical values. This is because compositions of maps in F
d
may have complex critical points but only real critical values. Proposition 3.1
is no longer true if we only require f to have real critical values, and this is
the reason why we need to assume that f have only real critical points (rather
than real critical values) in our main theorem. It is convenient to introduce
the following definition.
Definition 3.2. Let f and
˜
f be polynomials with real co efficients such
that all critical values belong to the real line. We say that they are strongly
combinatorially equivalent if they are Thurston combinatorially equivalent, and
there exists a real-symmetric homeomorphism H : C → C such that
˜
f ◦ H =
H ◦ f on the real axis.
By Proposition 3.1, if f and
˜
f have only real nondegenerate critical points,
and they are topologically conjugate on R, then they are strongly combinato-
rially equivalent.
4. Statement of the Key Lemma
In this section, we give the precise statement of our Key Lemma on puzzle
geometry. As we will need universal bounds to treat the infinitely renormal-
izable case, we shall not state this lemma for a general real polynomial which
RIGIDITY FOR REAL POLYNOMIALS
761
V
0
U
3
c
0
U
0
U
1
U
2
V
1
c
1
V
2
c
2
V
b−1
c
b−1
Figure 1: An example of a polynomial-like box mapping.
does not have a satisfactory initial geometry. Instead, we shall first introduce
the notion of “polynomial-like box mappings”, and state the puzzle geometry
for this class of maps. These polynomial-like box mappings appear naturally
as first return maps to certain puzzle pieces; see for example Lemma 6.7.
Definition 4.1. Let b ≥ 1 and m ≥ 0 be integers. Let V
i
, 0 ≤ i ≤ b − 1,
be topological disks with pairwise disjoint closures, and let U
j
, 0 ≤ j ≤ m, be
topological disks with pairwise disjoint closures which are compactly contained
in V
0
. We say that a holomorphic map
f :
m
j=0
U
j
∪
b−1
i=1
V
i
→
b−1
i=0
V
i
(1)
is a polynomial-like box mapping if the following hold:
• For each 1 ≤ j ≤ m, there exists 0 ≤ i = i(j) ≤ b − 1 such that
f : U
j
→ V
i
is a conformal map;
• For U equal to U
0
, V
1
, . . . , V
b−1
, there exists 0 ≤ i = i(U ) ≤ b − 1 such
that f : U → V
i
is a 2-to-1 branched covering.
The filled Julia set of f is defined to be
K(f ) = {z ∈ Dom(f) : f
n
(z) ∈ Dom(f) for any n ∈ N};
and the Julia set is J(f ) = ∂K(f).
An example of a polynomial-like box mapping is shown on Figure 1. In
fact, everything we do will go through in the case where critical points are
degenerate of even order. If b = 1, then such a map is frequently called
generalized polynomial-like.
