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COMPLEX DYNAMICS
HOLLY KRIEGER
These notes are compiled from the Part III Complex dynamics given in Lent 2020. Re-
porting of typos and mistakes to [email protected] will be rewarded with ac-
knowledgements, and (presumably) good mathematical karma.
Prerequisites: complex analysis is essential. I will also assume some basic knowl-
edge of hyperbolic geometry and Riemann surfaces, including the statement of
uniformization. I will try when possible to remind you of the relevant defini-
tions/theorems in use, and notes from the Cambridge Riemann surfaces course are
available from various online sources: one is http://qk206.user.srcf.net/notes/
riemann_surfaces.pdf. Finally, I assume you know the definition and basic proper-
ties of the Lebesgue measure on Euclidean space. No algebraic topology or differential
geometry is assumed. If you’re not certain that you have the prerequisites, or want
to flex your muscles before getting to the dynamics, have a go at Example Sheet 0.
Acknowledgements. This course owes a significant debt to the book of Milnor [Mi]
and follows it closely for the first 14 of 24 lectures. I also made substantial use of the book
of Branner and Fagella [BrFa] and the recent course of DeMarco (notes not available). My
sincere thanks go to the Cambridge students and faculty who attended this course in its
first instance, for catching mistakes and notational inconsistencies during the lectures and
example classes, and for their patience and assistance with correcting those errors.
1
2 HOLLY KRIEGER
***Lecture 1***
1. Overview, review
In this course we will study the iteration of holomorphic self-maps of Riemann surfaces.
Given a Riemann surface R and a holomorphic, non-constant map f : R 7→ R, we write
fn(z) := f ◦ f ◦ · · · ◦ f︸ ︷︷ ︸n times
for the nth iterate of f . We will study the behavior of fn as n→∞.
1.1. Classical inspiration: Newton’s method. Recall Newton’s method is an iterative
algorithm for numerical computation of the roots of polynomials, by finding tangent lines’
intersection with the real axis. Given a polynomial p(z), Newton’s method takes in an initial
guess z0 and iteratively computes
f(z) = z − p(z)
p′(z),
a self-map of the Riemann sphere. When and why does Newton’s method converge?
Figure 1. Newton’s method convergence domains for p(z) = z3 − 1: the
stable locus for f(z) = (2z3 − 1)/(3z2)
COMPLEX DYNAMICS 3
Figure 2. Newton’s method convergence domains for p(z) = z4− 2z2 + 9:
the stable locus for f(z) = (3z4 − 2z2 − 9)/(4z3 − 4z). Note in this case
Newton’s fails spectacularly on the real line (not a surprise as p is real with
no real roots!)
1.2. Where are we going? Our goal for this course relates to the Mandelbrot set.
Definition 1.1. Fix d ≥ 2. Given c ∈ C, write fc(z) = zd + c. The d-Mandelbrot set is
Md = {c ∈ C : |fnc (0)| 6→ ∞ as n→∞}.
Figure 3. Hello world! The case of d = 2.
Theorem 1.1 (McMullen [Mc]). The Mandelbrot set is universal for bifurcations.
Loosely: in any bifurcation locus, we see slightly distorted copies of the boundary of
some Md. In particular, we see distorted copies of the Mandelbrot set densely inside the
4 HOLLY KRIEGER
Mandelbrot set. We will explain what all this means.
1.3. Starting point: what are the Riemann surfaces? What are the possible self-
maps? The topology of a Riemann surface controls the possible holomorphic self-maps
which it admits. Recall:
Definition 1.2. A Riemann surface R is a connected, Hausdorff topological space, to-
gether with a collection of homeomorphisms
φα : Uα → Dα ⊂ C
with Uα open in R and Dα open in C, so that⋃α
Uα = R,
and if Uα ∩ Uβ 6= ∅, then φβ ◦ φ−1α is analytic on φα(Uα ∩ Uβ).
For a given α, the information of (Uα, φα) is a chart on the surface, and the compositions
φβ ◦ φ−1α are the transition functions. The collection of charts is called an atlas.
Theorem 1.2 (Uniformization theorem). Every Riemann surface is conformally isomorphic
to R/G where R is one of the three simply connected Riemann surfaces - C,C,D - and G a
subgroup of its automorphism group which acts freely and properly discontinuously on R.
Definition 1.3. A group G of homeomorphisms of a topological space X is said to act
properly discontinuously if every x ∈ X has an open neighborhood U so that g(U) ∩h(U) = ∅ for all g 6= h ∈ G, and freely if every the action of every non-identity element is
fixed point free.
We thus have a strong classification of all Riemann surfaces, particularly the non-hyperbolic
ones.
Universal cover C. The automorphisms of C are the Mobius transformations, each of
which has exactly 1 or 2 fixed points. Hence G consists of the identity map, and R = C.
COMPLEX DYNAMICS 5
Universal cover C. The automorphisms of C are the linear maps az+ b, as we proved.
If a 6= 1, there is a fixed point so G consists of translations, and so is a subgroup of C via
g 7→ g(0). As G acts properly discontinuously, it must consist of isolated points, so as shown
in example sheet 3, either G is trivial, Zw1, or a lattice Λ. So R is conformally equivalent
to C, C∗, or a complex torus C/Λ.
Universal cover D. The rest! Every Riemann surface which is not C, C∗, C, or C/Λis uniformized by D. A Riemann surface R is hyperbolic if it is uniformized by the disk,
and carries a unique metric so that the covering map is a local isometry with respect to the
Poincare metric
ds =2|dz|
1− |z|2on the disk.
Recall that the distance between any two points on the disk is the infimum along
smooth curves γ : [a, b]→ D which connect the two points of the length value
`(γ) =
∫ b
a
2
1− |γ(t)|2|γ′(t)| dt.
Using the origin and automorphisms, one computes that the distance function is
given by
d(x, y) =log(1 +R)
log(1−R),
where R =∣∣∣ y−x1−xy
∣∣∣.This is known as the hyperbolic or Poincare metric for R.
Now, what are the maps?
The study of self-maps of a hyperbolic Riemann surface is essentially the study of holo-
morphic self-maps of the unit disk. The proof of the proposition is an exercise (ES0):
Proposition 1.1. Let S = D/Γ be a hyperbolic Riemann surface and f : S → S a holo-
morphic self-map. Then f lifts to a holomorphic map F : D→ D, unique up to composition
with an element of Γ, and F induces a group homomorphism g : Γ→ Γ, so that
F ◦ γ = g(γ) ◦ F
for every γ ∈ Γ.
6 HOLLY KRIEGER
***Lecture 2***
Though we will focus on the dynamics of rational maps of the Riemann sphere, the
study of hyperbolic dynamics will be included as any local study of dynamics is necessarily
hyperbolic, and the following theorem provides a strong constraint on such maps.
Theorem 1.3 (Pick theorem). Let f : S → S be a holomorphic map of a hyperbolic
Riemann surface S with Poincare metric ρ. Then for all x, y ∈ S,
ρ(f(x), f(y)) ≤ ρ(x, y),
with strict inequality unless f lifts to an automorphism of D.
Proof. By definition of the Poincare metric, it suffices to prove the theorem for S = D; that
is, that
log(1 +R′)
log(1−R′)≤ log(1 +R)
log(1−R),
where
R =
∣∣∣∣ y − x1− xy
∣∣∣∣ and R′ =
∣∣∣∣∣ f(y)− f(x)
1− f(x)f(y)
∣∣∣∣∣ ,with strict inequality iff f is a disk automorphism. As log(1 + x)/ log(1 − x) is strictly
increasing, it suffices to show R′ ≤ R. This follows from Schwarz’s lemma: let
σ1(z) =y − z1− yz
and σ2(z) =f(y)− z1− f(y)z
be disk automorphisms. Then σ2 ◦ f ◦ σ−11 : D→ D and fixes 0, so by Schwarz’s lemma we
have
|σ2 ◦ f ◦ σ−11 (z)| ≤ |z|,
that is, ∣∣∣∣∣ f(y)− f(x)
1− f(y)f(x)
∣∣∣∣∣ ≤∣∣∣∣ y − x1− yx
∣∣∣∣by writing σ−1(z) = x. The statement on strict inequality follows from the corresponding
statement in Schwarz. �
Thus all self-maps of hyperbolic surfaces are contracting. This is in stark contrast
to the case of rational maps of the Riemann sphere; for example, the spherical metric
ds = 2|dz|/(1+|z|2) can be expanding or contracting for points of C under the map z 7→ z+1:
COMPLEX DYNAMICS 7
Proposition 1.2. Let f : C → C be a holomorphic, non-constant map. Then f is a non-
constant rational function; that is, there exist a1, . . . , am, b1, . . . , bn ∈ C and c ∈ Cx so
that
f(z) =c(z − a1) · · · (z − am)
(z − b1) · · · (z − bn).
Proof. Replacing by 1/f if needed, WLOG assume f(∞) ∈ C. By the identity principle,
f−1(∞) is a finite subset {b1, . . . , bn} ⊂ C of poles of f . Thus near each bj ∈ f−1(∞), we
have
f(z) =
∞∑i=−k
aj,i(z − bj)i.
Set Qj(z) =∑−1i=−k aj,i(z − bj)n. Then each Qj is a rational function, and g = f − (Q1 +
· · · + Qn) has only removable singularities, so g(C) ⊂ C. If g is not constant, then g(C) is
thus a compact, proper, and open subset of C, a contradiction. Thus g is constant and f is
rational as claimed. �
The dynamics of maps of this type will form the bulk of our study.
The case of universal covering C does indeed admit interesting dynamics, such as tran-
scendental maps like the exponential map z 7→ ez on C, or endomorphisms of elliptic curves
(complex tori), which descend from affine maps on C. However, we will not discuss these
examples in this class; the former because it is a complicated and independent field of study,
the latter because it is extremely specialized and not as interesting (from an analytic point
of view).
Lemma 1.1. Every Riemann surface admits a complete conformal metric.
Proof. You have already seen an explicit description of this metric in two of the three cases;
exercise (ES1). �
8 HOLLY KRIEGER
2. Families of holomorphic maps
As we are studying iteration of maps, we will often deal with sequences of holomorphic
functions. Certain iterative sequences have very predictable (“stable”) behavior; for exam-
ple, the self-map f(z) = z2 of the unit disk has iterates fn(z) = z2n
, so all points converge
to 0 under repeated application of f(z). To describe this behavior with a general notion of
stability, we will require two classical theorems from the theory of holomorphic functions of
domains of C which are not necessarily covered in a complex analysis course: Arzela-Ascoli
and Montel’s theorem. To introduce them, we need some definitions.
Definition 2.1. Let S, T be metric spaces and fn : S → T a sequence of continuous maps.
We say that {fn} converges locally uniformly if for all compact K ⊂ S and all ε > 0, there
exists N ∈ N so that
supz∈K
dT (fn(z), fm(z)) < ε
for all n,m ≥ N . We say that {fn} diverges locally uniformly if for all compact K ⊂ S and
compact K ′ ⊂ T , we have N ∈ N so that fn(K) ∩K ′ = ∅ for all n ≥ N .
Recall by the Weierstrass uniform convergence theorem that any locally uniformly con-
vergence subsequence will have a holomorphic limit.
You should consider locally uniform divergence to be convergence to infinity; in particular,
note that if T itself is compact, no sequence can diverge locally uniformly. As an example,
if f : Dx → Dx is any reasonable contracting map such as z 7→ z2, then the iterates of
f will converge on compact subsets to the constant function 0, which is not a function
taking values on this Riemann surface. While one-point compactifications play well with
holomorphic functions, the metrics do not necessarily do so.
2.1. Normality.
Definition 2.2. A family F of holomorphic functions f : S → T is normal if every sequence
has a subsequence which is either locally uniformly convergent or locally uniformly divergent.
Proposition 2.1. Normality is local; that is, if F is is normal on all elements of an open
cover of S, then F is normal on S.
Proof. Exercise (ES1). But note that normality depends on the topology, not the metric,
and so when we restrict to a disk we don’t need to be fussy about which distance function
we use to determine normality. �
The following definition and theorem provide a more hands-on normality test.
Definition 2.3. A family F of continuous functions on a domain U ⊂ C with values in a
metric space T is equicontinuous if for all ε > 0 there exists δ > 0 so that for all s, t ∈ Uand all f ∈ F ,
|s− t| < δ ⇒ dT (f(s), f(t)) < ε.
COMPLEX DYNAMICS 9
***Lecture 3***
Theorem 2.1 (Arzela-Ascoli). A family F : U → T of continuous maps on a domain
U ⊂ C to a metric space T is has the property that all sequences have a locally uniformly
convergent subsequence if and only if the following are both satisfied:
• F is equicontinuous on any compact subset of U , and
• for all z ∈ U , {f(z) : f ∈ F} lies in a compact subset of T .
Proof. We leave the easy (forward) direction as an exercise (ES1).
Let {zk} be a countable dense subset of U , and fix a sequence {fn} ⊂ F in the family.
Since the collection of images of any point takes values in a compact set, we may find an
array of integers nki so that
n11 < n12 < · · · < n1j < · · ·
n21 < n22 < · · · < n2j < · · ·
...
so that each row is contained in the previous one, and limj→∞ fnkj(zk) exists for each k ∈ N.
The diagonal sequence fnkkis then strictly increasing and for each k, eventually contained
in the kth row. Thus {gk := fnkk} is a subsequence of F which converges at all points zk.
Consider now a compact subset K ⊂ U ; we will show that {gk} converges uniformly on
K. Given ε > 0, we may by equicontinuity on K find δ > 0 so that for all x, y ∈ K and
all f ∈ F , |x − y| < δ implies dX(f(x), f(y)) < ε/3. Cover K by a finite number of δ/2
neighborhoods, and select one of the points zk from each neighborhood. For these finitely
many zk we can find N so that for all n,m > N we have dX(gn(zk), gm(zk)) < ε/3. As
every z ∈ K is within a δ-neighborhood of some zk, we have dX(gn(z), gm(z)) < ε for all
n,m > N and all z ∈ K. As these values live in a compact and therefore complete subset
of T , the subsequence {gk} is uniformly convergent on compact subsets of U and hence Fis normal as claimed. �
Corollary 2.1. If T is compact, a family of holomorphic maps from U to T is normal if
and only if it is equicontinuous on compact subsets.
Corollary 2.2. Let S, T be hyperbolic Riemann surfaces. Then the family F of holomorphic
maps from S to T is normal.
Proof. Note by Pick’s theorem, the family is equicontinuous. If there exists x ∈ S so that
{f(x) : f ∈ F} lies in some compact subset K ⊂ T , then again by Pick’s theorem, the same
is true for all y ∈ S. Thus by Arzela-Ascoli, the family is normal.
Otherwise, we may fix x ∈ S and y ∈ T and choose some infinite sequence of maps fn ∈ Fso that dT (fn(x), y) → ∞ as n → ∞. Fix a compact subset K ⊂ S and K ′ ⊂ T , noting
that dS(x,K) and dT (y,K ′) are bounded. Then by Pick’s theorem, fn(K) ∩K ′ = ∅ for n
sufficiently large, so {fn} diverges locally uniformly. �
In generality, this corollary is important enough that we will restate it as a theorem.
10 HOLLY KRIEGER
Theorem 2.2 (Montel’s theorem). Let S, T be Riemann surfaces with T hyperbolic. Then
the family of holomorphic maps from S to T forms a normal family.
Proof. If S is hyperbolic, we have the corollary above. If S is not hyperbolic, by uni-
formization, lifting, and Liouville’s theorem, any holomorphic map S → T is constant. Let
{fn} ⊂ F be a sequence of constant maps. If there exists an infinite subsequence with image
contained in some compact set, then we can extract a convergent subsequence and {fn} has
a locally uniformly convergent subsequence. Otherwise, only finitely many fn have image
intersecting any compact K ⊂ T , so the sequence is locally uniformly divergent. �
Example. Let f : C → C be a rational map and D ⊂ C an open disk. If the family of
iterates F := {fn |D: n ≥ 0} omits three values of C, then F is a normal family. For exam-
ple, the iterates of the map z 7→ z2 form a normal family on any disk contained in D or C\D.
2.2. Degree and ramification. We conclude the background section with some additional
remarks on holomorphic maps of domains in C.
Definition 2.4. A continuous map f : U → V of topological spaces is proper if for all
K ⊂ V compact, f−1(K) is compact in U .
Lemma 2.1. Let f : U → V be continuous. If U is compact and V Hausdorff, then f is
proper.
Proof. Let K be a compact subset of V . V Hausdorff ⇒ K closed, so V \K is open and
by continuity of f , f−1(V \K) is open in U . Thus f−1(K) = U \ f−1(V \K) is closed and
thus compact in U . �
In the following, let U, V be open in C, and suppose f : U → V is holomorphic and
non-constant.
Properness essentially allows us to promote statements about holomorphic mappings of
compact Riemann surfaces to equivalent statements for proper maps of arbitrary Riemann
surfaces. As an example, the following two propositions are easily proved if we take for
granted the corresponding statement for compact Riemann surfaces (which we do!):
Proposition 2.2. If f : U → V is proper, then f has a well-defined degree on U .
Theorem 2.3 (Riemann-Hurwitz). Let f : U → V be proper of degree d. Then
ξ(U) = dξ(V )−∑p∈U
(ep − 1),
where ep is the ramification index of f at p.
Remark. It may be useful to recall that any domain U in C has exhaustion by open
subsets with finitely many smooth boundary components. One way to see this in the case
that U is bounded is to subdivide C into δ-squares, and let Vδ be the union of all interior
squares of U . Smoothing the boundary of Vδ gives an open subset of U with finitely many
smooth boundary components, contained in U . For U unbounded, the same construction
will require intersecting with balls of increasing size to ensure the finiteness of boundary
COMPLEX DYNAMICS 11
components.
A small number of useful corollaries following directly from Riemann-Hurwitz:
Corollary 2.3. A degree d rational function f : C→ C has 2d− 2 critical points, counting
multiplicities.
Recall that while the derivative of a function is defined on charts, the vanishing of the
derivative is well-defined on the entire Riemann surface, and thus so are the critical points.
Corollary 2.4. If U and V have Euler characteristic 0 (for example, annuli or punctured
disks) then any proper holomorphic map f : U → V is unramified.
12 HOLLY KRIEGER
***Lecture 4***
3. Dynamics!
Definition 3.1. Let f : C→ C be a non-constant holomorphic map. The Fatou set of f is
F (f) := {z ∈ C : {fn} form a normal family on a neighborhood U of z}.
The Julia set is J(f) := C \ F (f).
You should have in mind that the Fatou set consists of points where long-term iterative
behavior is stable under perturbation, and the Julia set consists of unstable points.
Alert: now and forever more, I will consider only rational maps of degree at
least 2. In the case of Mobius transformations, the Fatou and Julia set can be
explicitly calculated (see ES1).
Example. We have already seen that the iterates of z 7→ z2 form a normal family on
any neighborhood contained in D or C \D, but never on the unit circle. Thus the Julia set
of z 7→ z2 is precisely the unit circle. This type of smooth Julia set is rare!
Here are some examples of Julia sets:
(a) (b)
Figure 4. Julia set for Newton’s method applied to p(z) = (z − 1) ∗ (z −c+ 0.5) ∗ (z + c+ 0.5) for c = 0.0051 + 0.033i is the boundary between the
various colors: orange/yellow and black points are all Fatou. A zoom at 0
is on the right in (B).
COMPLEX DYNAMICS 13
(a) z2 + .3144 + .2797i (b) z2 + i
(c) z2 + .0412 + i (d) z2 − .122 + .743i
(e) z2 − 0.592/z (f) z2 + 10−6/z3
Figure 5. Some interesting Julia sets! (E) due to Devaney, (F) to Mc-
Mullen. Note the similarity of the rabbit (D) to the zoom in the previous
Newton’s example.
14 HOLLY KRIEGER
3.1. Iteration and the Julia set.
Lemma 3.1. The Fatou and Julia sets of f are forward- and backward-invariant under f .
Proof. It suffices to prove that for any open set U ⊂ C, a sequence of iterates fnj converge
uniformly on compact subsets of U if and only if the sequence of iterates fnj+1 converges
uniformly on compact subsets of f−1(U). We have
supz∈K
d(fnj+1(z), fnk+1) = supw∈f(K)
d(fnj (w), fnk(w)),
and f is proper and continuous, so images and preimages preserve compactness, proving the
equivalence. �
We see immediately a first explanation for the self-similarity we noticed in the Julia sets:
if z ∈ J(f) and f ′(z) 6= 0, f provides a conformal isomorphism from a neighborhood N of
z to its image, with N ∩ J(f) sent to f(N) ∩ J(f).
Lemma 3.2. J(f) = J(fn) for any iterate fn of f , and similarly for the Fatou set.
Proof. By definition. �
The structure of the Julia set is strongly controlled by Montel’s theorem. We will see later
that the Julia set is non-empty; for the moment, I will phrase the statements conditionally.
For example, we have immediately that J(f) is the smallest totally f -invariant set which
contains at least 3 points, as the complement of any such set much by Montel’s be contained
in the Fatou set.
Theorem 3.1. Let z ∈ J(f) and U an open neighborhood of z. Then the union V :=
∪fn(U) of forward images of U contains all but at most two points of the Riemann sphere,
both of which are critical points in the Fatou set.
Proof. By Montel’s theorem, if the family {fn |U} omits three points of C, then it is normal,
proving the first statement. By definition, f(V ) ⊂ V , so any point z 6∈ V has all preimages
not in V . Considering ramification of f , the only possibility is 1 fully ramified point, 2 fully
ramified points, or 2 points mapping with full degree to each other. �
Corollary 3.1. If the Julia set contains an interior point, then J(f) = C.
Proof. By the previous theorem and forward-invariance, if the Julia set contains an interior
point, then J(f) contains all but at most two points of the Riemann sphere, so is dense in
C. However, the Julia set is clearly closed by definition, so is the full Riemann sphere. �
Remark. This absolutely can happen! We will understand why later in the course, but
for any t ∈ C \ {0, 1},
ft(z) =(z2 − t)2
4z(z − 1)(z − t)
is a rational map with J(f) = C.
COMPLEX DYNAMICS 15
3.2. Cycles and multipliers. In the example of z 7→ z2, we have very simple behavior
under iteration for points in the Fatou set: each such point converges to either 0 or∞ under
iteration. Generally, this type of iterative attraction is a local phenomenon.
Definition 3.2. Let z0 ∈ C. We say that z0 is periodic or in a periodic cycle for f if
fm(z0) = z0 for some m ∈ N. The minimal such m is the period of the cycle. If z0 has
period m, the derivative
λ := (fm)′(z0) =
m−1∏i=0
f ′(f i(z0))
is the multiplier of the cycle.
Note a simple computation with the usual chart shows that if f(∞) =∞ then
f ′(∞) = limz→∞
1/f ′(z)
.
Definition 3.3. Let z0 be a point of period m for f , with multiplier λ. We say z0 is
• superattracting if λ = 0
• attracting if |λ| < 1
• indifferent if |λ| = 1
• repelling if |λ| > 1.
16 HOLLY KRIEGER
***Lecture 5***
Definition 3.4. Suppose C := {z0, . . . , zm−1} is an attracting orbit of period m. The basin
of attraction for the cycle C is the set
A := {z ∈ C : limn→∞
fnm(z) ∈ C}.
Just as in the z 7→ z2 case, any attractor is in the Fatou set, and any repeller is in the
Julia set.
Theorem 3.2. Every attracting basin is contained in the Fatou set. Any repelling periodic
point is contained in the Julia set.
Proof. Since the J(fm) = J(f), we may WLOG assume that z0 has period 1; that is, z0
is a fixed point of f , with multiplier λ. Suppose first that |λ| < 1. By Taylor’s theorem,
for any |λ| < c < 1 we have |f(z) − z0| ≤ c|z − z0| for |z − z0| sufficiently small, so the
iterates of f converge uniformly on compact subsets of a small neighborhood of z0 to the
constant function z 7→ z0. Thus z0 is in the Fatou set. As the Julia set is closed and
forward-invariant, for any z ∈ J(f), any limit point of the sequence of iterates {fn(z)} is in
the Julia set as well, so the attracting basin of z0 is contained in F (f).
Suppose now that |λ| > 1. If z0 ∈ F (f), then the iterates of f form a normal family on a
neighborhood U of z0, and thus some subsequence converges to a holomorphic limit g on U .
However, the derivatives of the iterates at z0 satisfy (fn)′(z0) = λn so diverge to infinity, so
g cannot be holomorphic. Thus z0 ∈ J(f). �
Indifferent points may be in the Julia set or not: we will explore this in depth later. While
the derivative of a fixed point is a local computation, there is not complete independence in
the possible sets of multipliers, due to an important result known as the holomorphic fixed
point formula.
Definition 3.5. Let z0 be a fixed point of a rational map f of degree d. The residue index
of f at z0 is
if (z0) :=1
2πi
∫γ
dz
z − f(z),
where γ is any small positively oriented loop around z0.
The residue index is tightly tied to the multiplier of the fixed point, in the following way.
Lemma 3.3. Let λ be the multiplier of a fixed point z0 of f . If λ 6= 1, then
if (z0) =1
1− λ.
Proof. As the multiplier is coordinate-invariant and the residue index is translation-invariant,
we may WLOG assume z0 = 0. Expanding f as a power series, we can write
f(z) = λz + a2z2 + . . .
on a neighborhood of 0. Since λ 6= 1, we have z − f(z) = (1− λ)z(1 + . . . ), so that
1
z − f(z)=
1
(1− λ)z+ g(z)
COMPLEX DYNAMICS 17
for some holomorphic function g. Taking integrals completes the proof. �
Lemma 3.4. if (z) is coordinate-independent.
Proof. Exercise (ES1). By the preceding lemma, you need only consider fixed points of
multiplier 1. �
Theorem 3.3 (Holomorphic Lefschetz on the Riemann sphere). Let f : C→ C be a rational
map of degree at least 2. Then ∑z=f(z)
if (z) = 1.
Proof. By coordinate-independence, we may conjugate f if necessary to assume that f(∞) 6=∞. Let CR denote the circle of radius R about 0, positively oriented, and suppose R is
sufficiently large so that |z| < R for all fixed points z of f . We have via w = 1/z that∫CR
dz
z − f(z)=
∫−C1/R
−dww2(1/w − f(1/w))
=
∫C1/R
dw
w(1− wf(1/w)).
The non-zero poles of the integrand are the reciprocals of the fixed points, which by con-
struction do not lie interior to C1/R, so by the residue theorem,∑z=f(z)
if (z) =
∫CR
dz
z − f(z)= Resw=0
1
w(1− wf(1/w))= 1,
as claimed. �
3.3. The structure of the Julia set.
Corollary 3.2. A rational map f of degree d ≥ 2 has non-empty Julia set.
Proof. We have shown any repelling periodic point lies in the Julia set, so assume all fixed
points have multipliers |λ| ≤ 1. Suppose first that λ 6= 1 for any fixed point multiplier of f .
As the derivative of f(z)− z is f ′(z)− 1, it follows that there are d+ 1 fixed points for f .
As x 7→ 1/(1− x) sends the unit disk to the line <x = 1/2, the sum of the residues indices
has real part ≥ (d+ 1)/2, contradicting the fixed point formula.
Suppose now that λ = 1 for some fixed point of f . We will show that this fixed point lies
in the Julia set. In local coordinates, we have on a neighborhood of the point that f sends
w 7→ w + akwk + ... where ak 6= 0, so that
fn(w) = w + nakwk + ...
Thus the kth derivative of fn is k!nak, which diverges to ∞ as n → ∞. Thus the family
of iterates contains a sequence with no convergent subsequence, and the fixed point is not
Fatou. �
As a consequence, we see that J(f) is infinite, since we have shown that the Julia set
contains a fixed point z0, and that any point with finite grand orbit must be in the Fatou
set (exercise: ES1).
Remark 3.1. Note that by taking iterates in the argument above, we have in fact proved
that any periodic cycle with a root of unity multiplier is necessarily in the Julia set.
18 HOLLY KRIEGER
Note that it can happen that multiple fixed points prevents the existence of a repelling
fixed point. For example, in degree 2, the polynomial f(z) = z2 + 1/4 has two fixed points,
∞ and 1/2, the latter with multiplicity. However, we will eventually see that all but finitely
many periodic points of f are repelling.
COMPLEX DYNAMICS 19
***Lecture 6***
Proposition 3.1. If A ⊂ C is the attracting basin of a periodic orbit for f , then the Julia
set J(f) is the topological boundary ∂A of A.
Proof. Let U be any open neighborhood which intersects the Julia set. Since the forward
images of U intersect all but at most two points of the Riemann sphere, fn(U) ∩A 6= ∅ for
some n ∈ N. As A is closed under preimages by definition, U ∩ A is non-empty. As this
holds for any such U , J(f) ⊂ A. As A is contained in the Fatou set, J(f) ⊂ ∂A.
On the other hand, suppose z ∈ ∂A, and suppose some subsequence of iterates approaches
a holomorphic limit g on a neighborhood U of z. Then g |U takes a finite set of values on
U ∩A (namely, the cycle values) but is non-constant, a contradiction. Thus z is in the Julia
set of f . �
Finally, we will show that preimages of Julia set elements are dense in the Julia set. You
should have in mind here the example of z 7→ z2; for any α ∈ S1, the roots of z2n − α are
evenly spaced by a distance of normalized arc length 1/2n on the unit circle, and thus the
full set of preimages of α form a dense subset of S1.
Recall that we proved via Montel’s theorem that for any z ∈ J(f) and any open neigh-
borhood N of z, the union of forward images
U =⋃n∈N
fn(N)
contains all but at most two points in C, any of which must be critical points in the Fatou set.
In particular, if we fix z0, z1 ∈ J(f), then z1 is arbitrarily well approximated by preimages
of z0. For if not, then there is a neighborhood of z1 whose forward images do not contain
z0, a contradiction. Thus:
Corollary 3.3. For any z0 ∈ J(f), the set of all preimages
{z : fn(z) = z0 for some n ≥ 0}
form a dense subset of J(f).
J(f) is thus perfect: as J(f) is closed and infinite, it contains at least one non-isolated
point z0. The preimages of z0 form a dense, non-isolated subset of the Julia set, so no point
of J(f) is isolated.
4. Local dynamics - attractors
For the next few lectures we will focus on the local theory of complex dynamics around
a periodic cycle, which replacing by an iterate if necessary, we will assume is a fixed point.
As we saw in the case of the existence of an attracting basin determining the Julia set (as
its topological boundary), this local study is a key tool for the global study.
Conjugating by a Mobius transformation, we will in this section often assume WLOG
that f has a fixed point at 0.
20 HOLLY KRIEGER
Definition 4.1. A fixed point p of f is topologically attracting if there exists a neighborhood
U of p such that {fn} converges locally uniformly to the constant map z 7→ p on U .
Lemma 4.1. A fixed point p is topologically attracting if and only if it is attracting; that
is, iff the multiplier λ = f ′(p) at p satisfies |λ| < 1.
Proof. Taylor’s theorem in one direction, Schwarz’s lemma in the other. �
The following theorem is key to the study of polynomial and more generally attracting
rational dynamics.
Theorem 4.1 (Koenig’s linearization theorem). Let p be a fixed point of f with multiplier
λ satisfying |λ| 6= 0, 1. Then there exist a local holomorphic change of coordinate w = φ(z)
so that φ(0) = 0, and φ ◦ f ◦ φ−1(w) = λw for all w in some neighborhood of p:
U f(U)
C C
f
φ φ
·λ
The change of coordinate φ is unique up to multiplication by a constant.
Proof. WLOG assume p = 0. Suppose first that |λ| < 1, and find a constant c < 1 so that
c2 < |λ| < c. We can by Taylor’s theorem choose a disk of radius r > 0 of 0 so that for all
z ∈ D(0, r), |f(z)| ≤ c|z|. We thus for z ∈ D(0, r) have geometric convergence |fn(z)| ≤ rcn.
Again by Taylor expansion, we have a constant B > 0 so that |f(z)−λz| ≤ B|z|2 on D(0, r),
and thus for z ∈ D(0, r),
|fn+1(z)− λfn(z)| ≤ B|fn(z)|2 ≤ Br2c2n.
Thus wn = fn(z)/λn satisfy
|wn+1 − wn| ≤(Br2
|λ|
)(c2
|λ|
)n,
converging uniformly on compact subsets to 0. Thus the holomorphic functions z 7→ wn(z)
converge uniformly on compact subsets of Dr, to a holomorphic limit
φ(z) = limn→∞
fn(z)
λn.
The identity φ(f(z)) = λφ(z) follows immediately, and as the derivative of each z 7→ wn(z)
at the origin is 1, φ′(0) = 1 and thus φ has holomorphic inverse on a neighborhood of 0.
To see that φ is unique up to a constant, suppose that ψ was another linearizing change
of coordinate. Then φ(ψ−1(w)) = φ(ψ−1(λw)), as for all w ∈ ψ(U) we have
w λw
ψ−1(w) ψ−1(λw)
φ(ψ−1(w)) λφ(ψ−1(w)) = φ(ψ−1(λw))
·λ
ψ−1 ψ−1
f
φ φ
·λ
COMPLEX DYNAMICS 21
Expanding φ ◦ ψ−1 on a neighborhood of 0, we see that
λφ ◦ ψ−1(w) = λ(a1w + a2w2 + . . . ) = a1(λw) + a2(λw)2 + . . . ,
so that λak = λkak for all k ≥ 1. As |λ| 6∈ {0, 1}, we have ak = 0 for all k ≥ 2, and conclude
that φ ◦ ψ−1(w) = a1w, so that φ(z) = a1ψ(z) as claimed.
Finally, if |λ| > 1, apply the preceding proof to f−1 on a neighborhood of 0 on which it
is holomorphically defined. �
We may use the functional equation φ(f(z)) = λz to extend φ to the full basin, though
it will no longer necessarily remain conformal.
Corollary 4.1. Suppose p is an attracting fixed point of f with multiplier λ 6= 0 and basin
of attraction A. There is a holomorphic map φ : A → C so that the following diagram
commutes:
A A
C C
f
φ φ
·λ
which is biholomorphic on a neighborhood of p and unique up to multiplication by a
constant.
Proof. According to the linearization theorem, we may define biholomorphic φ on a neigh-
borhood U of p. By definition of A, for all z ∈ A, there exists n ∈ N so that fn(z) ∈ U .
Thus
φ(z) := limn→∞
φ(fn(z))
λn
satisfies the corollary. �
22 HOLLY KRIEGER
Figure 6. Preimages of circles under the Koenig map for z2 + 0.2i and
the fixed attractor.
We saw by the holomorphic fixed point formula that a rational map f : C → C cannot
have all of its fixed points attracting. In fact, a much stronger result holds, controlling
the total number of attracting cycles for a rational map; first, a preliminary definition and
lemma.
Definition 4.2. The immediate basin of an attracting cycle is the union of connected
component of the Fatou set containing all cycle elements.
COMPLEX DYNAMICS 23
***Lecture 7***
Theorem 4.2. Suppose f : C→ C has degree d ≥ 2. Then every attracting periodic cycle of
f has a critical point in its immediate basin. Consequently, f has at most 2d− 2 attracting
cycles.
Proof. WLOG assume the attractor is fixed and at 0. We will show that the Fatou compo-
nent U containing 0 must contain a critical point of f . Since the Julia set is infinite, U is
hyperbolic, so we may lift f : U → U to a holomorphic map F : D→ D of the unit disk, via
the universal covering map π : D → U . If U contains no critical point, then f ◦ π : D → U
is another covering map, so by universality we have a map G : D→ D so that the following
diagram commutes:
D D
U U
π π
G
f
Choosing π, F,G to fix 0, we see that G is inverse to F , and therefore F is a disk
automorphism and so f a local isometry for the hyperbolic metric, which is clearly false on
a neighborhood of the attracting fixed point 0. Thus U must contain a critical point. �
Corollary 4.2. Let f be a rational map. Then f has at most 4d− 4 non-repelling cycles.
Proof. Consider for any t ∈ C the map ft(z) = (1 − t)f(z) + tzd, where f has degree d.
For any cycle of period n, say containing a point α, we may find a holomorphic function
t 7→ α(t) on a neighborhood of 0, so long as α is not a repeated root of fn(z) − z; that is,
so long as the cycle does not have multiplier 1. In the case of multiplier 1, the same is true
after base change by a power of t.
We then also have a holomorphic map λ(t) sending α(t) to its period n multiplier. If λ(t)
is constant, it is either the constant 1, or we can extend α and λ throughout C; either way,
we have a contradiction at t = 1, as zd has no indifferent cycles.
Consider now the set
{θ ∈ S1 : |λ(εeiθ)| < 1}
for small ε. Since λ is conformal, the measure of this set is 1/2. Repeated this process for
all indifferent cycles, there exists a choice of direction θ so that we may perturb at least half
of the indifferent cycles to attracting cycles. Choosing the perturbation sufficiently small
so that the attracting cycles remain attracting, we have (N/2) + M ≤ 2d − 2 attracting
cycles for the perturbation, where N is the number of indifferent cycles and M the number
of attracting cycles. Thus f has at most N + M ≤ N + 2M ≤ 4d − 4 non-repelling cycles
as claimed. �
We can say something a bit more refined as an alternative approach to this proof, which
is an option in example sheet 1, but doesn’t cover the superattracting case.
24 HOLLY KRIEGER
Theorem 4.3. Suppose f(0) = 0 is attracting for f , with multiplier λ 6= 0, immediate basin
A0, and linearizing coordinate φ with local inverse ψ : D(0, ε) → A0. Then ψ extends by
analytic continuation to a maximal open disk D(0, r), extends homeomorphically over the
boundary circle ∂D(0, r), and the image ψ(∂D(0, r)) ⊂ A0 contains a critical point.
Remark. While the presence of an attracting cycle is a strong condition on a rational
map, actually determining whether an attracting cycle exists is difficult. In principle, we can
iterate the critical points to look for possible attracting cycles, but in reality, if the period
is large, this is impractical. As an example, it is open whether or not f3/2(z) = z2 − 3/2
has a (non-infinite) attracting cycle. Iterating the unique free critical point 0, the number
of digits of fn(0) grows like 2n, and we break our computers very quickly (and similarly
trying to solve for points of period n and test their derivatives).
Do not be fooled into thinking that the existence of a linearizing map through the full
basin of attraction guarantees any nice topological structure for the Julia set, such as con-
nectedness. In fact, disconnected Julia sets have large numbers of connected components:
Theorem 4.4. If a rational map f has disconnected Julia set, then J(f) has uncountably
many connected components.
Proof. Suppose that J(f) is not connected, and write J(f) = J0 ∪ J1 as two non-empty,
disjoint, compact subsets. As J(f) has no isolated points, J0 and J1 are infinite. Given
z ∈ J(f), we assign a sequence
β(z) = (βn(z))n∈N
where βn(z) = 0 if z ∈ J0, and otherwise βn(z) = 1. A connected component of J(f) cannot
have a disconnected forward iterate, so any two points in the same connected component
of J(f) have the same assigned sequence. We will show an uncountable number of such
sequences are realized.
Let (β0(z), . . . , βk(z)) be a finite piece of the sequence for z ∈ J(f). We will first show
there exists z′ ∈ J(f) so that for some n > k, βn(z) 6= βn(z′). Suppose not, so that all β(z′)
which start with (β0(z), . . . , βk(z)) satisfy β(z′) = β(z).
Let
Uz,k := {z ∈ C : f i(z) 6∈ J1−βi(z) for all 0 ≤ i ≤ k},
noting each Uz,k is open and contains the Fatou set. Some subsequence (βnj (z)) is constant,
say taking the value of β0(z). Thus if w is a Julia point of Uz,k, since it has the same β-
sequences as z, fnj (w) ∈ Jβnj(z) i.e. fnj (w) ∈ Uz,0. Since all Uz,k contain the Fatou set,
the same holds for Fatou points of Uz,k, and so fnj (Uz,k) ⊂ Uz,0. Thus a subsequence of
the iterates form an family of maps fnj : Uz,k → Uz,0; as both these spaces are hyperbolic
(recall each Ji is infinite), the family is normal and thus has a convergent subsequence,
contradicting that the neighborhood Uz,k has non-empty intersection with the Julia set.
Thus every finite sequence realized by a Julia point can be extended in at least two ways,
and the set of all such sequences is therefore uncountable. �
COMPLEX DYNAMICS 25
As a concrete example, consider f(z) = (z2 − 1)/(z2 − c) with c = 1/2 + i, which has a
unique critical orbit and a fixed attractor. We always have a choice of positive or negative
preimage, so disconnecting J(f) by the real axis, the map
z 7→ β(z)
described above in fact bijects the Julia set to the space of all infinite sequences of zeros
and ones, so that application of f corresponds to the shift operator.
Figure 7. The uncountable Julia set for f(z) = (z2 − 1)/(z2 − c) with
c = 1/2 + i. Note the isomorphic extension of the linearizing coordinate
until we hit the “first” critical point at 0.
Figure 8. Attraction of a critical point: the same example, with the orbit
of 0 in white, as it is attracted to a fixed point.
26 HOLLY KRIEGER
***Lecture 8***
5. Local dynamics - superattractors
We turn now to a local study which is crucial for understanding polynomial dynamics
and the Mandelbrot set: the study of superattracting periodic cycles, which by the product
rule are cycles which contain a critical point of the rational map.
Theorem 5.1 (Bottcher 1904). Suppose f(0) = 0 is holomorphic, with local expansion
f(z) = amzm + am+1z
m+1 + · · · .
Then there exists a holomorphic local change of coordinate φ with φ(0) = 0 so that φ(f(z)) =
φ(z)m. φ is unique up to multiplication by an (m− 1)st root of unity.
Proof. The proof is very similar to Koenig’s theorem, so I merely sketch it and leave the
details for you to fill in. First, we may up to (global) conjugacy assume that am = 1, by
considering α(z/α) for some (m− 1)st root α of am. We thus have
f(z) = zm(1 + h(z))
for some holomorphic h(z) vanishing at 0. We thus have a holomorphic logarithm of 1+h(z)
on a neighborhood of 0, so we may write
1 + h(z) = ek(z)
on a neighborhood U of 0, and f(z) = zmek(z). By induction, we have - defined on the same
neighborhood U - a sequence of holomorphic maps kn(z) so that
fn(z) = zmn
ekn(z).
Choose then the branch of the mnth root of fn(z) so that
φn(z) := fn(z)1/mn
= z(1 +O(z)).
It can be shown that φn(z) converges uniformly on compact subsets of U to a holomorphic
map φ, which satisfies the requirements of the theorem.
For uniqueness, use the Taylor expansion as in the linearizing case. �
We see that, unlike the linearizing coordinate for a non-superattracting attracting basin,
because we are taking roots in this construction there is generally no hope of extending the
coordinate to the entire basin of attraction. That said, we can always take roots in R, so
the modulus of φ can always extend.
Corollary 5.1. Let f(0) = 0 be a superattracting fixed point for f , with basin of attraction
A, and let φ be a Bottcher coordinate on a neighborhood of 0. Then z 7→ |φ(z)| extends
uniquely to a continuous map |φ| : A → [0, 1) satisfying the identity |φ(f(z))| = |φ(z)|m for
all z ∈ A.
Proof. Given z ∈ A, set |φ(z)| := |φ(fn(z))|1/mn
, where n is chosen sufficiently large so
that fn(z) lies in the neighborhood of definition for the Bottcher coordinate. Continuity,
functional equation, and uniqueness are immediate. �
COMPLEX DYNAMICS 27
Remark. We can just as well for f(∞) = ∞ choose a Bottcher coordinate for which
φ(∞) = ∞, still conjugating to a power map on a neighborhood of ∞, by taking the
reciprocal of the usual map. I will often refer to this as the Bottcher coordinate in this case:
the choice will be obvious in context.
5.1. Attraction example: Newton’s method. We now develop in detail an important
and motivating example of the global theory. Recall that Newton’s method for a polynomial
p(z) of degree d ≥ 2 concerns the iteration of the auxiliary rational map of degree d
f(z) = z − p(z)
p′(z).
If p has distinct roots, we see immediately that the d distinct roots of p are fixed. As
f ′(z) =p(z)p′′(z)
p′(z)2,
the roots of p are superattracting points for f , and that ∞ is a fixed point with derivative
d/(d− 1) > 1; that is, a repelling fixed point.
If the only Fatou components of f are the elements of the attracting basins of the roots
of p, then Newton’s method does a relatively good job of finding a zero of the polynomial
p, since it will only fail on the Julia set. However, this need not be the case! While we
have accounted for all the fixed points of f , there could for example be an attracting cy-
cle of some higher period. As the only non-fixed critical point of f is 0, it will necessarily
be attracted to that cycle, and thus Newton’s method will fail on an open neighborhood of 0.
For example, the family
pc(z) := (z − 1)(z − c+ 0.5)(z + c+ 0.5)
has Newton maps
fc(z) := z − (z − 1)(z − c+ 0.5)(z + c+ 0.5)
3z2 − c2 − 0.75,
and 0 is the unique critical point of fc(z) which is not fixed for all choices of c ∈ C (i.e.
all members of the family). Here is a picture which colors a parameter c ∈ C green if 0 is
attracted to 1, blue if 0 is attracted to −a− 1/2, and red if 0 is attracted to a− 1/2.
28 HOLLY KRIEGER
Note that not all parameters take a color red, green, or blue. Let’s zoom in on one of
those regions in the lower left.
Figure 9. Looks a bit familiar...
It isn’t just the parameter space that looks familiar. If we choose one of these maps by
fixing c, say one in here:
COMPLEX DYNAMICS 29
...then we see lots of points around 0 which for this specific parameter do not converge
to a root of pc(z) under iteration.
Figure 10. Rabbit rabbit.
5.2. Polynomial maps.
30 HOLLY KRIEGER
Definition 5.1. Let p(z) = adzd+ · · ·+a1z+a0 be a polynomial of degree d ≥ 2. The filled
Julia set of p is
K(p) := {z ∈ C : pn(z) 6→ ∞ as n→∞}.
Corollary 5.2. Let p(z) be a polynomial of degree d ≥ 2. Then J(p) = ∂K(p).
Proof. As∞ is a totally ramified fixed point for p, it is a superattracting, and thus attracting,
point. The complement of the filled Julia set for p is
B(∞) := {z ∈ C : fn(z)→∞ as n→∞},
which is precisely the basin of attraction for the fixed point at ∞. Since the boundary of
any full basin is the Julia set, J(p) = ∂K(p). �
Definition 5.2. Let f(z) be a polynomial of degree d ≥ 2. The Green’s function associated
to f is
Gf (z) := limn→∞
log+ |fn(z)|dn
,
where log+ x = max{0, log x} for any x ≥ 0.
Lemma 5.1. The Green’s function Gf associated to a polynomial f satisfies:
(1) Gf is continuous everywhere, and harmonic on C \K(f)
(2) Gf (z) = log |z|+O(1) as |z| → ∞(3) G(z)→ 0 as z → K(f)
(4) Gf (f(z)) = dGf (z)
Gf (z) is uniquely characterized by properties (1), (2), and (4). Additionally, for all z 6∈ K(f),
Gf (z) = log |φf (z)|,
where |φf | is the extended Bottcher coordinate on the basin of infinity.
COMPLEX DYNAMICS 31
***Lecture 9***
Proof. Consider the function log+ |f(z)|−d log+ |z| from C→ R≥0. This functions is clearly
continuous and thus bounded by some constant C. We thus have∣∣∣∣ log+ |fn(z)|dn
− log+ |fn−1(z)|dn−1
∣∣∣∣ ≤ C
dn
for all n, so that for all m < n, we may use a telescoping sum∣∣∣∣ log+ |fn(z)|dn
− log+ |fm(z)|dm
∣∣∣∣ ≤ n−1∑k=m
∣∣∣∣ log+ |fk+1(z)|dk+1
− log+ |fk(z)|dk
∣∣∣∣≤
n−1∑k=m
C
dk+1
≤ C
dm(d− 1).
Thus Gf is a uniform limit of continuous functions, hence continuous. As |fn(z)| remains
bounded on the filled Julia set, we have Gf (z) = 0 for all z ∈ K(f), and property (3) follows
immediately. For z 6∈ K(f), let U be a simply connected neighborhood of z with compact
closure disjoint from K(f), so that there exists n ∈ N so that |fm(U)| ≥ 1 for all m ≥ n,
and thus fm(z) is a the logarithm of a holomorphic function on U and log |fm(z)|/dn thus
harmonic for all m ≥ n. As we have uniform convergence to Gf on U by the telescoping
argument, Gf is harmonic on C \K(f), completing the proof of property (1).
Taking m = 0 in the telescoping sum, we have for |z| > 1 that∣∣∣∣ log+ |fn(z)|dn
− log |z|∣∣∣∣ ≤ C
d− 1,
proving (2). The functional equation (4) is clear from definition.
To prove uniqueness, suppose that H(z) has properties (1), (2), (4). Then the difference
G(z) := Gf (z) − H(z) is continuous and bounded on the Riemann sphere, and satisfies
G(f(z)) = dG(z), so that for all z, dnG(z) is bounded as n → ∞. Thus G(z) = 0 for all z
and so Gf (z) = H(z).
Finally, noting that the function log |φf (z)| satisfies condition (1), (2), and (4) completes
the proof. �
Remark. The Green’s function is also known as the potential function associated to
the compact set K(f); note that by the lemma, this function is determined by K(f) only,
though our definition uses the polynomial f .
Example. One easily computes that for the polynomial p(z) = zd, we have Gp(z) =
log+ |z|.
32 HOLLY KRIEGER
5.3. Critical points and superattractors. In the attracting case, any attracting cycle
attracts a critical point. In the superattracting case, this is true in the trivial sense. Whether
or not any *other* critical point lies in the basin determines how far the Bottcher coordinate
extends.
Theorem 5.2. Suppose f(0) = 0 is a superattracting fixed point with Bottcher coordinate
φ for f at 0 . There exists a unique open disk Dr of maximal radius 0 < r ≤ 1 so that
the inverse ψ of φ extends holomorphically to ψ : D(0, r)→ A0, where A0 is the immediate
basin of the superattractor. If r = 1, then ψ maps the unit disk isomorphically onto A0, and
0 is the only critical point in the immediate basin. If r < 1, then there is at least one other
critical point in A0, lying on the boundary of ψ(D(0, r)).
The proof will be worked through as an exercise (ES2).
We can see precisely where the injectivity fails if we draw the level curves of φ carefully
(in particular, by drawing curves with values taken by the orbit of a critical point):
Figure 11. Critical orbit level curves for z2 + 1/2. The light orange lem-
niscate containing the critical point 0 is mapped two to one by z2 + 1/2 to
the curve bounding the bright yellow region.
The following corollary is immediate, and will be key to our study of the Mandelbrot set:
Corollary 5.3. Let fc(z) = z2 + c, with c 6∈ M (recall: the forward iterates of 0 are thus
converging to ∞). Then c is in the basin of ∞, and the Bottcher coordinate extends to a
neighborhood of c.
We can additionally deduce a useful alternative characterization of the Mandelbrot set.
First we need a topological proposition (note to 2020 students: I didn’t do this proof in
class, so it is not examinable).
Proposition 5.1. A closed subset of the sphere is connected if and only if every component
of its complement is simply connected.
COMPLEX DYNAMICS 33
Proof. If S is a closed, disconnected subset of the sphere, let U and V disconnect S, and
S1 := S ∩U c and S2 := S ∩ V c. As these are closed (hence compact) subsets of the sphere,
they are a positive finite distance δ apart. Taking for example the boundary of a net of
squares of diameter δ/3 covering any component of S1, we may then find a Jordan curve
γ in the complement of S so that one component of C \ γ contains points of S1, while the
other contains points of S2. Thus γ is not contractible in the complement, and so some
complementary component is not simply connected.
On the other hand, if we have a non-contractible curve in a complementary component
of S, then both components of its complement intersect S, so S is disconnected by the
complement of the curve. �
34 HOLLY KRIEGER
****Lecture 10****
Proposition 5.2. Let f be a polynomial of degree d ≥ 2. If all the finite critical points of
f are contained in K(f), then both K(f) and J(f) are connected. If at least one critical
point belongs to the basin of infinity, then both K(f) and J(f) are disconnected.
Proof. By the preceding theorem, if no finite critical points lie in the basin, then the Bottcher
coordinate extends to an isomorphism
C \K(f)→ C \ D,
so the complement of K(f) is simply connected. By the topological proposition, K(f) is
connected.
To prove that J(f) is connected it remains to show that every bounded Fatou component
is simply connected. Let U be a bounded Fatou component, and suppose that γ is a simply
closed curve in U . Then the bounded component V of C \ γ consists of points which do
not go to infinity under iteration, by the maximum modulus principle. Thus V contains no
Julia points, and so V ⊂ U . Thus U is simply connected.
On the other hand, if the Julia set is connected, so is K(f), so the basin of infinity A is
simply connected, so by uniformization, a topological disk. Thus by Riemann-Hurwitz, we
have
1 = χ(A) = dχ(A)−∑p∈A
ep − 1.
Since∞ has ramification index d, it is the unique critical point in A. So if the basin contains
a finite critical point, J(f) and K(f) are disconnected. �
Remark. We know that J(f) disconnected ⇒ J(f) has uncountably many components.
Refining the kneading argument using the level curves of the Green’s function for values of
iterates of a critical point in the basin shows the same for K(f), if f has a critical point in
the basin of infinity.
Corollary 5.4. A parameter c ∈ M if and only if the corresponding Julia set J(fc) (resp.
filled Julia set K(fc)) is connected.
Proof. 0 is the only non-fixed critical point for fc(z) = z2 + c, so c ∈ M iff 0 is not in
the basin of infinity iff the Bottcher coordinate is an isomorphism i.e. K(f) and J(f)
connected. �
6. Local dynamics - repellers
Remark. By Koenig linearization, if f(z0) = z0 has multiplier |λ| > 1 and λ 6∈ R, then
the Julia set contains an infinite sequence of points lying in the conformal image under the
linearizing map of the logarithmic spiral
S := {z0 + e2πt(a+i) : t ∈ R},
COMPLEX DYNAMICS 35
where λ = eR+iθ and a = R/θ. While this isn’t a proof, this suggests we might find spiraling
behavior in the Julia set; and by density of preimages, if there is spiraling somewhere, there
is spiraling everywhere.
There is no analogy of a ’repelling basin’; indeed, since we know any neighborhood of a
Julia point has non-normal iterates, there cannot be. However, we can extend the linearizing
map very easily.
Corollary 6.1. Suppose z0 is a repelling fixed point for the rational map f . Then there
exists a holomorphic map ψ : C→ C with ψ(0) = z0, so that the diagram
S S
C C
f
·λ
ψ ψ
commutes, with ψ holomorphic on a neighborhood of 0. ψ is unique up to pre-composition
with a scaling; that is, w 7→ ψ(cw) for c 6= 0.
Proof. Let φ be the Koenig linearizing map, and let ψ be a holomorphic inverse for φ on
a neighborhood of 0. For any w ∈ C, find n sufficiently large so that w/λn lies in this
neighborhood, and define
ψ(w) = fn(ψ(w/λn)).
The rest is a check. �
7. Local dynamics - parabolics
Definition 7.1. An indifferent periodic point of f is parabolic if the multiplier λ is a root
of unity.
Recall we have shown that parabolic cycles lie in the Julia set J(f): it follows that they
cannot be locally linearizable. In this section, we will make some progress on describing the
local dynamics despite that fact. Model: z 7→ z + z2 at 0: attracting direction, repelling
direction. We will focus first on the case of λ = 1.
Let
f(z) = z + azn+1 + . . .
on a neighborhood of 0, with n+ 1 ≥ 2 the multiplicity of the fixed point. We have
f(z) = z(1 + azn + . . . ),
and so the local behavior of z depends on the argument of azn.
Definition 7.2. In the above setting, let v0 be any complex number satisfying navn0 = 1,
and define the vector v0 to be the vector to v0 based at 0. Define for k ∈ {1, . . . , 2n− 1}
vk := eπik/nv0.
The vectors v0,v2, . . . ,v2n−2 are repelling vectors for the parabolic fixed point, and v1,v3, . . . ,v2n−1the attracting vectors for the parabolic fixed point.
36 HOLLY KRIEGER
***Lecture 11***
We describe now the local dynamics in terms of the attracting and repelling vectors,
using branches of the substitution
w = φ(z) =−1
nazn,
which sends our attracting and repelling vectors to unit vectors in the positive and negative
real directions, respectively. Let c = −1/na, and write Ri for the ray determined by vector
vi, 0 ≤ i ≤ 2n− 1. Let ∆i denote the sector
∆i := {reiθvi : r > 0, |θ| < π/n}.
Then φ(z) := c/zn maps each ∆i biholomorphically onto a slit plane, omitting the positive
real axis if i is even and the negative real axis if i is odd. We thus have for each i a
holomorphically define inverse ψi : C \ R(−1)i → ∆i of φ. We are going to consider the
function Fj := φ ◦ f ◦ ψj on ∆j .
Since f(z) = z(1 + azn + o(z)), we have
f ◦ ψj(w) = (c/w)1/n(1 + ac/w + o(1/w))
COMPLEX DYNAMICS 37
as |w| → ∞, and composing with φ(z) = c/zn, we have for Fj := φ ◦ f ◦ ψj that
Fj(w) = w(
1 + ac
w+ o(1/w)
)−n= w
(1 +−nacw
+ o(1/w)
).
Since c = −1/na, this gives
Fj(w) = w + 1 + o(1) as |w| → ∞.
Choose a number R > 0 so that for all 0 ≤ j ≤ 2n− 1 we have
|Fj(w)− w − 1| < 1/2 for all |w| > R,
so that in particular, we have
<(Fj(w)) > <(w) + 1/2,
and so with z = ψj(w) sufficiently small we have
<φ(f(z)) > <φ(z) + 1/2.
If j is odd (so ∆j contains an attracting vector, with repelling boundaries), then Fj maps the
right half plane HR := {z : <(z) > R} into itself, and so f maps its image Pj(R) := ψj(HR)
into itself, and the iterates of f restricted to Pj(R) converge uniformly to the constant map
0. Pj(R) is known as an attracting petal, and is thus Fatou.
In fact these points describe all iterates attracted to 0, for if limn→∞ fn(z) = 0, then we
eventually have <φ(f(z)) > <φ(z)+1/2, so eventually <φ(fn(z)) > R, and zm must belong
to one of the attracting petals Pj(R). In fact, writing wk := φ(fk(z)) for such a point z, we
have <wk →∞ as k →∞, and so wk+1 − wk → 1 as k →∞. So the average
wk − w0
k→ 1
as well, so wk ∼ k as k →∞; that is, fk(z) ∼ vj as k →∞, where the iterates are eventually
in the attracting petal Pj(R). We thus have a well-defined notion for all points attracted to
a parabolic fixed point of a direction of attraction.
Finally, applying this procedure to f−1, we obtain also repelling petals, on which back-
ward orbits of a branch of f−1 converge to the parabolic fixed point.
Definition 7.3. Let v be an attracting vector of a parabolic fixed point of multiplier λ = 1.
Then the parabolic basin of attraction associated to v is the set of points which converge
under iteration to the fixed point along direction v, and the immediate basin associated to v
is the connected component of the Fatou set containing v. For λq = 1, we will use the same
definitions, applied to fq.
Lemma 7.1. Suppose λ = e2πip/q is a primitive qth root of unity, and f(0) = 0 with
multiplier λ. Then the multiplicity of 0 as a fixed point of fq is congruent to 1 mod q.
Proof. Let n + 1 be the multiplicity of 0 as a fixed point of fq. Then multiplication by λ
permutes the n attracting vectors, so n ≡ 0 mod q i.e. n+ 1 ≡ 1 mod q. �
38 HOLLY KRIEGER
Here, for example, we see a parabolic fixed point of multiplier λ = e2πiθ, θ = 2/5, along
with the orbit of 0 to illustrate the map on the parabolic basins.
Figure 12. A parabolic fixed point for z2 + c, c ' 0.448 + 0.53i.
Note that the picture is somewhat similar in the case of a fixed point of multiplier 1 but
the action of f is different on the basins: it fixes the petals rather than rotating them.
Figure 13. The filled Julia set of z + z7, and the shape of an orbit in an
attracting petal near a repelling vector.
COMPLEX DYNAMICS 39
We will not again go through another coordinate construction proof. However, be aware
of the statement of the following theorem.
Theorem 7.1. Suppose P is an attracting basin of f for the parabolic fixed point z0. There
exists a conformal embedding α : P → C, unique up to composition with a translation, so
that
α(f(z)) = 1 + α(z)
for all α ∈ P ∩ f−1(P). This map extends uniquely to a map A → C on the attracting basin
of the parabolic point, which necessarily contains a critical point of f .
40 HOLLY KRIEGER
***Lecture 12***
8. Irrationally indifferent cycles
It remains to deal with the (complicated!) case when |λ| = 1 is not a root of unity. First,
a classification theorem.
Theorem 8.1 (Classification of hyperbolic self-maps). Let f : U → U be a holomorphic
map of a hyperbolic Riemann surface. Then exactly one of the following four possibilities
holds:
• f has an attracting fixed point in U , and all orbits in U converge to this point,
uniformly on compact subsets.
• f escapes U ; that is, for every compact K ⊂ U there is nK ∈ N so that K∩fn(K) = ∅for n ≥ nK .
• f has two distinct periodic points in U . In this case, some iterate of f is the identity
map on U .
• U is isomorphic to a disk, punctured disk, or an annulus, and f conjugates under
this map to an irrational rotation.
Proof. Denote by d the Poincare distance on U . If for any p ∈ U ,
limn→∞
d(fn(p), p) =∞,
then by Schwarz-Pick, all compact subsets diverge and we are in the second case (“escape”).
So suppose this doesn’t happen. For any fixed p0, then, some subsequence of iterates of p0
converges. Passing to this subsequence, since fn forms a normal family, we find a holomor-
phic limit function g of iterates of f so that g has a fixed point p ∈ U . Note that as a limit
of iterates of f , g commutes with f , and so f(p) is also a fixed point of g.
By Schwarz-Pick, either f is strictly distance-decreasing, or a local isometry. In the first
case, g is also distance-decreasing, so cannot have two fixed points; in particular, f(p) = p,
and p must be attracting as f is distance-decreasing, so we are in the first case.
Suppose now that f is a local isometry for d. Then the multiplier of g at p is necessarily
of the form e2πiα. Choose a sequence of integers m so that e2πimα → 1 as m → ∞. The
subsequence of iterates gm then form a normal (non-diverging) family so have holomorphic
limit h (again of iterates of f !) with a fixed point of multiplier 1. Lifting h to the universal
cover D, by Schwarz’s lemma, h is the identity map.
Note then that f is necessarily a conformal automorphism of U , as some sequence of
iterates limits to the identity map. If U is simply connected, it follows from Schwarz that
f is a rotation, and we are done. So suppose U is not simply connected, and there is a
non-trivial subgroup Γ of Aut(D) so that U = D/Γ. Write F for the disk automorphism
which is the lift of f , choosing F (0) = 0 so that F (z) = e2πiαz. For each j ∈ N, F j induces
a map gj : Γ→ Γ satisfying
F j ◦ γ = gj(γ) ◦ F j .
COMPLEX DYNAMICS 41
Since Γ is discrete, F j close to the identity implies gj(γ) = γ; that is, the jth iterate of F
commutes with Γ for j >> 1. But each element of Γ is then a rotation, since for each γ ∈ Γ,
we have
e2πiαj ◦ γ(0) = γ(e2πiαj0) = γ(0),
so γ(0) is the unique fixed point of the rotation F j i.e. equals 0. So Γ is a discrete
group of rotations, and so generated by a single element. Thus the complement of U has
two components, and U is either a punctured plane, punctured disk, or annulus. As U is
hyperbolic, only the latter two are possible. �
Corollary 8.1. Let f be a rational map of degree d ≥ 2 with indifferent fixed point z0.
Then z0 ∈ F (f) if and only if f is locally linearizable at z0.
Proof. If f is locally linearizable and |λ| = 1, we have a neighborhood of z0 which is mapped
to itself, so the iterates of f form a normal family on this neighborhood. On the other hand,
if z0 is Fatou, then by the classification theorem the Fatou component containing z0 is
necessarily of the fourth kind, and a disk, so we have a local linearization. �
Definition 8.1. We say an irrationally indifferent fixed point of f is Siegel (resp. Cremer)
if is is locally linearizable (resp. not locally linearizable). That is, Siegel points are Fatou,
and Cremer points are Julia.
Definition 8.2. A Fatou component containing a Siegel periodic point is a Siegel disk.
Note that f is conjugate to an irrational rotation on the full component.
Linearizability is in large part a condition related to the Diophantine approximability
properties of λ.
Definition 8.3. Let θ be an irrational number. We say that θ is Diophantine to order κ if
there exists ε > 0 so that for all rational numbers p/q we have∣∣∣∣θ − p
q
∣∣∣∣ < ε
qκ.
Theorem 8.2 (Liouville). Suppose θ is algebraic of degree d. Then θ is Diophantine to
order d.
Proof. Let f(x) = adzd+· · ·+a0 be a polynomial of degree d with integral coefficients so that
f(θ) = 0. We may factor out linear factors to assume that f is non-zero on rational numbers.
Let M = maxx∈(θ−1,θ+1) |f ′(x)|. By the mean value theorem, we have for |θ− p/q| < 1 that
|f(p/q)| = |f(p/q)− f(θ)| ≤M |θ − p/q|.
On the other hand, since the coefficients of f are integral, we have |f(p/q)| ≥ 1/qd. Thus
we have for ε < 1/M the necessary inequality. �
Theorem 8.3 (Siegel). Suppose θ is Diophantine of order κ < ∞, and suppose f has a
fixed point at 0 with multiplier λ = e2πiθ. Then f is locally linearizable at 0.
This theorem is a hassle to prove and we will prove a slightly different result, following
Milnor.
42 HOLLY KRIEGER
We note first a simple inequality connecting the approximation of 1 by the multiplier
with the approximation by rational numbers of its argument. Given n ∈ N, let m be the
integer minimizing |nθ −m|. We have
|λn − 1| =√
2− 2 cos(2πnθ) =
√4 sin2(πnθ) = 2| sin(πnθ −mπ)|,
via the trig identity 1− cos 2θ = 2 sin2 θ. As |nθ−m| < 1/2, we can approximate the latter
by π|nθ −m|, so there are constants c1, c2 so that
c1|nθ −m| ≤ |λn − 1| ≤ c2|nθ −m|.
COMPLEX DYNAMICS 43
***Lecture 13***
Theorem 8.4 (Cremer). Suppose that f(z) is a rational map of degree d ≥ 2 with f(0) = 0
and the multiplier λ = e2πiθ of f at 0 satisfying the following: there exist infinitely many
pairs of coprime integers (p, q) = 1 so that∣∣∣∣θ − p
q
∣∣∣∣ ≤ 1
qdq.
Then 0 is a Cremer fixed point of f .
Proof. Note first that the condition on θ implies that for some constant C and infinitely
many q ∈ N,
|λq − 1| ≤ C|qθ − p| ≤ C
qdq−1,
so that
lim inf |λq − 1|1/(dq−1) = 0.
First we prove the case when f is a monic polynomial,
f(z) = λz + · · ·+ zd,
so that
fn(z) = λnz + · · · zdn
.
For any n ∈ N, the points of period dividing n are roots of the polynomial
(λn − 1)z + · · ·+ zdn
= fn(z)− z.
Therefore the product of the dn − 1 non-zero points of period dividing n is ±λn − 1. Thus
at least one of these points must satisfy
|z|dn−1 ≤ |λn − 1|.
Choosing n among the infinite set of q ∈ N satisfying the inequality of the hypothesis, we
see that there is a sequence of periodic points of f converging to 0, all but finitely many of
which must be repelling. So 0 ∈ J(f), and f cannot be locally linearizable there.
For the general rational map case, since 0 is not critical, we can find a preimage z1 6= 0 of
0 under f . Conjugating by a Mobius transformation, assume that f(∞) = f(0) = 0. Write
f as a quotient of polynomials, we must then have (up to a conjugation by scaling) that
f(z) =ad−1z
d−1 + · · ·+ λz
zd + · · ·+ 1,
so that fn(z) has the form
fn(z) =bnz
dn−1 + · · ·+ λnz
zdn + · · ·+ 1,
so that the period n points equation has the form
0 = z(zdn
+ · · ·+ 1)− bnzdn−1 + · · ·+ λnz = z(zd
n
+ · · ·+ (1− λn)).
We thus have the same conclusion on the size of some root as in the polynomial case, and
conclude as above that 0 ∈ J(f) and so f is not locally linearizable there. �
44 HOLLY KRIEGER
However, Siegel maps not only exist, but are in a Lebesgue sense generic.
Proposition 8.1. Let fλ(z) := λz + z2. Then for (Lebesgue) almost every λ ∈ S1, 0 is a
Siegel point of f .
Proof. For each λ ∈ D, let σ(λ) be the radius of the largest disk on which there is a linearizing
map ψλ with
ψλ(λw) = fλ(ψλ(w)),
taking σ(λ) = 0 if no such disk exists. The function λ 7→ σ(λ) cannot be continuous if Siegel
maps exist in this family (since σ takes the value 0 at any parabolic or Cremer point), but
it is upper semicontinuous. Recall:
Definition 8.4. A real valued function σ on a metric space X is upper semicontinuous if
for all σ0 ∈ R,
{x ∈ X : σ(x) ≥ σ0}is closed; equivalently, if
lim supx→x0,x 6=x0
σ(x) ≤ σ(x0)
holds for all x0 ∈ X.
Note that σ(λ) is uniformly bounded (by 2) for all λ ∈ D, since |z| > 2⇒ |z+λz2| > |z|,so z lies in the basin of infinity and cannot be in the linearization image. Fix σ0, and
consider a convergent sequence λk in
{x : σ(x) ≥ σ0}.
The set of all holomorphic functions ψ : D(0, σ0) → D(0, 2) forms a normal family, so we
may pass to a convergent subsequence of the λk on the same neighborhood. This limit must
be a linearizing map on D(0, σ0) for so the limit λ of the λk is in {x : σ(x) ≥ σ0} as well,
and so the set is closed and σ upper semicontinuous.
Recall that for λ ∈ D \ {0}, the Koenig map is defined in the attracting basin of 0 by
φλ(z) = limn→∞
fnλ (z)
λn,
and we showed that the limit converges locally uniformly. Thus φλ(z) is a holomorphic
function of λ as well, and since the critical point −λ/2 is in the attracting basin,
η(λ) := φλ(−λ/2)
is a holomorphic function of λ ∈ D \ {0}. In fact, as asserted in class (proof ES1), |η(λ)| is
precisely the value σ(λ) for all λ 6= 0. Since σ(0) = 0, upper semicontinuity gives η(λ)→ 0
as λ→ 0 and so η is holomorphic throughout D.
Consider now for a fixed λ0 ∈ S1 the limit
limλ→λ0,|λ|<1
η(λ).
By upper semicontinuity at λ0, this limit is defined and equal to 0 if and only if σ(λ0) = 0.
COMPLEX DYNAMICS 45
To complete the proof, I claim that the set E of λ0 such that this limit exists and equals
0 has Lebesgue measure 0 on the unit circle.
Lemma 8.1. Suppose that f : D→ C is bounded and holomorphic, and non-zero on D\{0}.Then the radial limit
limr→1
f(reiθ)
exists and takes the value 0 on a set of Lebesgue measure 0.
Proof. Scale f WLOG to assume f(D) ⊂ D. Let E be the set of angles θ so that the limit
exists and is 0, and suppose the Lebesgue measure L(E) is positive. Fix δ, ε > 0 and let
E(ε, δ) be the set of all angles λ ∈ S1 such that
|f(rλ)| < ε whenever 1− δ < r < 1.
These form a family of Lebesgue-measureable sets, so for each fixed ε, their Lebesgue mea-
sures satisfy
L(E(ε, δ))→ L(E) as δ → 0.
Suppose towards contradiction that L(E) > 0, so then we may find δ > 0 so that L(E(ε, δ)) >
L(E)/2.
Now, suppose the zero of f at 0 has order k. We then have f(z)/zk holomorphic and
non-zero on the disk, so that there exists a holomorphic logarithm g(z) = log(f(z)/zk) on D.
Therefore log |g(z)| is harmonic, so satisfies the mean-value property for harmonic functions
log |g(0)| = 1
2π
∫ 2π
0
log |g(reiθ)| dθ
for any 0 < r < 1. Therefore this integral is independent of choice of r. Since
1
2π
∫ 2π
0
log |reiθ|k dθ = k log r,
we have
1
2π
∫ 2π
0
log |f(reiθ)| dθ = k log r + C
for some constant C; in particular, the limit as r → 1 of this integral exists and is finite.
On the other hand, we have by estimates that
1
2π
∫ 2π
0
log |f(reiθ)| dθ < L(E) log ε
2→ −∞,
a contradiction. �
Thus almost all points are Siegel, as claimed.
�
46 HOLLY KRIEGER
Note nonetheless that there are plenty of Cremer points: one can show directly that
the set of λ satisfying the Cremer restriction is both uncountable and dense in S1, using
Cremer’s result. The state of the art (Petracovici, Perez-Marco, Yoccoz, Brjuno) describes
in reasonable detail when an indifferent fixed point is linearizable in terms of the continued
fraction expansion of the angle, but it is still not completely classified.
COMPLEX DYNAMICS 47
****Lecture 14****
Recall that any neighborhood N of an element z ∈ J(f) has the property that
J(f) ⊂ ∪∞n=1fn(N);
by compactness of J(f), in fact a finite subset will do. This allows us to control both Siegel
and Cremer points in terms of the critical points of f .
Definition 8.5. The (forward) orbit of a point z0 under f is the sequence
{fn(z0)}n∈N.
A critical orbit is the forward orbit of any critical point of f .
Before we can describe how critical points interact with Cremer points and Siegel disks,
we need to refine our information about repelling points in the Julia set.
Theorem 8.5. Repelling cycles are dense in J(f); that is, for any rational map f of degree
d ≥ 2, the Julia set is the topological closure of the repelling periodic points.
Proof. Recall that J(f) has no isolated points, so WLOG we may exclude any finite number
of points. Let z0 be any point of J(f) which is not a critical value and not a fixed point, so
that z0 has d distinct preimages z1, . . . , zd. On any sufficiently small neighborhood N of z0
we can then find holomorphic inverses φ1, . . . , φd of f so that φi(z0) = zi. Suppose that for
all z ∈ N and all n ∈ N, fn(z) 6∈ {z, φ1(z), φ2(z)}, and consider for each n ∈ N the function
gn(z) =(fn(z)− φ1(z))(z − φ2(z))
(fn(z)− φ2(z)(z − φ1(z)).
Then gn : N → C \ {0, 1,∞}, so the sequence {gn(z)} forms a normal family on N . Thus
fn(z) does as well, contradicting the hypothesis that N intersects the Julia set. So for some
z ∈ N and some n ∈ N, either fn(z) = z or fn(z) = φj(z) with j ∈ {1, 2}. So z is a point
of period n or n+ 1. Since all but finitely many periodic points are repelling, shrinking N
we see that the repelling periodic points are dense in J(f) as claimed. �
Corollary 8.2. Let U be an open neighborhood intersecting J(f). Then for all n sufficiently
large, J(f) ⊂ fn(U).
Proof. Since repelling cycles are dense in J(f), we may choose z0 ∈ U repelling of period
k, and via the linearizing coordinate a neighborhood z0 ∈ V ⊂ U so that U ⊂ fk(U). Then
V ⊂ fk(V ) ⊂ f2k(V ) ⊂ ... and the union of these iterates cover J(f), so by compactness of
J(f), fnk(V ) contains J(f) for n sufficiently large. Since J(f) is f -invariant, the same is
therefore true for all fm(V ) with m > nk, proving the corollary. �
Proposition 8.2. Suppose z0 is a Cremer point of f , or a boundary point of a Siegel disk.
Then every neighborhood of z0 contains infinitely many distinct critical orbit points.
Proof. First we suppose that z0 is a Cremer point; WLOG, z0 = 0 is fixed by f . Suppose
z0 ∈ D is an open disk with no critical orbit elements, so we may define a branch gn of
f−n on D which fixes 0, for all n ∈ N. Then gn(D) contains no preimage of a critical
point; thus is contained in a hyperbolic domain, unless f has two totally ramified critical
48 HOLLY KRIEGER
points. In that case, we have seen that f is conjugate to z 7→ z±d, so f2(z) = zd2
and f
has only repelling periodic points, a contradiction. So all gn map into a fixed hyperbolic
domain, and thus {gn} forms a normal family. Let g be a limit of a subsequence, noting that
g(0) = 0. As (fn)′(0) ∈ S1 for all n ∈ N, we may shrink D to assume g is a conformal iso-
morphism onto its image. Let D′ be a strictly smaller subdisk of D, and U ′ := g(D′). Then
infinitely many iterates of U ′ map into D. On the other hand, U ′ is a neighborhood which
intersects the Julia set, so by the preceding corollary, eventually covers J(f), a contradiction.
Now suppose that z0 lies on the boundary of a Siegel disk ∆, with f(∆) = ∆. By the
same construction, if there is a neighborhood D with no critical orbit points, we may find
(this time choosing the branch which maps D ∩∆ into ∆) a normal family gn of inverses of
iterates of f , which again are therefore normal and converge to a holomorphic limit map.
By the same argument, any subdisk D′ must have g(D′) contained in the Fatou set, as
infinitely many iterates of f map g(D′) into D. This is impossible as g(D′) contains Siegel
boundary (hence Julia) points. �
COMPLEX DYNAMICS 49
***Lecture 15***
9. Quasiconformal surgery and Herman rings
We are close to a complete classification of periodic Fatou components, via the classifica-
tion theorem of hyperbolic maps. That classification permits three types of behavior for an
irrationally rotated component, but in fact, the punctured disk can never happen (for the
puncture must itself also be Fatou). We have seen the disk via Siegel disks. What about
the annulus?
Definition 9.1. A Fatou component U of f is a Herman ring if U is conformally isomorphic
to an annulus, under which isomorphism f corresponds to an irrational rotation.
Note that by the maximum-modulus principle, no polynomial admits a Herman ring
component. We will spend the next few days learning about quasiconformal maps, qua-
siconformal surgery, and proving that Hermann rings exist. Idea: glue analytic functions
with Siegel disks along invariant curves in the disks. Obstruction: the identity principle.
Solution: quasiconformal surgery.
Now is the right time for some conformal basics of annuli. Let AR denote the standard
annulus of modulus R > 1:
AR := {z ∈ C : 1 < |z| < R}.
Lemma 9.1. If AR is conformally isomorphic to AS, then R = S.
Proof. Suppose f : AR → AS is a conformal isomorphism. Postcomposing with z 7→ S/z if
necessary, assume that lim zn ∈ S1 ⇒ limφ(zn) ∈ S1. Let γt denote the positively oriented
circle of radius t about 0, for 1 < t < R, and denote by Bt the region interior to f(γt).
Green’s formula on the plane tells us that if f(z) = u(x, y) + iv(x, y), we have∫∂Bt
f(z) dz =
∫∂Bt
(u+ iv) dx+ (−v + iu) dy =
∫ ∫−vx + iux − uy − ivy dA;
in particular,
Area(Bt) =1
2i
∫∂Bt
z dz =1
2i
∫|z|=t
fdf.
Let f =∑anz
n be the Laurent series for f on AR. Then for z = teiθ we have
df = f ′(z)dz = i∑n∈Z
nan(teiθ)n,
so1
2i
∫|z|=t
fdf =1
2i
∫ 2π
0
f(teiθ)f(teiθ)dθ = π∑n∈Z
n|an|2t2n.
Taking t→ 1, we see that ∑n∈Z
n|an|2 = 1,
50 HOLLY KRIEGER
so that
Area(Bt)− πt2 = πt2∑n∈Z
n|an|2(t2n−2 − 1).
As each term on the right is non-negative, so is the left-hand side, and taking t → R, we
see that the area of f(AR) = AS is at least πR2 − π, so that S ≥ R. Applying the same
argument to the inverse map, S ≤ R and thus they are equal. �
Definition 9.2. Suppose U is a domain conformally isomorphic to AR for some R > 1.
The modulus of the topological annulus U is
Mod(U) =1
2πlogR.
Definition 9.3. An orientation-preserving homeomorphism f : R→ S of Riemann surfaces
is K-quasiconformal if for all conformal annuli A ⊂ R,
1
KMod(A) ≤ Mod(f(A)) ≤ KMod(A).
We will need a more hands-on definition in the special case of a continuously differen-
tiable homeomorphism. We will not prove the equivalence of these definitions (see Branner-
Fagella), but every quasiconformal map we actually construct will have continuous deriva-
tives.
Definition 9.4. Suppose that U, V are domains in C, and that f : U → V is orientation-
preserving C1 homeomorphism. The complex dilatation of f at z0 ∈ U is
µf (z0) :=fzfz,
defined almost everywhere. f is K-quasiconformal for K ≥ 1 if
|µf (z0)| ≤ K − 1
K + 1
for (Lebesgue) almost every z0 ∈ U . f is quasiconformal if it is K-quasiconformal for some
finite K.
Recall that we have
fz =∂f
∂z=
1
2
(∂f
∂x− i∂f
∂y
)and
fz =∂f
∂z=
1
2
(∂f
∂x+ i
∂f
∂y
).
A more natural interpretation of quasiconformality - and the dilatation - occurs in the
tangent space. Given f which is real-differentiable at z0 ∈ C, we obtain the induced deriv-
ative map
Df(z0) : C→ Cwhich sends z 7→ fz(z0)z+fz(z0)z, a R-linear map. This linear map sends ellipses to ellipses,
and K(z0) := (1 + |µ(z0)|)/(1 − |µ(z0)|) is simply the ratio of major to minor axis for the
ellipse pre-images of circles centered at 0 in C, viewed as the tangent space of R at z0. Note
that f orientation-preserving implies that the dilatation is in D wherever it is defined, as
COMPLEX DYNAMICS 51
orientation is preserved when detDf = |fz|2 − |fz|2 > 0.
We will use some basic facts about quasiconformality:
• f is conformal ⇔ f is 1-quasiconformal ⇔ f is quasiconformal and fz exists and is
0 almost everywhere (the interesting implication here is Weyl’s lemma).
• If a homeomorphism f has continuous derivatives, then f is quasiconformal on any
compact set.
• K-quasiconformality is local.
• (subtle!) Suppose that γ ⊂ U is a quasiconformal image of a line (key example: any
conformal circle). Then a homeomorphism f : U → V is K-quasiconformal on U if
and only if f is K-quasiconformal on U \ γ.
The key to the utility of quasiconformality in holomorphic dynamics is that last item
above, which allows us to paste together quasiconformal maps along nice smooth curves,
and that they are an immensely flexible class of maps. In fact, the (seemingly much larger)
class of dilatation functions are all achieved by quasiconformal maps.
Theorem 9.1 (Measurable Riemann Mapping Theorem). Let µ be a bounded, measurable
function on C satisfying ||µ||∞ < 1. Then there is a unique quasiconformal homeomorphism
φ : C→ C fixing the points 0, 1,∞, such that µφ = µ almost everywhere.
In fact, if µ has dilatation bounded by κ ≤ 1, then we may construct a family of quasi-
conformal maps φt : C→ C for t ∈ D(0, 1/κ) so that
µφt= tµ,
with φt normalized as above, and with φt(z) a holomorphic function of t for each z ∈ C.
52 HOLLY KRIEGER
***Lecture 16***
An example MRMT application that we will make use of is a key lemma of Shishikura [Sh],
which is a nice packaging of the quasiconformal surgery process of Douady and Hubbard.
First, a definition:
Definition 9.5. Let U be a domain in C. g : U → C is quasiregular if g = f ◦φ, where f is
analytic and φ quasiconformal. Note then g is locally quasiconformal at all but finitely many
points (the φ-preimages of the critical points of f); in fact, that statement is an equivalent
definition of quasiregularity.
Write f∗µ for the pullback dilatation µ(f(z)).
Proposition 9.1. Let f : C → C be a quasiregular map, and φ a quasiconformal home-
omorphism on C with complex dilatation µφ. Suppose that µφ is f -invariant; that is,
µφ(f(z)) = µφ(z) almost everywhere. Then g = φ ◦ f ◦ φ−1 is a rational map.
Proof. Chain rule, or proof by picture, shows that Dg sends circles to circles away from its
critical points:
So g is locally 1-quasiconformal i.e. holomorphic away from the critical points. These
points are removable singularities, so g is rational. �
Lemma 9.2 (Fundamental lemma of quasiconformal surgery, following Shishikura). Let
g : C→ C be quasiregular. Suppose for 1 ≤ i ≤ m there are disjoint open sets Ei of C and
quasiconformal mappings Φi : Ei → C, and an integer N ≥ 0 so that the following hold:
• g(E) ⊂ E where E := E1 ∪ · · · ∪ Em,
• Φ ◦ g ◦ Φ−1i is analytic on Φi(Ei), where Φ is defined to restrict to Φi on each Ei,
• µg = 0 almost everywhere on C \ g−N (E).
Then there exists a quasiconformal homeomorphism φ of C such that φ◦g ◦φ−1 is a rational
function.
Proof. Let µΦ denote the complex dilatation of Φ. We will use µΦ to construct a new dilata-
tion function µ which is g invariant as follows: define µ := µΦ on the set E, and for all n ∈ N,
recursively define µ on g−n(E) so that µ(z) = µ(g(z)). Finally, on C\∪g−n(E), define µ ≡ 0.
COMPLEX DYNAMICS 53
By the definition of µ and the second assertion, we have g∗µ = µ. Note that by the first
assertion, we have
E ⊂ g−1(E) ⊂ g−2(E) ⊂ · · · ⊂ g−N (E) ⊂ · · · ⊂⋃n
g−n(E),
and to prove µ is bounded it suffices to show that µ is bounded on g−N (E), since after that
point the pullback doesn’t change the bound on the dilatation, by the third assertion. Since
g and Φ are quasiregular, µ is uniformly bounded by some (K − 1)/(K + 1) on g−N (E)
(exercise (ES3): quasiconformality is multiplicative for compositions, and K = KΦKNg ).
Thus by the MRMT, there is a K-quasiconformal map φ of C so that µ = µφ almost
everywhere. By the proposition, the conjugate f := φ ◦ g ◦ φ−1 is a rational function.
�
Let’s explain the point. What we are doing here is gluing together g, restricted to C \E(where it is basically analytic), to Φ ◦ g ◦ Φ−1
i on E (which really is analytic), pulling back
the latter to its full backwards orbit. We can’t just hack the sphere up and do this and
expect a rational map (identity principle!), but Shishikura is asserting that the measurable
Riemann mapping theorem basically shows that it is enough to cook up a conjugate with
trivial dilatation to obtain a rational map which does behave this way (I am hand-waving
here, ignoring the critical points).
So! Assuming this lemma, let’s sketch Shishikura’s construction of a Herman ring now,
in a special case. Let f(z) = λz + z2, with 0 a Siegel point of multiplier λ = e2πiθ, and
call the Siegel disk of f Sf . Let g(z) = f(z) = λz + z2 be the rational map which, note,
has a fixed Siegel point at 0 of multiplier λ2 = e−2πiθ. Choose a circle C about 0 of radius
0 < r < 1 in D, and write γf and γg respectively for the corresponding f (resp. g) invariant
curve in the Siegel disk Sf (resp. Sg).
We are going to construct a quasiconformal map Ψ : C→ C so that:
• Ψ(γf ) = γg• Ψ ◦ f = g ◦Ψ on γf• Ψ is conformal outside a neighborhood of Sf ∩Ψ−1(Sg)
then use this Ψ to construct our rational map which acts as f outside of γf and as g inside.
54 HOLLY KRIEGER
Figure 14. Shishikura’s construction, illustrated by Dominguez and Fagella
Write Bf for the component of C \ γf contained in Sf , and define similarly Bg. Since
λz = λz, we have via the linearization coordinates and complex conjugation a homeomor-
phism ψ : γf → γg such that ψ ◦ f = g ◦ ψ. Each component of C \ γf is a conformal disk,
so we may find a conformal isomorphism Ψ′ : C \ γf → C \ γg sending Bf to C \ Bg and
vice versa. We may then on an annular neighborhood of γf modify and extend this to a
quasiconformal homeomorphism Ψ : C→ C which agrees with ψ on γf . We are using here
a little interpolation lemma: that smooth, orientation-preserving homemorphisms between
the boundary curves of annuli may be linearly interpolated to a quasiconformal (in fact,
C1) homemorphism between the annuli - see ES3!
Define then h : C→ C to agree with f on C \Bf , and to take the values
h(z) = Ψ−1 ◦ g ◦Ψ for all z ∈ Bf .
Then h is quasiregular, and the sets
E0 = C \Bfand
E1 = B1 ∩Ψ−1(Sg)
together with the maps Φ0 := id and Φ1 := Ψ satisfy the hypotheses of the fundamental
lemma, with N = 1. So there exists a quasiconformal mapping φ so that H := φ ◦ h ◦ φ−1
is a rational map. Write A := φ(Sf ∩ Ψ−1(Sg)). Then the dynamics of H on the annular
neighborhood A is conjugate to an irrational rotation. On the other hand, A cannot belong
to a disk or punctured disk component, as the interior of the annulus contains infinitely
many periodic points for H (namely, images of periodic points for g) so infinitely many
points of J(H). So A is necessarily a Herman ring.
COMPLEX DYNAMICS 55
***Lecture 17***
In the Herman ring construction of the preceding lecture, we can track exactly what
happens to the critical points of f and g; for each boundary curve of the Herman ring, there
must be some critical orbit accumulating. The images of infinity and 0 remain critical as
well. So we have 4 critical points for the new map H, which is therefore of degree 3.
There are other known methods of constructing Herman rings (they often arise, for ex-
ample, in Blaschke products), but Shishikura’s process is reversible, so they all arise from
a gluing of Siegel disks. Note that this construction absolutely kills any misconception you
might’ve had connecting the existence of a Herman ring to any particular type of periodic
point behavior. Nonetheless, Herman rings boundaries interact with critical orbits in the
same way as Siegel disk boundaries.
Proposition 9.2. If U is a Herman ring of f , then every boundary point of U belongs to
the closure of the orbit of some critical point of f . This boundary consists of two connected,
infinite components.
Proof. Just like the Siegel case - exercise (ES3). �
10. The classification of Fatou components
We nearly complete our global study with a fundamental statement on rational maps,
whose proof is part of a Part III essay if you are interested.
Theorem 10.1 (Sullivan, No Wandering Domains). Let f be a rational map of degree d ≥ 2.
Then every Fatou component U of f is preperiodic; that is, there exist integers m,n so that
fn+m(U) = fm(U). In particular, every Fatou component is either a branch cover or a
conformal copy of some periodic Fatou component, according to whether U, f(U), . . . fm(U)
contains a critical point of f or not.
Sketch. The proof is a quasiconformal one. For any Fatou component U , we can
construct for any dilatation function on U some map which preserves that dilatation. If
U wanders, we can use f to spread that function through the entire forward orbit of U
to obtain a dilatation which is f -invariant on the grand orbit of U , and 0 elsewhere, so
f -invariant everywhere. Thus MRMT gives us a quasiconformal conjugacy to some rational
map, which preserves this new dilatation function. Doing this for each choice of dilatation,
we obtain an infinite dimensional family of degree d rational maps, but the family clearly
has finite dimension 2d+ 1, a contradiction.
Theorem 10.2. Let U be a Fatou component of f , where f has degree d ≥ 2 with f(U) = U.
Then U is one of the following:
• the immediate basin of an attracting fixed point
• the basin of a petal of a parabolic fixed point
• a Siegel disk
• a Herman ring
We will prove this via the following lemma:
56 HOLLY KRIEGER
Lemma 10.1 (Snail Lemma). Suppose that f(z) = λz + a2z2 + · · · with |λ| = 1 but λ 6= 1,
and that there exists a domain V so that f(V ) ∩ V 6= ∅. Then fm(V ) cannot converge
uniformly to 0.
Assuming the lemma, here is the proof.
Proof of classification of fixed Fatou components. By the classification of self-maps of hy-
perbolic surfaces, there are only four possibilities for f : U → U :
• U contains an attracting fixed point
• all orbits of U escape to the boundary
• f is an automorphism of finite order
• f is conjugate to an irrational rotation of a disk, punctured disk, or annulus.
Clearly the third case cannot happen; nor can the punctured disk case, since the puncture
would be necessarily Fatou. So it remains to show that in the second case, all orbits are
attracted to a parabolic fixed point.
First we will show that all orbits are attracted to a fixed point. Given z0 ∈ U , choose a
path γ in U from z0 to z1 := f(z0), and continue this path inductively with γ(t+1) = f(γ(t)).
By hypothesis, the orbit of z0 converges to infinity with respect to the hyperbolic metric on
U , so tends to the boundary of U (with respect to the spherical metric). Thus the spherical
diameter of the sets γ([n, n+ 1]) must tend to 0 as n→∞. Since f(γ(t)) = γ(t+ 1), every
accumulation point of this path is a fixed point for f . By Schwarz-Pick, such an accumula-
tion point is unique, and all orbits in U converge to it.
Since this boundary fixed point is Julia, it cannot be attracting or Siegel; since it attracts
all points of U , it cannot be repelling. Thus it is either parabolic or Cremer, and we need
to show it is not Cremer. To do so, we will show that a Cremer point cannot be a limit of
an f -invariant domain.
To complete the proof of the theorem, note that the domain U satisfies f(U) ∩ U 6= ∅,so we cannot by the snail lemma have escape to a fixed boundary point unless that point
has multiplier 1. It follows that U is the component of a parabolic basin of attraction,
completing the proof. �
COMPLEX DYNAMICS 57
***Lecture 18***
To prove the lemma, we first need a classical distortion estimate, known as the Koebe
one quarter theorem.
Theorem 10.3. Suppose f : D→ C is injective, holomorphic, fixes 0, and satisfies f ′(0) =
1. Then D(0, 1/4) ⊂ D.
Proof. Recall our previous use of Green’s theorem to estimate the area of conformal images
of annuli. A similar computation yields that if g(z) = 1/z + b0 + b1z + ... is injective in
D, then∑n|bn|2 ≤ 1, for given any 0 < r < 1, the area of the complement of the image
Dr := C \ g(D(0, r) is computed to be
−1
2i
∫∂D(0,r)
g dg = π
(1
r2−∑
n|bn|2r2n
).
As this is non-negative, taking r → 1 gives the claimed inequality.
Consequently, if
f(z) = z +
∞∑n=2
anzn
satisfies the hypotheses of the theorem, then
g(z) = 1/√f(z2) = 1/z − a2z/2 + . . .
satisfies the area estimate; in particular, |b1|2 = |a2|2/4 ≤ 1 i.e. |a2| ≤ 2.
Finally, fix a point c ∈ C, and suppose c 6∈ f(D). Then consider the function
gc(z) =cf(z)
c− f(z)= z +
(a2 +
1
c
)z2 + · · · .
This function is an injective holomorphic map of the unit disk fixing 0 with derivative 1 at
0, so ∣∣∣∣1c∣∣∣∣ ≤ | − a2|+
∣∣∣∣a2 +1
c
∣∣∣∣ ≤ 2 + 2 = 4,
completing the proof. �
Remark. Note that this bound is optimal, as shown by the function f(z) = z(1−z)2 .
Lemma 10.2 (Snail Lemma). Suppose that f(z) = λz + a2z2 + · · · with |λ| = 1 but λ 6= 1,
and that there exists a domain V so that f(V ) ∩ V 6= ∅. Then fm(V ) cannot converge
uniformly to 0.
Proof. Suppose not. By the parabolic petal theorem, λ must be an irrational rotation. We
may iterate V so that f is injective on V , and restricting to a neighborhood of a simple path
in V between p, f(p) ∈ V , assume that V is simply connected. Fix 0 6= ζ ∈ V . Consider the
sequence
φn(z) :=fn(z)
fn(ζ)
58 HOLLY KRIEGER
of holomorphic functions on V . Let σ1 be an analytic isomorphism from D to V which sends
0 to ζ, and let σ2(z) = z − 1. The derivative of σ2 ◦ φn ◦ σ1 at 0 is φ′n(ζ)σ′1(0). Define
gn(z) :=σ2 ◦ φn ◦ σ1(z)
φ′n(ζ)σ′1(0).
Then gn never takes the value −1/φ′n(ζ)σ′1(0), but gn : D → C fixes 0, with g′n(0) = 1, so
by Koebe’s theorem, gn takes all values in the disk of radius 1/4 about 0. Thus we have
4 > |φ′n(ζ)σ′1(0)|,
and the gn and the φn form a normal family on V .
Suppose there is a subsequence nk so that φ′nk(ζ)→ 0, and extract a limit function φ by
passing to another subsequence. As local degree is computed by an integral for non-constant
functions, all φn injective implies either φ is injective or constant. Since φ(ζ) = 1, if φ is
constant, it is the constant value 1, and we have |φn(z)− 1| < δ for n sufficiently large and
all z ∈ V . As fn(V ) → 0, for n sufficiently large we would have fn+1(V ) ≈ λfn(V ), so a
contradiction to fn+1(V )∩ fn(V ) 6= ∅. So φ is nonconstant, and so φ′(ζ) 6= 0, contradicting
φ′nk(ζ) → 0. Thus φ′n(ζ) is uniformly bounded away from 0 for n sufficiently large, say by
ρ′. Considering again Koebe’s applied to the maps gn, we that φn(V ) contains a disk of
radius 14φ′n(ζ)σ′1(0) about 1, so for ρ = ρ′σ′1(0)/4 > 0 we see that fn(V ) contains a disk of
radius ρ about 1 for all n sufficiently large. Hence fn contains a disk of radius centered at
fn(ζ) of radius ρ|fn(ζ)|.
Since θ is irrational, we may choose N sufficiently large so that an annulus containing S1
is covered by the disks of radius ρ/2 centered at e2πimθ for 0 ≤ m ≤ N , where λ = e2πiθ.
Then for n sufficiently large, the disks of radii ρ|fm(ζ)| centered at fm(ζ) for n ≤ m ≤ n+N
cover an open annulus, for we may on a small neighborhood of 0 find a constant C so that
|f j(z)− λjz| ≤ C|z|2
holds, and for n sufficiently large, the orbit of fn(ζ) will lie in this neighborhood and satisfy
C|fn(ζ)|2 < ρ|fn(ζ)|/2. Thus U := ∪fn(V ) contains a punctured neighborhood of 0 by the
maximum-modulus principle. By the classification theorem, we conclude that 0 is a Siegel
point, f is conjugate to an irrational rotation on U , contradicting fn(V )→ 0.
�
This concludes our ‘classical’ complex dynamical study! From now on, we will be working
towards universality.
COMPLEX DYNAMICS 59
***Lecture 19***
11. MRMT and Straightening
Definition 11.1. Suppose that U ⊂ V are simply connected domains in C with U ⊂ V and
each with analytic Jordan curve boundaries. A holomorphic map f : U → V is polynomial-
like if it is proper. In this case, we define the filled Julia set of f to be
K(f) :=⋂n≥1
f−n(U),
the set of points in U which have well-defined forward orbits.
Example. Certainly any polynomial f : C → C is polynomial-like, and the notion of
filled Julia set is the usual one. The same is true if we restrict f to f−1(V ) for any suffi-
ciently large disk V .
Example. Consider the restriction of the cubic polynomial fε(z) = εz3 + z2 for small ε.
The critical points are 0,−2/3ε, the latter of which is in the basin of infinity, since
−2
3ε7→ 4
27ε27→≈ 1
ε57→≈ 1
ε147→ · · · .
Let γ be the lemniscate through −2/3ε which is a level curve for the Green’s function. Then
if U is the component of C \ γ containing 0, f : U → f(U) is polynomial-like of degree 2.
Figure 15. Escaping critical orbit level curves for f(z) = z3/8 + z2.
Example. The Newton’s example we saw previously has polynomial-like behavior on a
neighborhood of 0. Let c = 0.333i, and
Nc(z) := z − (z − 1)(z − c+ 1/2)(z + c+ 1/2)
(z − 1)(z − c+ 1/2) + (z − 1)(z + c+ 1/2) + (z − c+ 1/2)(z + c+ 1/2).
60 HOLLY KRIEGER
Figure 16. The yellow circles approximate first preimages of the white
circle, the red circles the second preimages of the white. The action of f is
indicated by the white arrows.
In this example, call the region interior to the red circle U , and to the white circle V .
Then U ⊂ V , and the second iterate of Nc is a polynomial-like map from U to V . In
this case, the filled Julia set of N2c on U is the (N2
a -forward invariant) closure of the Fatou
component containing 0.
Our goal now is to show that when we see polynomial-like behavior, we see copies of filled
Julia sets that look like those in polynomials.
Definition 11.2. A topological space A is locally connected if for all a ∈ A and all open
sets a ∈ V ⊂ A, there exists a connected, open set with a ∈ U ⊂ V .
COMPLEX DYNAMICS 61
***Lecture 20***
Example: The circle is locally connected, and the continuous image in C of a compact lo-
cally connected set is locally connected. In particular, any Jordan curve is locally connected.
Non-example: the example for connected not path-connected works here:
{(x, sin(1/x)) : x ∈ (0, 1]} ∪ {(0, 0)}.
Theorem 11.1 (Caratheodory + continuity theorem). Suppose f : D → U is conformal,
where U is a bounded domain in C. Then TFAE:
• f extends continuously to D• ∂U is the continuous image of a circle
• ∂U is locally connected
• C \ U is locally connected
If ∂U is an analytic curve, then f extends to an open neighborhood of D.
The theorem is based on using conformal estimates to prove uniform continuity in the
disk, and the final statement a Schwarz reflection argument. We will not prove it here: see
e.g. Pommeranke.
Definition 11.3. Two polynomial-like maps f : U → V and g : U ′ → V ′ are hybrid equiv-
alent if there exist neighborhoods W,W ′ of the filled Julia sets K(f) and K(g) respectively
and a quasiconformal map φ : W →W ′ satisfying g = φ ◦ f ◦ φ−1 with ∂φ = 0 on K(f).
By definition, a hybrid conjugacy satisfies φ(K(f)) = K(g) and φ(∂K(f)) = ∂K(g).
Note that, as the conjugacy is a conformal map on the interior of K(f), multipliers are
preserved under hybrid conjugacy for periodic Fatou points (attracting, superattracting,
Siegel).
Theorem 11.2 (Straightening, Douady-Hubbard [DoHu]). Suppose that f : U → V is a
polynomial-like map of degree d ≥ 2. Then there exists a quasiconformal map φ : C → Cwith µφ = 0 on K(f) so that φ−1 ◦ f ◦ φ is a polynomial.
Proof. Choose ρ > 1, and let R : C \ V → C \ D(0, ρd) be a uniformizing map which
fixes ∞. By Caratheodory++, R extends continuously to the boundaries. Choose a lift
ψ : ∂U → C(0, ρ) so that R(f(z)) = ψ(z)d for all z ∈ ∂U .
Let A0 and A(ρ) denote the annuli V \ U and {ρ < |z| < ρd} respectively. As shown in
ES3, linear interpolation (applied to the inverse maps) provides an extension φ : A0 → A(ρ)
of ψ,R which is C1, so quasiconformal. Define the map F : C → C by gluing the map
z 7→ zd into V \ U . More precisely, let
F (x) =
f(z) z ∈ UR−1(φ(z)d) z ∈ V \ UR−1(R(z)d) z ∈ C \ V
Let
Φ(z) =
{φ(z) z ∈ V \ UR(z) z 6∈ V
62 HOLLY KRIEGER
recalling that conformal circles are quasiconformally removable. I claim that the quasiregular
map F and the quasiconformal map Φ satisfy the fundamental lemma of quasiconformal
surgery, where E = C \ U . Since the image of R−1 is the complement of V , its clear that
F (E) ⊂ E. Given z ∈ A(ρ), φ−1(z) ∈ V \ U so
F (φ−1(z)) = R−1(φ(φ−1(z))d) = R−1(zd),
and so
Φ ◦ F ◦ Φ−1(z) = zd
is analytic. If |z| > ρ, then R−1(z) 6∈ V , so
F (Φ−1(z)) = R−1(R(R−1(z))d) = R−1(zd),
and again Φ ◦ F ◦ Φ−1(z) = zd. So the second condition is satisfied. Finally, F agrees with
f on U = C \ E so is conformal there.
Thus by the fundamental lemma, F is quasiconformally conjugate to a rational map for
which∞ is totally invariant, i.e. a polynomial, and (as seen in the proof of the fundamental
lemma), has 0 dilatation on U , so is a hybrid conjugacy.
�
In fact, the polynomial of the straightening theorem is unique whenever K(f) is con-
nected:
Proposition 11.1. Suppose f, g are polynomials which are hybrid equivalent, and K(f) is
connected. Then f and g are conjugate by an affine map.
Proof. Write φf , φg for the Bottcher maps for f, g respectively. Since K(f) (and hence also
K(g)) are connected, φg ◦φ−1f provides a conformal isomorphism from C\K(f)→ C\K(g).
As f and g are hybrid equivalent, we have 1-quasiconformal homeomorphism from a neigh-
borhood W of K(f) to a neighborhood W ′ of K(g).
Since log |φf (z)| → 0 as z → K(f), there exists δ > 0 so that the curve
Cδ := {z ∈ C \K(f) : log |φf (z)| = δ
is contained in W . Since K(f) is connected, φf is an isomorphism from Cδ to a circle, so
the curve is a conformal circle and thus quasiconformally removable.
Pasting then φg◦φ−1f and the hybrid conjugacy along this curve, we obtain a 1-quasiconformal
map which conjugates f and g; that is, a rational map. Since ∞ is fixed by this map, it is
affine. �
Let’s take a step back to admire our (their) handiwork. We’ve just shown, for example,
that the polynomial-like map we constructed from that Newton map is quasiconformally
conjugate to some degree 2 polynomial, and so the filled Julia set of the polynomial-like
map is quasiconformally identified with the filled Julia set of some degree 2 polynomial.
Since the Newton polynomial-like map had a fixed attractor, so does the polynomial whose
filled Julia set it is a copy of! For my choice of a, the critical point was itself fixed, so the
map is conjugate to z2, and so the filled Julia set is quasiconformally a disk.
COMPLEX DYNAMICS 63
We also understand now why some Newton Julia sets have rabbits - because they have
an iterate which restricts to a polynomial-like map which is quasiconformally conjugate to
the rabbit polynomial.
Figure 17. A Newton rabbit.
64 HOLLY KRIEGER
***Lecture 21***
12. Holomorphic families of dynamical systems
Definition 12.1. Let U ⊂ C be a domain, and fix d ≥ 2. A collection {fλ : λ ∈ U} of
rational maps of degree d is a holomorphic family of rational maps (over U) if the map
f : U × C→ C
defined by (λ, z) 7→ fλ(z) is holomorphic in (λ, z).
Note that a function f(λ, z) in two complex variables is holomorphic if it has a convergent
power series expansion; equivalently, if it is continuous and holomorphic in each variable;
that is,∂f
∂λ=∂f
∂z= 0.
It follows that the coefficients of the maps are holomorphic in λ; however, note that
the converse is not so, as z 7→ λz3 + z2 doesn’t describe a holomorphic family. Indeed, as
we approach (0,∞) along the curve λz = −1, f(λ, z) = 0, while f(0, z) = z2 →∞ as z →∞.
Example. We have now considered many times the example fc(z) = z2 + c, which forms
a holomorphic family over the domain C, and similarly with gλ(z) := λz + z2.
Definition 12.2. We say that a holomorphic family f over U of degree d is critically
marked if for each 1 ≤ i ≤ 2d− 2, there exist holomorphic maps
ci : U → C
so that for each λ ∈ U , {c1(λ), . . . , c2d−2(λ)} is the set of critical points for fλ.
Example. As any constant map is holomorphic, the family fc(z) = z2 + c is critically
marked by the functions c1(λ) =∞ and c2(λ) = 0.
Definition 12.3. The family is said to be critically stable if
{λ 7→ fnλ (ci(λ))}n≥0
forms a normal family on U for each 1 ≤ i ≤ 2d − 2. Define the stable locus S(f) of the
family to be the set of z ∈ U so that these functions form a normal family on a neighborhood
of z, and the bifurcation locus B(f) to be the complement of the stable locus in U .
Proposition 12.1. Let f be the family fc(z) = z2 + c over C. Then the bifurcation locus
B(f) is the boundary ∂M of the Mandelbrot set.
Proof. Recall that
M := {c ∈ C : |fnc (0)| remains bounded under iteration }.
Note first that |c| > 2⇒ c 6∈M . For if |c| > 2, then we have inductively that
|fnc (0)| ≥ |c|(|c| − 1)2n−2
→∞.
COMPLEX DYNAMICS 65
On the other hand, if |c| ≤ 2, then we have for any |z| = 2 + δ inductively that
|fnc (z)| ≥ 2 + 4nδ,
so c ∈M⇒ |fnc (z)| ≤ 2 for all n ≥ 0. Therefore
M = {c ∈ C : |fnc (0)| ≤ 2 for all n ≥ 0}.
Now, the two marked critical points for fc are c1 ≡ ∞ and c2 ≡ 0. Since fc(∞) =∞ for all
c ∈ C, the family c 7→ fnc (c1(c)) is constant, so trivially normal. On the other hand, we have
just shown that the family c 7→ fnc (0) is uniformly bounded for c ∈ M, so a normal family
on any open subset of M i.e. on the interior of M. On C \ M, any subsequence of the
family converges locally uniformly to the constant ∞. Therefore no open disk intersecting
∂M can have c 7→ fnc (0) as a normal family, since any holomorphic limit would take the
constant value ∞ on an open neighborhood in but not all of the disk. So ∂M = B(f) as
claimed. �
Definition 12.4. A holomorphic motion of a set S in C is a family {φλ : S → C : λ ∈ D}of injections so that φ0 is the identity, and λ 7→ φλ(s) is holomorphic for each s ∈ S.
We will need the following foundational technical result on holomorphic motions:
Theorem 12.1 (The λ-lemma, MSS). A holomorphic motion of a set S ⊂ C has a unique
extension to a holomorphic motion of S. The extended motion gives a continuous map
φ : D× S → C.
Remark. In fact, for each λ ∈ D, φλ : S → C extends to a quasiconformal map of the
sphere to itself, with dilatation bounded by (1 + |λ|)/(1− |λ|). This takes a decent bit more
work to prove, and we won’t do it.
Proof. The idea here is that the injectivity of the holomorphic motion gives disjointness of
the images, which forces equicontinuity. Let’s make it precise. If S has only finitely many
points, we are done, so assume S is infinite.
Let ρ denote the hyperbolic metric on C \ {0, 1,∞}. For any fixed B, the hyperbolic ball
ρ(z, w) ≤ B has complex diameter which converges to 0 as |z| → 0; more precisely, there
exists a continuous function η so that
|w| ≤ η(B, |z|)
for all ρ(w, z) ≤ B, and for any fixed B, η(B, |z|)→ 0 as |z| → 0.
We will apply this fact to the cross-ratio function
gx,y,z,w(λ) := (φλ(x), φλ(y), φλ(z), φλ(w)),
where
(z1, z2, z3, z4) =z3 − z1
z3 − z2
z4 − z2
z4 − z1.
Fixing any four distinct x, y, z, w ∈ S (and suppressing notational dependence), this gives a
holomorphic function g : D→ C \ {0, 1}, which by Pick’s theorem is non-increasing for the
respective hyperbolic metrics ρD and ρ. In particular,
ρ(g(λ), g(0)) ≤ ρD(λ, 0) = log1 + |λ|1− |λ|
.
66 HOLLY KRIEGER
Thus for all 0 < r < 1, the discussion above implies that
|g(λ)| ≤ η(M(r), |(x, y, z, w)|)
for all |λ| ≤ r. Therefore each φλ is uniformly continuous on S, and so extends continuously
to the boundary of S.
Given λ ∈ S \ S, the extension φλ is a uniform limit of holomorphic functions, so holo-
morphic. In addition, the map φλ must be injective, as for any fixed x, y, we can compose
with the holomorphic Mobius coordinate
A(z) =z − φλ(x)
z − φλ(y)
to a holomorphic motion which sends x constantly to 0 and y constantly to∞. This extends
injectively if and only if the original motion extends injectively, completing the proof. �
COMPLEX DYNAMICS 67
***Lecture 22***
In the definition of the Mandelbrot set, our analysis that a polynomial has connected
Julia set iff all finite critical points remain bounded allowed us to replace the ‘critical point’
definition with a more natural dynamical definition of the Mandelbrot set which considered
only the Julia sets. We can similarly relate critical stability to motions of Julia sets.
Definition 12.5. If {fλ : λ ∈ D} is a holomorphic family of rational maps, we say that
the Julia sets are moving holomorphically in the family if there is a holomorphic motion
{φλ : J(f0)→ J(fλ)} of homeomorphisms so that
φλ(f0(z)) = fλ(φλ(z))
for all z ∈ J(f0). That is, we have a topological conjugacy on J(f0) to J(fλ) which moves
holomorphically with λ.
Theorem 12.2 (MSS). Let {fλ : λ ∈ D} be a holomorphic family of rational maps of
degree d ≥ 2. The family fλ is critically stable if and only if the Julia sets are moving
holomorphically.
Proof. Suppose the Julia sets are moving holomorphically, and fix any three points x0, y0, z0 ∈J(f0) not in the forward orbit of any critical point ci(0) of f0; by the conjugacy provided
by the holomorphic motion, the (distinct) points xλ := φλ(x0), yλ := φλ(y0), zλ := φλ(z0)
are disjoint from the critical orbits of fλ. The family of Mobius transformations
Aλ :=z − xλz − zλ
· yλ − zλyλ − xλ
is holomorphic, so we conclude that λ 7→ Aλ(fnλ (ci(λ))) is a normal family for each i, so the
same for fnλ (ci(λ)). So f is critically stable.
Now suppose that f is a critically stable family over the unit disk. For each critically
marked point ci, let gi : D → C be a holomorphic limit of a subsequence of iterates at ci.
We will show first that there is a uniform (in λ) bound on the period of any attracting cycle
for an element of the family. Suppose that λ0 ∈ D has an attracting cycle, which attracts
the critical point ci(λ0), so that fm0 (gi(λ0)) = gi(λ0) is in an attracting cycle. Since the
multiplier of the cycle is not 1, the periodic cycle moves holomorphically on a neighborhood
of λ0, and for sufficiently small motion, remains attracting. Since the Koenig linearizing
map moves holomorphically with λ, the critical point ci(λ) is attracted to this cycle for all
λ near λ0. Thus we have a neighborhood in D so that
fmλ (gi(λ) = gi(λ);
by the identity principle, this relation holds throughout D. Therefore if the critical point
ci(λ) is attracted to a periodic point, that periodic point must be in the cycle of gi(λ), which
has period at most m. Since there are only finitely many critical points, we have a uniform
upper bound M on the period on any attracting cycle throughout the disk, as claimed.
It follows that any repelling cycle of period > M for f0 can be followed holomorphically
throughout the disk, and remain repelling, since otherwise the cycle would become parabolic,
68 HOLLY KRIEGER
and so have a perturbation with an attracting cycle of period > M . Thus the set of repelling
points of period > M for f0 admit a holomorphic motion, noting that injectivity follows
from multiplier 6= 1. Additionally, this motion satisfies for a point z of period N that
φλ(fN0 (z)) = φλ(z) = fNλ (φλ(z)),
and since f is a bijection on the cycle, φλ(f0(z)) = fλ(φλ(z)). By the λ-lemma and density
of repelling cycles in Julia sets we have a holomorphic motion of J(f0), and the conjugacy
requirement follows from continuity of the motion and conjugacy on the repelling points of
high period. �
Example. We have seen that the boundary of the Mandelbrot set is the bifurcation
locus for the family z 7→ fc(z) = z2 + c. The collection of parameters c ∈ C so that fc has
a finite attracting cycle is contained in the stable locus, for if fc0 has an attracting cycle of
period n, then so do all nearby fc, and the attracting cycle and Koenig map are holomorphic
functions of c. The map c 7→ fnc (0) is thus uniformly bounded on a neighborhood of c0,
hence a normal family. As a particular case, the set of parameters c which have a fixed
attracting point is the main cardioid of the Mandelbrot set, and is really a cardioid, as the
fixed points are the roots of
z2 − z + c = 0,
so their multipliers (which are simply the points multiplied by 2) are the roots of
1
4w2 − 1
2w + c = 0,
and so satisfy
c =2w − w2
4.
Thus the set of parameters c for which some fixed point multiplier has modulus less than 1
is the image of the unit disk under the map
w 7→ 2w − w2
4,
indeed a cardioid. Identifying D with the main cardioid under this map (which is a conformal
isomorphism on D) we see that restricting to the main cardioid thus produces a holomorphic
family which is critically stable. Thus by MSS, the Julia sets are moving holomorphically;
in particular, since c = 0 is inside this component, all Julia sets in this component are
topological circles. By the remark, they are in fact quasiconformal circles.
More generally, we define:
Definition 12.6. Let U be a connected component of the interior of the Mandelbrot set.
We say U is hyperbolic if U contains a parameter c0 so that fc0 has a finite attracting cycle.
Lemma 12.1. Suppose U is a hyperbolic component of M, containing a map fc0 with an
attracting point of period N . Then all elements of U correspond to maps with an attracting
cycle of period N , and the map ρU : U → D sending c to the multiplier of this cycle is an
analytic map.
COMPLEX DYNAMICS 69
Proof. Let fc0 with c0 ∈ U have an attracting point z0 of period N and multiplier λ0.
By the implicit function theorem, we have an analytic homeomorphic extension z(c) on a
neighborhood of c0 so that z(0) = z0 and z(c) is a point of period N for fc. It follows that
the multiplier map c 7→ fNc (z(c)) is analytic as well. Since any two points in the component
are connected by a path, on which the period of the (unique!) attracting cycle is necessarily
constant, all component elements have a fixed period for that attracting cycle, and the
multiplier map is well-defined and analytic on the entire component U . �
Note that as a polynomial in c, z(c), the multiplier map is also proper. In fact, with some
work one can show it provides a conformal isomorphism on the component, just as in the
main cardioid case.
70 HOLLY KRIEGER
***Lecture 23***
Remark. All known connected components of the interior of the Mandelbrot set are
believed to consist of maps with an attracting cycle, but it is not known whether these are
the only components which occur!
One of the biggest open problems in complex dynamics is exactly this.
Conjecture 12.1 (Density of hyperbolicity). The interior of the Mandelbrot set consists
of hyperbolic components.
13. Universality of the Mandelbrot set
We have seen that the straightening theorem explains to us why we see polynomial Julia
sets in rational Julia sets, when they have some polynomial-like restriction. Our next goal
is to prove straightening in families, to explain why we see something like the Mandelbrot
set in any holomorphic family of polynomial-like maps in degree 2.
Definition 13.1. A holomorphic family of polynomial-like maps is a collection
fλ : Uλ → Vλ
of polynomial-like maps of degree d ≥ 2, parametrized by λ ∈ D, for which
U := {(λ, z) ∈ C2 : z ∈ Uλ}
and
V := {(λ, z) ∈ C2 : z ∈ Vλ}are open complex manifolds, and the map f : U → V defined by (λ, z) 7→ (λ, fλ(z)) is proper
and holomorphic.
Definition 13.2. Let F be a holomorphic family of polynomial-like maps. The connected-
ness locus M(F) is
M(F) := {λ ∈ D : K(fλ) is connected}.
Example. By definition, the connectedness locus of the polynomial-like family fc(z) :=
z2 + c is the Mandelbrot set.
In a holomorphic family of polynomial-like maps, it may not make sense to ask about
the iterates of critical points forming a normal family, since iterates are not necessarily well-
defined for all points. However, the equivalence with holomorphic motion of the Julia set
suggests the following definition.
Definition 13.3. Let F be a holomorphic family of polynomial-like maps of degree d. The
stable locus of the family is the set of points λ ∈ D where the Julia set ∂K(fλ) locally moves
holomorphically. The bifurcation locus of the family is the complement of the stable locus.
Lemma 13.1. Let F be a holomorphic family of polynomial-like maps of degree d with
M(F) compact in D. Then the bifurcation locus B(M(F)) is ∂M(F), the boundary of the
connectedness locus.
COMPLEX DYNAMICS 71
Proof. Same as for the Mandelbrot set! Exercise (ES4). �
To any fλ a holomorphic family F of polynomial-like maps of degree d = 2, we have by
the straightening theorem a quasiconformal homeomorphism φλ which conjugates f to a
polynomial of degree 2, and satisfies ∂φλ = 0 on the filled Julia set K(fλ). If the filled Julia
setK(fλ) is connected, this polynomial is unique up to affine conjugacy, so a unique φλ which
conjugates fλ to z2 + cλ for some cλ ∈ C. In fact, since the conjugacy homeomorphically
identifies K(fλ) and the filled Julia set of z2 + cλ, we have cλ ∈M . Define then
Φ : M(F)→M
to be the map λ 7→ cλ.
We can finally explain (partially) why we see the Mandelbrot set everywhere in this
theorem of Douady-Hubbard.
Theorem 13.1 (Douady-Hubbard [DoHu]). Suppose that F is a holomorphic family of
polynomial-like maps in degree d = 2, and let Φ : M(F) → M be the straightening map
described above. Then Φ is continuous on M(F) and holomorphic on the interior of M(F).
Proof. ***We will simplify the proof by assuming density of hyperbolicity. This is not nec-
essary, but simplifies the technology needed by a reasonable bit.***
Holomorphicity in the interior. In the case of λ so that fλ has an attracting cycle
(which since d = 2 is necessarily unique, as it attracts the only free critical point), just as
in the Mandelbrot case, we have a local conformal isomorphism given by the multiplier of
the attracting cycle. Since the cycle period and multiplier are preserved by φλ, this map
factors through M(F) →M → D, where the last arrow is the relevant multiplier map for
the component of the image Φ(λ). Therefore the map M(F)→M is analytic at any point
λ ∈ D so that fλ has an attracting cycle.
Continuity on the boundary: Siegel approximation.
Lemma 13.2. Given λ0 ∈ ∂M(F), there exists a sequence λn → λ0 in ∂M(F) so that fλn
has a Siegel disk.
Proof. In our proof of holomorphic Julia motion, we proved that a uniform upper bound
on the period of an attracting cycle for fλ on any neighborhood U of λ0 guarantees the
Julia sets move holomorphically on that neighborhood. Thus λ0 in the bifurcation locus
of F implies that there is no such upper bound on any neighborhood of λ. We may then
find a sequence λ′n → λ, necessarily in M(F), so that λ′n has an attracting cycle of period
≥ n. Since there can be only one attracting cycle at a time, these cycles become repelling
and hence indifferent near λ0 as well. We can choose then a sequence λn → λ0 so that fλn
has an indifferent cycle; by density of Siegel parameters in S1, we can choose them to have
Siegel disks. �
72 HOLLY KRIEGER
Remark. Note that while we only proved density of Siegel parameters in S1 for the
family λz + z2, that argument required only (1) a uniform bound on the radius of the
linearizing map for fixed attractors in the family, and (2) a holomorphically moving critical
point. We have (1) by compactness of the connectedness locus, and (2) by uniqueness of
the critical point. In particular, we are not circularly invoking continuity of straightening.
COMPLEX DYNAMICS 73
***Lecture 24***
Given λn → λ0 so that fλnhas a period ≥ n, Siegel disk, we have fλn
∈ B(F). For since
the cycle and its multiplier can be continued analytically in a neighborhood of λn, we have
arbitrarily close parameters where the cycle is repelling, so Julia, while at λn it is strictly
in the interior of the filled Julia set, so the Julia sets cannot move holomorphically on any
neighborhood of λn. Since the Siegel multiplier is preserved by the straightening map, each
Φ(λn) is by the same argument a parameter in the bifurcation locus of the quadratic poly-
nomial family; that is, in ∂M.
Normality of the family of straightening maps. Claim: the straightening maps
for λ ∈ M(F) form a normal family of maps. Since the F is holomorphic, we can in the
proof of the straightening theorem construct the initial maps on the boundary of the an-
nulus V \ U to vary smoothly with λ as well, and similarly with the interpolating maps on
annuli. Thus the (continuous!) derivatives of the interpolating map of annuli are uniformly
bounded on compact subsets of D; in particular, on M(F), the straightening maps φλ are
all K-quasiconformal for some uniform finite K. They are therefore equicontinuous, so by
the Arzela-Ascoli theorem, are a normal family.
Consider now the straightening maps φ(λn) for this sequence of parameters. As the
straightening maps for M(F) form a normal family, we have a subsequence φλnkwhich con-
verges locally uniformly to some φ0; as M is compact, we may WLOG assume that Φ(λnk)
approaches a limit c0. By the conjugacy relation, the limit map φ0 provides a quasiconfor-
mal conjugacy between fλ0and z2 +c0. Since c0 is a limit of points in ∂M, c0 ∈ ∂M as well.
Quasiconformal uniqueness on ∂M . If the limit φ0 was a hybrid conjugacy, we’d have
Φ(fλ0) = c0 by the lemma on hybrid conjugacy for polynomials with connected filled Julia
sets. While quasiconformal conjugacy class is not enough to specify a Mandelbrot parameter
(indeed, we have seen that in the hyperbolic components, all Julia sets are quasiconformally
conjugate), it is sufficient on the boundary.
Lemma 13.3. Suppose c, c′ ∈ C with c ∈ ∂M and z2+c, z2+c′ quasiconformally conjugate.
Then c = c′.
Proof. Exercise (ES4). �
In either case, we conclude that Φ(λ0) ∈ ∂M. Now to prove that any sequence λn → λ0
satisfies Φ(λn)→ λ0, again appeal to normality to find a subsequence of the φλnwhich have
a limit providing a quasiconformal conjugacy of z2 + Φ(λ0) and the quadratic polynomial
z2 + lim Φ(λn); then by the lemma, these agree.
�
You are well-equipped now to read McMullen’s completion of the universality of the
Mandelbrot set [Mc] if interested in the full statement; indeed, the preceding theorem is the
primary technical input to universality.
74 HOLLY KRIEGER
Theorem 13.2 (McMullen [Mc]). Let F be a holomorphic family of rational maps over
D. Then the bifurcation locus B(F) is empty, or contains a quasiconformal copy of ∂Md,
where
Md := {c ∈ C : K(zd + c) is connected }.For a generic family with non-empty bifurcation locus, B(F) contains a copy of ∂M.
Finally, we complete our discussion with the following tantalizing hint into the connection
between the dynamical plane and the parameter plane.
Theorem 13.3. The Mandelbrot set is connected.
Proof. We will construct a map which identifies C \M with the complement C \ D using
the Bottcher coordinates. First, fix c ∈ C. If U is a domain in the basin of infinity so that
for all n ∈ N, we can choose a holomorphic branch of fnc (z)1/2n
on U , then the function
φ(z) := limn→∞
fnc (z)1/2n
satisfies the properties of the Bottcher coordinate, so long as our branches are chosen com-
patibly. More precisely, let z ∈ B(∞) so that Gc(z) > Gc(0). Then there is a simply
connected neighborhood U of z so that fn is a conformal isomorphism on U for all n ∈ N,
and never takes the value√−c or 0 on U . Thus we may for all n ∈ N define holomorphic
branches of
log
(1 +
c
fnc (z)2
),
as z 7→ 1+ cfnc (z)2 is a conformal isomorphism from U to a simply connected domain uniformly
(in N) bounded away from 0 and ∞. So
φN (z) = z
N∏n=0
(1 +
c
fnc (z)2
)1/2n−1
= fNc (z)1/2N
is holomorphic (and conformal) on U , where we choose the root that fixes 1. φN approaches
a holomorphic limit, as with the binomial expansion we have
|φN+1(z)− φN (z)| = |φN+1(z)| ·∣∣∣∣1− φN (z)
φN+1(z)
∣∣∣∣ .|φN+1(z)| is uniformly bounded on U as already noted, and
1−(
1 +c
φN (z)2N+1
)1/2N+1
=
∞∑k=1
2−(N+1)(2−(N+1) − 1) · · · (2−(N+1) − kk!
· ck
φN+1(z)2N+1k.
Thus in modulus this difference is no larger than∞∑k=1
∣∣∣∣ c
φN+1(z)2N+1
∣∣∣∣ ,which uniformly converges to 0 as N →∞. Thus
φ(z) := limn→∞
fnc (z)1/2n
indeed defines a holomorphic function on Gc(z) > Gc(0) as claimed, which must therefore
be the Bottcher coordinate φc by uniqueness in degree 2.
COMPLEX DYNAMICS 75
Define now
Φ(c) := φc(c) = c
∞∏n=0
(1 +
c
fnc (c)2
)1/2n−1
.
The same convergence estimates prove that Φ is holomorphic in c for all c 6∈ M, so Φ
provides a holomorphic map from C \D→ C \D. As Φ(c)/c→ 1 as c→∞, this map has a
removable singularity at∞, and can be extended with Φ(∞) =∞. I claim this extended map
is proper. If c→ c0 ∈ ∂M, then c→ K(fc), so Gc(c)→ 0 and consequently |Φ(c)| → 1. As
the local degree at∞ of Φ is 1, Φ has global degree 1 and provides a conformal isomorphism
as claimed.
Thus the complement of the Mandelbrot set is simply connected and so the Mandelbrot
set is connected. �
Remark. Note however that it is very easy to artificially produce connectedness loci
which are not connected. For example, gλ(z) := z2 + λ2 + 2 has a connectedness locus
consisting of two disconnected homeomorphic copies of the Mandelbrot set, since λ2 + 2
provides a double cover of C with branching outside the Mandelbrot set.
Figure 18. The connectedness locus (in black) of the family z2 + λ2 + 2.
We conclude with the statement (if not the motivation) of the most important open
question in the complex dynamics of one variable.
Conjecture 13.1 (MLC). The map Φ : C\M → C\D extends continuously to the boundary
∂M .
By work of Douady-Hubbard [DoHuO], MLC is known to be a strictly stronger conjecture
than density of hyperbolicity.
76 HOLLY KRIEGER
References
[BrFa] B. Branner and N. Fagella. Quasiconformal Surgery in Holomorphic Dynamics. Cambridge Studies
in Advanced Mathematics, Cambridge University Press (2014).
[DoHu] A. Douady and J. H. Hubbard. On the dynamics of polynomial-like mappings. Ann. Sci. Ecole
Norm. Sup. 18 (1985), 287–343.
[DoHuO] A. Douady and J. H. Hubbard. Etude dynamique des polynomes complexes, I, II. Publ. Math. Or-
say (1984-1985). English version available at http://pi.math.cornell.edu/~hubbard/OrsayEnglish.pdf
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