We now prove some results about angles (and components) in M which will be useful when considering cluster points in Chapters 3, 4 and 5. We first consider the rabbit components ofM; that is, maps belonging to the hyperbolic components which bifurcate directly off of the main cardioid.
Proposition 1.8.1. A hyperbolic component has internal address1→nif and only if it is a rabbit component.
Proof. If H is a rabbit component of period n then it bifurcates from the main cardioid. Then there are no other hyperbolic components on the combinatorial arc between H and the main cardioid and so the internal address (using the third characterisation) is 1→n.
Now suppose H has internal address 1 → n. Then by Lemma 1.7.14, since
nis divisible by 1, Hmust be a bifurcation from a component of period 1, which is the main cardioid.
Lemma 1.8.2(See also [Wit88], Claim 10.1.1). Letf be ann-rabbit. Thenz∈J(f)
is biaccessible iff it is a pre-image of theα-fixed point.
Recall that the width of a sector is defined to be the difference between the angles of the two parameter rays bounding it. The following result is Proposition 2.4.3 of [Sch94]. The width of a hyperbolic component will be the width of the wake which is formed by the two parameter rays landing at its root point.
Proposition 1.8.3. Given a hyperbolic component of period m and width δ, the width of its p/n-subwake is
(2m−1)2
2nm−1 δ. (1.3)
Corollary 1.8.4. The wakes of an n-rabbit component is narrow. That is, the width must be 1/(2n−1).
Proof. The rabbit component will bifurcate off of the main cardioid by Proposi- tion 1.8.1, and lie in thep/n-subwake. Hence it has width
(2−1)2
2n−1 =
1 2n−1.
The following corollary will be of use in Chapter 5, as it tells us which maps have a period two orbit with a given combinatorial rotation number.
Corollary 1.8.5. Let H be a period 2n hyperbolic component which bifurcates off of the period 2 hyperbolic component of M. Then there exists precisely one period
2n component H′ in the wake of H
Proof. We first calculate the width δH of the wake of the hyperbolic componentH. The width of the period 2 component is 1/3, and so 1.3 gives us
δH= (22−1)2 22n−1 · 1 3 = 3 22n−1.
Therefore there are precisely two rays of angle with denominator 22n−1 between the rays landing at the base of H. Hence there must exists precisely one period 2n
componentH′ in the wake ofH.
By considering the limbs which bifurcate off of the period 2 component, we realise we have enough information to calculate the angled internal address of the second component H′ that lies in the wake of the component of period 2n which bifurcates off of the period 2 component.
Proposition 1.8.6. Let Hbe a period2nhyperbolic component with angled internal address
11/2 →2p/n →2n,
withp coprime to n. Then the other component H′ of period 2n which is contained in the wake ofH has angled internal address
11/2 →2p/n →(2n−1)1/2 →2n.
Proof. Since H′ is contained in the wake of H, we know that the angled internal address begins
We will now show that S2 = 2n−1, which will prove the proposition. Note that
if S2 = 2n then, by Lemma 1.7.14, H′ bifurcates from the period 2 hyperbolic
component, contradicting the assumption onH′ being in the wake (and not equal to) the componentH which bifurcates from the period 2 component. If S2 = 2m
for somem6=n, then this would represent a bifurcation of the period 2 component into the period 2m component (Lemma 1.7.14), which contradicts the assumption thatH′ is in the wake of the period 2ncomponent. It follows that S
2 is odd.
Since S2 is odd, it is congruent to 1 mod 2. Furthermore, we notice that
2∈orbρ(1). Now using Lemma 1.7.13, we can substitute in what we already know
to get
n= S2−1 2 + 1, which when rearranged yields
S2= 2(n−1) + 1 = 2n−1
as required.
Lemma 1.8.7. If(T, f) is the Hubbard tree of an n-rabbitf, then all the points in the critical orbit off are endpoints of T.
Proof. The proof is trivial. f has internal address 1p/n → n, so the Hubbard tree
contains a fixed point withnglobal arms. Each of these global arms has an endpoint, which must belong to the critical orbit. But there are onlynelements in the critical orbit, so each one must be the endpoint of one of these global arms.
1.8.1 Higher Degree Cases
In Chapter 4, it will be necessary to generalise the notion of a n-rabbit to higher degree cases. A lot of terminology from the symbolic dynamics of quadratic poly- nomials carries over the higher degree case, as was shown by Lau and Schleicher in [LS94]. In this section we will briefly state the results we need.
We first comment that a Multibrot set Md will be the connectedness locus
of polynomials of the formz7→zd+c, as an analogous definition toM. That is Md={c∈C: zd+c has connected Julia set}.
Figure 1.7 shows the degree 3 Multibrot set. We note that, in the non-quadratic case, the current characterisations of the (angled) internal address is not enough to uniquely define the hyperbolic components. For example, the degree 3 multibrot
set contains two degree 2 components, which each have angled internal address 11/2 → 2. To fix this problem, we need to introduce the concept of “sectors” of hyperbolic components. Since we are not concerned with uniqueness in this thesis, we omit this discussion.
Figure 1.7: The degree 3 Multibrot set.
The hyperbolic components inMd haved−1 root point. One of these is the
landing point of precisely two parameter rays; this will be known as theprincipal
root point. The other root points will be the landing point of one parameter ray, and will be callednon-principal root points.
Recall that an n-rabbit was defined to be a map belonging to a hyperbolic component which bifurcates off of the cardioid in M. We will similarly define an
n-rabbit in degree d to be a map which belongs to a hyperbolic component which bifurcates off of the (unique) period 1 component of Md. By a similar proof to
Proposition 1.8.1, such a map will have internal address 1 → n. Furthermore, it has angled internal address 1p/n → n and so, by the characterisation of internal
addresses, has a fixed point with combinatorial rotation numberp/n.
Figure 1.8: A degree 3 rabbit (corresponding to the parameter rays 1/26 and 3/26) and the external rays landing on the twoβ-fixed points and theα-fixed point.
Suppose f is a degree d n-rabbit. Then the landing point of the external rays of angles k/(d−1) will be fixed points. There exists one more (finite) fixed point, which will be called theα-fixed point, in keeping with the terminology from the quadratic case. Figure 1.8 shows a degree 3 rabbit.
One final important point should be borne in mind. In the quadratic case, all root points of Fatou components are principal, meaning they are the landing point of two or more external rays. However, in the higher degree d case, we find that each Fatou component hasd−2 non-principal root points. If the Fatou component
U is periodic, there exists d−1 fixed points on the boundary of U. One of these is principal and the otherd−2 are called non-principal and are the landing point of only one external ray. We will find that this extra complication makes the discussion slightly more difficult in Chapter 4 and also allows us to construct the example
constructed in Appendix E.