Las funciones de los usos atípicos de la puntuación en guiones de CMHW y

In this section, we briefly describe some terminology which allows us to discuss the structure of Julia sets. Some of this will be a preliminary to the study of Hubbard trees in Section 1.7.

1.6.1 Internal rays

In this section we describe how to construct an analogue to external rays that exist inside the Fatou components of a rational map. We assumef is a rational map with (pre)periodic critical points.

Let

φa(z) = z−a

1−¯az (|a|<1).

We remark that φa will map the unit disk onto itself. We say that a map of the

form

f(z) =e2πitφa1(z)φa2(z)· · ·φad(z) (t∈[0,1)) (1.2)

is aBlaschke product of degree d. We then have the following ([Mil06], page 162).

Lemma 1.6.1. A rational map of degree d carries the unit disk onto itself if and only if it is a Blaschke product.

We use this result to construct internal rays inside the Fatou components of a degree d bicritical rational map F with disjoint periodic superattracting cycles. The construction can be carried out in more general cases but we restrict ourself only to the result required in this thesis. Given a periodic Fatou componentU, the first return map F◦n to the component is a degree d covering with precisely one critical point cU. There exists a Riemann map Φ : D → U such that Φ(0) = cU.

Then we have a commutative diagram U F◦n D Φ o o B=Φ−1_◦F◦n_◦Φ U D Φ o o .

The map B is a rational map which will map the unit disk onto itself by a degree

d covering, and so by Lemma 1.6.1 it is a Blaschke product, and so has the form (1.2). Furthermore, the only fixed point is 0, so φaj = z for j = 1, . . . , d and

so B(z) = e2πitzd for some t ∈ [0,1). We can normalise (by composition with a rotation) so thatB is in fact equal to the mapz7→zd_{on the disk. Define the radial}

arcs

rθ={re2πiθ : 0≤r <1} ⊂D.

Then the internal ray of angle θis the arc Φ−1(rθ)⊂U.

We make some observations about internal rays. Firstly, each internal ray has the centre as an endpoint, and the internal ray of anglesk/(d−1),k= 0,1, . . . , d−2 will be fixed under the first return map to the component, F◦n. Assuming the boundary of U is locally connected (which it will be in all cases we will consider), we can also discuss the landing of internal rays in the same way as with external rays. These landing points will belong to the Julia set of the mapF.

Given a periodic cycle of superattracting basins, we can define the internal rays of each basin individually. However, we can construct the rays in such a way so that, ifF(U) =V in the cycle, then the internal ray of angle θ inU will map onto the internal ray of angle θ inV. This can be done by, if necessary, composing the Blaschke products with rotations so that this agreement is achieved. Furthermore, if U′ is a pre-periodic Fatou component that maps onto a periodic superattracting basin, then there exists an integerk so thatF◦k_(U′_{) is a memberU of the periodic} superattracting cycle. The map F◦k_|

U′ is a homeomorphism, and so we can define the internal ray of angleθinU′ to be the pre-image (underF◦k) of the internal ray of angleθinU. Since all Fatou components are pre-periodic for hyperbolic rational maps, this defines the notion of internal rays for all Fatou components.

Now suppose that z is a periodic point in J(F) of period p which lies on the boundary of the periodic Fatou components U1, U2, . . . , Un. Then zwill be the

landing point of precisely one internal ray from each Ui. The map F◦p is a local

the combinatorial rotation number atz. Furthermore, ifz is also the landing point of external rays, the combinatorial rotation number defined for internal rays is the same as that defined for external rays (Proposition 1.4.4).

1.6.2 Regulated arcs

Supposef is a polynomial with locally connected (and hence path connected) Julia set. Sometimes we will want to construct paths in the filled Julia setK(f). It will aid us if we have a canonical way of making these paths. If we have J(f) =K(f), then the path between two pointsx, yinJ(f) is uniquely defined. However, if this is not the case, we need to decide how we will define the arc [x, y]. The problem occurs when we pass through the Fatou components, but fortunately the internal rays give us a way of passing through them in a way which can be consistently defined.

Definition 1.6.2. Letx, y∈J(f). The arc [x, y] will be called regulated if, for each Fatou component U, U ∩[x, y] is contained in the union of (at most) two internal rays ofU.