COSTOS DIRECTOS Mano de Obra

γ e γ c1 c2 p1 p1 q1 q1 p2 q2 p1 p1 q1 q1 p2 q2

Figure 6.1. A map with a Levy cycle.

post(f) = post(F) _{⊂ C}, and consider tiles and flowers for (F,_C). Let δ0 be as in

(5.14) for the mapF. Then we can decomposeγ1into finitely many arcsα1, . . . , αl

such that diam(αj)< δ0forj= 1, . . . , l. By Lemma 5.34 (i) every connected subset

ofF−k_(α

j) is contained in a k-flower Wjk forj= 1, . . . , l. It follows that for each

k∈Nthe curveγk is contained in the union ofl k-flowersW1k, . . . , Wlk.

To reach a contradiction, assume now thatf, and hence alsoF, is expanding. Then we have diam(γk)≤ l X j=1 diam(Wjk)≤2l max X∈Xkdiam(X)→0 ask→ ∞.

On the other hand, we can find a finite open coverU ofS2consisting of simply connected regionsU each of which contains at most one postcritical point off (for example, the 0-flowers form such a cover). By what we have just seen, we can find k_∈ Nsuch that diam(γk) is smaller than the Lebesgue number of U. Thenγk is

contained in a setU _{∈ U}. Since U is simply connected and contains at most one postcritical point of f, the curveγk is peripheral. This is a contradiction showing

thatf is not expanding.

Example 6.11. We now present an example of a Thurston map f with a Levy cycle. The mapf will have no periodic critical points, and so it provides an example of a Thurston map without periodic critical points that is not expanding, contrasting Proposition 2.3. Since Levy cycles persist under Thurston equivalence, f is also not equivalent to any expanding Thurston map.

For the construction we start with a topological sphereS2 _{that is a pillow (see}

Section A.10). Similar to Section 1.1, the pillow is obtained by gluing two unit squares together along their boundaries. As before, we color one side (i.e., one square) of the pillow white, and the other black.

The black side is divided horizontally into two rectangles, one of which is colored white and the other colored black. The white side of the pillow is subdivided into four quadrilaterals, two white and two black ones as indicated on the left in Figure 6.1. Here we have cut the pillow along three sides so that we obtain a rectangle as shown in the picture. The symbols in the picture indicate which sides have to be identified to recover the pillow.

Nowf is constructed by mapping each white quadrilateral homeomorphically to the white face, and each black quadrilateral to the black face of the pillow. Here f maps vertices to vertices. In Figure 6.1 we have marked two vertices of each quadrilateral (on the left), as well as two vertices of the pillow (on the right) by a black or white dot to indicate the correspondence of vertices under f. Finally,

we require that f agrees on sides shared by two quadrilaterals. The map f thus defined is indeed a Thurston map (because it realizes a two-tile subdivision rule; see Chapter 12 and in particular Proposition 12.3). The postcritical points correspond to the vertices of the pillow. The mapf has two critical pointsc1 and c2 and the

following ramification portrait:

c1 3:1 //p1 //q1 c2 3:1 //p2 //q2 .

Thus f has no periodic critical points, its signature is (3,3,3,3), and f has a hyperbolic orbifold.

We consider the Jordan curve γ _⊂ S2

\post(f) as indicated on the right in Figure 6.1. One of the components γe of f−1_{(γ) is shown on the left (the other}

components of f−1_{(γ) are not shown). Clearly, eγ is isotopic rel. post(f) to γ.}

Furthermore, the degree off:γe_→γ is equal to 1, and so Γ =_{γ_}is a Levy cycle. Lemma 6.10 implies thatf is not expanding, and, by our earlier discussion, no Thurston map equivalent tof is expanding.

6.3. Latt`es-type maps and expansion

We know (see Theorem 3.1 (i) and (i’)) that every Latt`es map is expanding. This is not always true for Latt`es-type maps, but it is easy to decide when this is the case. The relevant condition is based on the following definition.

LetL: R2 _→R2 _{be a linear map. We call it expanding if|λ| >1 for each of}

the two (possibly complex) eigenvaluesλofL. Proposition6.12. Let f: S2

→S2 _{be a Latt`es-type map and L=L}

A be the linear part of an affine map A as in Definition 3.3. Then f is expanding (as a Thurston map) if and only if Lis expanding (as a linear map).

For the proof we require two lemmas. Lemma 6.13. Suppose L: R2

→ R2 _{is an expanding linear map. Then there} exist constants ρ >1 andn0∈Nsuch that

(6.5) _|Ln_(v)

| ≥ρn |v_|

for allv_∈R2 _{and all n}

∈Nwithn_≥n0.

Here_|v_|=px2_+y2_{denotes the usual Euclidean norm ofv= (x, y)∈R}2_.

Proof. The eigenvalues ofLare the two (possibly identical) rootsλ1, λ2∈C

of the characteristic polynomial P(λ) = det(L₋λidR2) of L. We may assume |λ1| ≤ |λ2|. Since L is expanding we have|λ1| > 1. The polynomialP has real

coefficients, and soλ2=λ1 ifλ1 is not real.

There exists a basis ofR2_{consisting of two linearly independent vectorsv} 1, v2∈

R2_{such that the linear mapLhas a matrix representation with respect to this basis}

of one of the following forms:

M1=|λ1| cosθ ₋sinθ sinθ cosθ ,whereθ_∈R, M2= λ1 0 0 λ2 , orM3= λ1 1 0 λ1 .