8. PROBLEMATICAS SECTORIALES
8.3. SECTOR DEPORTE RECREACIÓN Y CULTURA
3.4. Latt`es-type maps
We now consider Latt`es-type maps f: S2 → S2 as in Definition 3.3. If f is
such a map, then there exists a crystallographic group G acting on R2 ∼= C, a
G-equivariant affine mapA:R2→R2, and a branched covering map Θ :R2→S2
induced byG such thatf◦A= Θ◦A. Then f is continuous (see Lemma A.22). Since Θ is induced by G, the quotient space R2/G is homeomorphic to S2 which
implies thatGis not isomorphic toZ2.
In explicit constructions of Latt`es-type maps one usually turns this around and starts with a crystallographic groupGnot isomorphic toZ2 and anG-equivariant
affine map A: R2
→ R2. Then S2 = R2/G is a topological 2-sphere and the
quotient map Θ :R2
→R2/G∼=S2is a branched covering map induced byG. The
G-equivariance of A ensures that this map descends to the quotient R2/G ∼=S2
and so there exists a continuous map f: S2
→S2 such that f
◦A = Θ◦A (see Lemma A.24). As the considerations below will show, the additional condition that the linear partLAofA(see (3.4)) satisfies det(LA)>1 ensures thatf is a Thurston
map.
Indeed, let Gtr be the subgroup of translations in G. We know that then
T2 =R2/G
tr is a (topological) 2-torus. We denote byπ:R2 →T2=R2/Gtr the
quotient map.
The argument in the proof of the implication (ii) ⇒(iii) in Theorem 3.1 (see Section 3.1) shows that A and Θ descend to mapsA and Θ onT2. In this proof
the maps were assumed to be holomorphic, but this played no role in the existence proof for A and Θ. So we obtain continuous mapsA:T2 →T2 and Θ :T2→S2
such thatA◦π=π◦Aand Θ = Θ◦π. Note that as a composition of the covering mapπ:R2→T with the homeomorphismA: R2 →R2, the mapπ◦A:R2→T2
is a covering map. This combined with the last relations implies thatAand Θ are branched covering maps (see Lemma A.16 (ii)). As we discussed, it follows thatA is a (topological) torus endomorphism.
Similar to (3.10), one can summarize the relations of these maps in the following commutative diagram: (3.20) R2 A // π Θ R2 π Θ T2 A // Θ T2 Θ S2 f //S2.
Since Θ and Θ◦Aare branched covering maps, andf◦Θ = Θ◦A, the mapf is also a branched covering map (see Lemma A.16 (i) and (ii)).
It is easy to see that deg(f) = deg(A) (see the beginning of the proof of Lemma 3.12), but the degree off can also be computed fromA.
Lemma 3.16. Let f:S2
→S2 be a Latt`es-type map, and suppose A: R2 →R2 is an affine map and A a torus endomorphism as in (3.20). LetLA be the linear part of A. Thendeg(f) = deg(A) = det(LA).
In particular, if f:Cb →Cb is a Latt`es map and A(z) =αz+β is as in Theo- rem 3.1(ii), thendeg(f) =|α|2.
Proof. Letf:S2→S2 be a Latt`es-type map, and supposeGis a crystallo-
graphic group andAan affine map as in Definition 3.3. Then we have a commuta- tive diagram as in (3.20) and we know that deg(f) = deg(A). So we have to verify that deg(A) = det(LA). Essentially, this follows from standard facts about degrees
of torus endomorphisms as discussed in more detail in Section A.8. Indeed, let Γ ⊂R2 ∼=Cbe the underlying lattice of G. Then G
tr consists of
all translations of the form u∈R2 7→τ
γ(u) := u+γ, where γ ∈Γ. Accordingly,
we can identify the torusT2 =R2/G
tr with the quotientR2/Γ and can think of
the lattice Γ as representing the fundamental group ofT2(see the discussion after
Lemma A.25).
Now an elementary computation shows that A◦τγ◦A−1=τLA(γ)
for each γ∈R2, and in particular for eachγ
∈Γ. SinceA is a lift ofA to R2, it
follows that LA is the unique map induced by A on the fundamental group Γ of
T2 (see Lemma A.25 (iii)). Now Lemma A.25 (iv) implies deg(A) = det(L A) as
desired.
Iff is a Latt`es map andA(z) =αz+β as in Theorem 3.1 (ii), then in complex notationLA(z) =αz forz∈C. For the determinant ofLAconsidered as aR-linear
map, we have det(LA) = |α|2. So it follows from the first part of the proof that
deg(f) = det(LA) =|α|2 as claimed.
Proof of Proposition 3.5. Let f: S2 → S2 be a Latt`es-type map, and
G, A, and Θ be as in Definition 3.3. Then we have a diagram as in (3.20). Here det(LA)>1 by assumption which by Lemma 3.16 translates into deg(f) =
deg(A) = det(LA)≥2. We conclude thatf is a quotient of a torus endomorphism
(see Definition 3.4).
So we can apply Lemma 3.12 and it follows thatf is a Thurston map without periodic critical points. It remains to show thatf has a parabolic orbifold.
For this we verify the criterion in Lemma 3.13 with the branched covering map Θ : T2→S2 as provided by (3.20). So supposex, y∈T2 and Θ(x) = Θ(y). Since
π:R2→T2is surjective, there existu, v∈R2 withπ(u) =xandπ(v) =y. Then
Θ(u) = (Θ◦π)(u) = Θ(x) = Θ(y) = (Θ◦π)(v) = Θ(v).
Since Θ is induced by the crystallographic group G, there exists g ∈Gwith v = g(u). Now Θ = Θ◦g and
deg(π, u) = deg(π, v) = deg(g, u) = 1. We conclude that
deg(Θ, y) = deg(Θ, y)·deg(π, v) = deg(Θ, v) = deg(Θ, v)·deg(g, u) = deg(Θ, u) = deg(Θ, x)·deg(π, u) = deg(Θ, x)
3.4. LATT`ES-TYPE MAPS 71
Corollary3.17. Letf: S2→S2be a Latt`es-type map with a crystallographic group G and a branched covering map Θ : R2 → S2 induced by G as in Defini- tion 3.3, and letOf = (S2, αf)be the associated orbifold off. Then for u∈R2 we have
αf(Θ(u)) = deg(Θ, u) = #Gu.
Proof. Letu ∈R2. Then deg(Θ, u) = #G
u for u∈ R2 as follows from the
second equality in (3.7) and the uniqueness statement in Proposition 3.9 (it is also easy to see this directly).
If we use the notation as in the previous proof, then the considerations there show that ifp:= Θ(u), then the degree of Θ in each point of the fiber Θ−1(p) is the same and is equal to deg(Θ, u). So by Lemma 3.12 (iii) we also have αf(Θ(u)) =
deg(Θ, u).
We know that the type of a crystallographic group G is determined by the orders of the point stabilizers #Gu, u∈R2. Moreover, if Θ : R2 →S2 is induced
by G, then the map Gu ∈ R2/G
7→ Θ(u) ∈ S2 is a bijection (it is actually a
homeomorphism; see the discussion after Proposition 3.9). So it follows from the corollary that the signature ofOf corresponds to the type of the crystallographic
groupG. For example, ifGis of type (2222), then the signature ofOf is (2,2,2,2).
Of course, the corollary also applies when f: Cb → Cb is a Latt`es map and Θ : C→ Cb is a holomorphic map induced by G. Then the statement shows that αf=α, whereαis as in Proposition 3.9, and that Θ is the (holomorphic) universal
orbifold covering map of (Cb, αf).
Since a Latt`es-type map has parabolic orbifold and no periodic critical points, we know by Proposition 2.14 that the orbifold of each such map has one of the signa- tures (2,2,2,2), (2,4,4), (3,3,3), or (2,3,6) (this also follows from Corollary 3.17). The last three cases lead to nothing new and essentially give Latt`es maps.
Proposition 3.18. Let f: S2
→ S2 be a Latt`es-type map with orbifold sig- nature(2,4,4),(3,3,3), or (2,3,6). Thenf is topologically conjugate to a Latt`es map.
To prove this statement we need a lemma that gives a criterion when an R- linear mapL:C→CisC-linear. Here theR-linearity orC-linearity forLof course means that L(z+w) =L(z) +L(w) and L(λz) =λL(z) for allz, w ∈ Cand all λ∈Ror allλ∈C, respectively.
Lemma 3.19. Let L: C → C be an R-linear map with det(L) > 0. Suppose there exist ζ ∈ C\R and η ∈ C with L(ζz) = ηL(z) for all z ∈ C. Then L is
C-linear.
Proof. Since L is R-linear, there exist unique numbers a, b ∈ C such that L(z) =az+bzforz∈C. Then the determinant ofL(as anR-linear map) is given by det(L) =|a|2− |b|2>0. It follows thata6= 0.
Now for allz∈Cwe have
L(ζz) =ζaz+ζbz=ηL(z) =ηaz+ηbz, and so
Sincea6= 0, the first equation impliesζ=η. Then the second equation combined with the fact thatζ /∈Rgivesb= 0. HenceL(z) =az forz∈C. This shows that
LisC-linear.
Proof of Proposition 3.18. We know that there exists a crystallographic groupG, a branched covering map Θ :R2
→S2induced byG, and aG-equivariant
affine homeomorphismA:R2
→R2with det(L
A)>1 such thatf arises as in (3.5).
By conjugation with a suitable map in Aut(C), we may assume that G is one of the groups Ge listed in Theorem 3.7 (see the discussion preceding (3.15)). ThenG is not isomorphic to Z2, becauseC/G∼=S2.
It follows from Proposition 3.9 that we can find a homeomorphismϕ:S2→Cb
such that ϕ◦Θ :R2∼=C→Cb is a holomorphic map. So if we replace the original
map Θ withϕ◦Θ andf with its conjugateϕ◦f◦ϕ−1, then we are further reduced
to the case thatS2=Cb and that Θ is holomorphic (a related argument was given
in the beginning of Section 3.3). It is enough to show that then f:Cb → Cb is a Latt`es map.
By assumption the signature of the orbifold off is (2,4,4), (3,3,3), or (2,3,6). By Corollary 3.17 this means that G is a crystallographic groupGe of type (244), (333), or (236) in Theorem 3.7. In these cases, G contains a rotation g0 of the
form z∈C7→g0(z) =ζz, whereζ =e2πi/n is a primitiven-th root of unity with
n∈ {3,4,6}. In particular,ζ ∈C\R. Since the homeomorphismApasses to the quotient Cb ∼=C/G, this map is G-equivariant (Lemma A.24) and so there exists g1∈Gsuch that
(3.21) A◦g0=g1◦A.
LetL=LAbe the linear part ofA. This is anR-linear map satisfying det(L)>0.
Since G consists of orientation-preserving isometries, the linear part of g1 is C-
linear, as for every map inG. Comparing linear parts of the maps in (3.21), we see that there existsη∈C,|η|= 1, such that
L(ζz) =ηL(z)
for allz ∈C. This shows that L satisfies the hypotheses of Lemma 3.19 and we conclude thatLisC-linear. HenceAcan be written in the formA(z) =αz+β for z∈C, whereα, β∈C, α6= 0. In particular,A is holomorphic, and it follows that
f is indeed a Latt`es map.
By Proposition 3.18 only Latt`es-type maps whose orbifolds have signature (2,2,2,2) give genuinely new maps beyond Latt`es maps. We summarize some facts about these maps in the following discussion. For specific maps see Examples 6.15 and 16.8.
Example 3.20 (Latt`es-type maps with signature (2,2,2,2)). We know that each such Latt`es-type map arises from a crystallographic group Gof type (2222) (see Corollary 3.17). In this case, it is elementary to check that the isometries in G remain isometries on R2 not only if we conjugate them by an isometry on
R2, but even if we conjugate them by an affine homeomorphism h: R2 → R2.
It follows that the class of crystallographic groups G of type (2222) is preserved under conjugation by such a homeomorphismh. Similarly, the class of orientation- preserving affine homeomorphismsA: R2 →R2 is preserved under conjugation by
3.4. LATT`ES-TYPE MAPS 73
of Latt`es-type maps with orbifold signature (2,2,2,2) that the underlying lattice Γ ofG is equal to the integer lattice Γ =Z2 and that the groupGconsists of all
isometriesg:R2→R2 of the form
(3.22) u∈R2
7→g(u) =±u+γ, whereγ∈Γ =Z2.
The quotient spaceR2/Gis a 2-sphereS2. Indeed, one can identifyR2/Gwith
a pillow ∆ (see Section A.10) in the following way (the ensuing discussion is closely related to the more general considerations in Section 3.5). Let
R:= [0,1]×[0,1/2],
S := [0,1/2]×[0,1/2], and S′:= [1/2,1]×[0,1/2].
Then R =S∪S′ is a fundamental domain (see Section A.7) for the action of G
on R2, i.e., R contains a representative from every orbit, and this representative
is unique in R if it lies in the interior of R. So R2/G is obtained from R by
identifying certain points on the boundary of R. For this one folds the rectangle R along the line ℓ ={(x, y)∈ R2 : x= 1/2
} and identifies corresponding points on the boundaries of the two squaresS and S′ under this folding operation; so for
example the point (0, t) ∈ R2 is identified with (1, t) ∈ R2 for t ∈ [0,1/2]. The
pillow ∆ obtained in this way is our quotient spaceR2/G.
Let Θ be the map that sends the squareS ⊂R2by the identity map toS ⊂∆.
This maps extends by successive reflections in a natural way to a continuous map Θ : R2
→ ∆. If g ∈G, then Θ maps g(S) by an isometry to one side of ∆, and g(S′) to the other side of ∆. Note that Θ is the same map as in Section 1.1 and corresponds to the quotient mapR2
→R2/G under the identification ∆∼=R2/G.
The map Θ is induced by G, and from the geometric description one easily sees that Θ :R2
→∆∼=R2/Gis a branched covering map (this also follows from
Proposition 3.9). Its critical points are the corners of the squaresg(S) andg(S′),
g ∈G, i.e., the points in 1 2Z
2. We have deg(Θ, u) = #G
u = 2 for u∈ 12Z2 and
deg(Θ, u) = #Gu= 1 foru∈R2\21Z2. Note that the set Θ(12Z2) consists precisely
of the four corners of ∆. We define a ramification function α on ∆ that assigns the value 2 to each of these corners, and 1 to all other points of ∆. In this way we obtain an orbifold (∆, α) with signature (2,2,2,2). If we use some conformal identification ∆∼=Cb (as discussed in Section 1.1), then Θ : R2∼=C
→∆∼=Cb is the (holomorphic) universal orbifold covering map of this orbifold (see Theorem 3.10). Now let A(u) = LA(u) +u0, u ∈ R2, be an orientation-preserving affine ho-
meomorphism. Then A induces a map on the quotient R2/G if and only if A is
G-equivariant, or equivalently ifA◦g◦A−1∈Gfor eachg∈G(see Lemma A.24).
Exactly as in Proposition 3.14, this is the case if and only if 2u0∈Γ andLA(Γ)⊂Γ.
Since Γ =Z2 the latter condition is true precisely if the matrix representing
LAwith respect to the standard basis inR2has integer coefficients. So we conclude
that the homeomorphism A induces a map onS2 =R2/G precisely if A has the
form (3.23) A(u) = a b c d x y +1 2 x0 y0 foru= x y ∈R2,
where a, b, c, d, x0, y0 ∈ Z and det(LA) = ad−bc ≥ 1. Here the last inequality
Letf:S2 → S2 be the branched covering map induced by A as in (3.23). If
det(LA) = 1, thenA−1 also is of the form (3.23), and sof has a continuous inverse
induced byA−1. In this casef:S2→S2is a homeomorphism.
If det(LA) ≥ 2, then deg(f) = det(LA) ≥ 2 by Lemma 3.16. Then f is a
Latt`es-type map, and we know that in this case the orbifold of f has signature (2,2,2,2) (see Corollary 3.17).
In the construction above, we can also use some other branched covering map Θ :R2
→S2induced byG. As follows from the uniqueness part in Proposition 3.9,
this leads to the same class of Latt`es-type maps up to topological conjugacy. By the previous discussion we established the following statement.
Proposition3.21. LetGbe the group consisting of all isometriesg:R2→R2 of the form (3.22), andA:R2→R2be an affine orientation-preserving homeomor- phism as in (3.23).
Then A descends to a map f:S2 → S2 on the quotient R2/G ∼= S2, i.e., if
Θ :R2→R2/G∼=S2 is the quotient map, thenΘ◦A=f◦Θ.
If ad−bc = 1, then f is a homeomorphism that is orientation-preserving. If
ad−bc≥2, thenf is a Latt`es-type map whose orbifold has signature(2,2,2,2). Moreover, every Latt`es-type map with orbifold signature (2,2,2,2) is topologi- cally conjugate to a mapf obtained in this way.
An obvious question is when a Latt`es-type mapf is Thurston equivalent to a rational map. This is always true if the signature ofOf is equal to (2,4,4), (3,3,3),
or (2,3,6), because thenf is even conjugate to a Latt`es map (Proposition 3.18). IfOf has signature (2,2,2,2), the question is answered by the following state-
ment which can be seen as complementary to Thurston’s characterization of rational Thurston maps with hyperbolic orbifold (Theorem 2.18).
Theorem3.22 (Rationality of Latt`es-type maps). Letf:S2→S2be a Latt`es- type map with orbifold signature (2,2,2,2) and A:R2
→R2 be an affine map as in Definition 3.3 with linear partLA. Then f is Thurston equivalent to a rational map if and only ifLAis a real multiple of the identity map onR2or the eigenvalues of LA belong toC\R.
We will not prove this here, but refer to [DH93, Proposition 9.7] for an essen- tially equivalent statement.
In the final part of this section we provide the proof of Proposition 3.6 that characterizes Latt`es-type maps up to Thurston equivalence.
Proof of Proposition 3.6. Suppose first thatf is a Thurston map that is Thurston equivalent to a Latt`es-type mapg. Thenf andg have the same orbifold signature (Proposition 2.15). Since g has a parabolic orbifold and no periodic critical points by Proposition 3.5, the same is true for the mapf as follows from Propositions 2.14 and 2.9 (ii).
For the converse direction, suppose that f is a Thurston map with parabolic orbifoldOf and no periodic critical points. We know that then the signature ofOf
is (2,2,2,2), (2,4,4), (2,3,6), or (3,3,3).
In the last three cases the map has precisely three postcritical points. As we will see later (Theorem 7.2), every Thurston mapf with three postcritical points is Thurston equivalent to a rational map R. Then the signatures of the orbifolds of f and R are the same (Proposition 2.15), and so R is a Thurston map with a
3.4. LATT`ES-TYPE MAPS 75
parabolic orbifoldORand no periodic critical points. By Definition 3.2 the mapR
is a Latt`es map, and the statement follows in this case.
So we are left with the case where Of has signature (2,2,2,2). We may as-
sume that f is defined on Cb. Let α =αf be the ramification function of f, and