PROCESO DE ELABORACIÓN DEL MATERIAL

In this section we will generalise the results of the previous section to non-self maps of compact, simply connected, locally connected sets in the real plane (also known asnon- separating Peano continua). Note that in our terminology simply connected always implies connected.

We denote the Riemann sphere by ˆC. If X ⊂ Cˆ is compact, connected and non-

separating, then its complement U = _Cˆ \ X is simply connected (an open set U ⊂ _Cˆ is simply connected if and only if bothUand ˆC\Uare connected).

The classical Riemann mapping theorem states that ifU _{( C}is non-empty, simply connected and open, then there exists a biholomorphic map fromUonto the open unit disk

D = {z ∈ C : kzk < 1}, known as theRiemann map. We will make use of the following

stronger result by Carath´eodory (see [63, Theorem 17.14]).

Theorem 4.12 (Carath´eodory’s theorem). If U _{( C}ˆ is non-empty, simply connected and open,_Cˆ \U has at least two points, and additionally∂U (or_Cˆ \U) is locally connected, then the inverse of the Riemann map,φ:D →U, extends continuously to a map from the closed unit disk_Donto U.

As in the case of Jordan domains, we will need a requirement on the homeomor- phisms to be orientation-preserving. To make sense of this notion for more general subsets of the plane, we cite the following result, simplified by our assumption of a non-separating and locally connected set (see also Oversteegen and Tymchatyn [77]).

Theorem 4.13(Oversteegen and Valkenburg [78]). Let X ⊂ _Cˆ be compact, simply con- nected and locally connected, and f: X →Y ⊂ _Cˆ a homeomorphism. Then the following are equivalent:

(1) f is isotopic to the identity on X;

(2) there exists an isotopy F: ˆC×[0,1]→Cˆ such that F0=id_Cˆ and F1|X = f ;

(3) f extends to an orientation-preserving homeomorphism of_Cˆ;

(4) if U =_Cˆ \X and V =_Cˆ \Y, then f induces a homeomorphism f from the prime endˆ circle of U to the prime end circle of V which preserves the circular order.

To explain assertion (4) in the above theorem, letφ:D→Uandψ:D→V be the

extended inverse Riemann maps given by Carath´eodory’s theorem, and denoteS1 = ∂_D. Then the homeomorphism ˆf:S1 →S1is said to beinduced by f, if

ψ|_S1◦ fˆ= f|∂U◦φ|S1.

More details can be found in [78], for an introduction to Carath´eodory’s theory of prime ends the reader is referred to Milnor’s book [63, Chapter 17].

Furthermore, we will make use of the following notation: Forε >0, we denote the closedε-neighbourhood of a setX⊂_R2by

Xε={z∈_R2: inf

x∈Xkx−zk ≤ε}.

In the proof of our main theorem, we will considerε-neighbourhoods of disconnected com- pact sets. A key fact we need is that any two given connected components of such a set are separated by theε-neighbourhood of the set, ifε > 0 is chosen sufficiently small. This is the following technical lemma, whose proof can be found in Section 4.4.

Lemma 4.14. Let X be a compact subset of_Rn,n ∈_N. LetC={Ci}i∈I be the collection of

connected components of X and assume|C| ≥ 2. Let C,C0 ∈ Cbe two distinct connected components. Then for ε > 0 sufficiently small, C and C0 lie in two distinct connected components of Xε.

We are now ready to prove our main results.

Proof of Theorem 4.2. LetCbe a connected component ofX∩Ywhich contains a periodic orbit of f. We will prove that f has a fixed point in C. The strategy of the proof is to construct Jordan-domain neighbourhoods ofXandY and to extend f to a homeomorphism between these neighbourhoods, so that Theorem 4.10 can be applied to the extension. The

existence of a fixed point for the extension will then be shown to imply the existence of a fixed point for f inC.

For fixed ε > 0, we first construct a closed neighbourhood Nε ofX with bound- ary Γ = ∂Nε a Jordan curve such that Nε\X has a foliation{γz}z∈Γ with the following

properties:

• eachγzis an arc connecting the pointz∈Γwith a pointx(z)∈∂X;

• γz∩∂X={x(z)}andγz∩Γ ={z}for eachz∈Γ;

• S

z∈Γγz=Nε\X;

• ifz,z0, then eitherγz∩γz0 =∅, orx(z)=x(z0) andγ_z∩γ_z0 ={x(z)} ⊂∂X;

• each arcγzlies within anε-ball centred at its basepointx(z)∈∂X:γz⊂ Bε(x(z)).

Observe that by the last property,Nεis included in theε-neighbourhoodXεofX.

We identifyR2 = CandC ⊂ Cˆ in the usual way and denoteU = Cˆ \X. ThenU

satisfies the hypotheses of Theorem 4.12, so there exists a continuous mapφ: D→U(the

extended inverse Riemann map), whose restriction to the open unit diskDis a conformal

homeomorphism from D to U and φ(∂D) = ∂U. The map φ can be chosen such that

φ(0)=∞. SinceDis compact,φis uniformly continuous, so we can findδ=δ(ε)>0 such

that for allx,y∈_Dwithkx−yk< δ,kφ(x)−φ(y)k< ε.

Assume without loss of generality thatδ <1 and setAδ={x∈_C: 1−δ≤ kxk ≤1} andNε =φ(Aδ)∪X. ThenNεis a closed neighbourhood ofX, whose boundaryΓ =∂Nε= φ({x ∈_D : kxk = 1−δ}) is a Jordan curve. We can then construct the foliation{γz}z∈Γof

Nε\Xby taking the images of radial lines inAδ(see Figure 4.4): γz=φ({r·v: 1−δ≤r ≤1}), wherev=

φ−1_z

kφ−1_zk.

All the required properties ofNεand its foliation{γz}then follow becauseφmaps

Aδ\∂Dbijectively and uniformly continuously ontoNε\X; in particular, the last required

property follows by the choice ofδ=δ(ε).

Next, since f is a homeomorphism,Y = f(X) is simply connected, compact and also locally connected (see [63, Theorem 17.15]). Hence, withV = Cˆ \Y andφ:D →V

the corresponding extended inverse Riemann mapping, we can repeat the same construction as before to obtain a closed neighbourhoodMεofY, such thatΘ =∂Mεis a Jordan curve and{θz}is a foliation ofMε\Xwith the same properties as{γz}.

By choosingδ > 0 small enough such that both constructions work with this given value, we get that Nε = X∪φ(Aδ) and Mε = Y ∪ψ(Aδ). Denote by{ls : s ∈ S1} the

Figure 4.4: (Proof of Theorem 4.2) The extended inverse Riemann mapφ:D→Cˆ \int(X)

maps the annulusAδ={x∈_C: 1−δ≤ kxk ≤1}ontoNε\int(X) such that∂Dis mapped

onto∂X. Each leaf γzof the foliation of Nε\int(X) is the image underφof a radial line

segmentls⊂Aδ.

foliation ofAδby radial lines such thatγz = φ(ls) forz = φ((1−δ)·s) andθz0 = ψ(l_s) for

z0=ψ((1−δ)·s).

From Theorem 4.13 we get thatf induces a homeomorphism ˆf:S1→S1such that ψ|_S1◦ fˆ= f|∂U◦φ|S1.

Hence we can extend f:X →YtoFε:Nε→ Mεby mapping each arcγz ⊂Nε\Xto the

corresponding arcθz0 ⊂ Mε\X. More precisely, letH: A_δ → A_δbe the homeomorphism

which maps the radial line segmentls linearly to the radial line segment lfˆ(s) (that is, in

polar coordinatesH= id|_[1−δ,1]× fˆ). Then define the homeomorphic extension of f toNε by setting Fε(x)=          f(x) ifx∈X, ψ◦H◦φ−1_{(x) ifx∈N}ε_\X.

Note that for 0 < ε0 < ε, we can repeat the above construction to obtain Jordan domains Nε0 andMε0 and a homeomorphismFε0: Nε

0

→ Mε0, and moreoverFε(x) = Fε0(x) for all

x∈Nε0 ⊂Nε.

Every connected component of X ∩Y is a subset of a connected component of Nε∩Mε. LetKεbe the connected component ofNε∩Mεwhich containsC.

Claim 1. Fε0has a fixed point in Kε 0

for everyε0 ∈(0, ε].

This follows from the main result of the previous section: the periodic point for f inCis also a periodic point forFε0inKε

0

. By Theorem 4.10,Fε0has a fixed point inKε 0

.

Claim 2. Fεhas a fixed point in Kε∩(X∩Y).

sequence of fixed points (pk)k∈N such that pk ∈ N

ε/k_∩Mε/k_{. By passing to a convergent}

subsequence, we get thatpk →pask → ∞, withp∈Kε∩(X∩Y) andFε(pk)=Fε/k(pk)=

pk. By continuityFε(p)= p, which proves the claim.

Claim 3. Fεhas a fixed point in C.

By Claim 2 we know thatFε(and henceFε0 forε0∈(0, ε]) possesses a fixed point

p∈Kε∩(X∩Y). Ifp<C, thenplies in another connected component ofKε∩(X∩Y), say

C0. We can now apply Lemma 4.14 toKε∩(X∩Y) and its connected components. Since Nε0 ⊆ Xε0 andMε

0

⊆ Yε0, we obtain that forε0 > 0 sufficiently small, p _< Kε 0

. By again applying Theorem 4.10 and Claim 2, we get thatFε0 has a fixed point p0 ∈ Kε

0

∩(X∩Y), which is also a fixed point forFε.

We can iterate this argument and obtain a sequence ofFε-fixed points (pk)k∈Nsuch

thatpk ∈K1/k∩(X∩Y). As in the proof of the previous claim, by passing to a convergent

subsequence,pk → pask→ ∞, we get anFε-fixed pointp ∈Kε∩(X∩Y). If plies in a

connected component ofKε∩(X∩Y) distinct fromC, then, by Lemma 4.14,p < Kε 0

for sufficiently smallε0. But by construction,p∈Kε0 for allε0 ∈(0, ε], sop∈C, finishing the proof of Claim 3.

Claim 3 implies that f has a fixed point inC, and Theorem 4.2 is proved.

Proof of Corollary 4.3. IfC is a connected component ofX∩Y which contains a point x such that fn(x) ∈ C for all n ∈ _N, then exactly as in the proof of Theorem 4.2, we can construct an extensionF of f to a closed neighbourhoodNεofX, for smallε >0. Using Corollary 4.11 instead of Theorem 4.10, we get that the extension has a fixed point. The same argument then shows that one such fixed point lies inC, as required.