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Fixed Point Theorem for Non-Self Maps of Regions in the Plane

Let X and Y be compact, simply connected and locally connected subsets of R^2, and let f : X ->Y be a homeomorphism isotopic to the identity on X. Generalizing Brouwer's plane translation theorem for self-maps of the plane, we prove that f has no recurrent (in particular, no periodic) points if it has no fixed points.


Introduction and Statement of the Result
Since Brouwer's proof of his plane translation theorem [4], many alternative proofs of the theorem and its key ingredient, the translation arc lemma, have been given (for several more recent ones, see Brown [5], Fathi [7], Franks [8]). The following is a concise formulation of its main statement.
Theorem A (Barge and Franks [1]). Suppose f : R 2 → R 2 is an orientation-preserving homeomorphism of the plane. If f has got a periodic point then it has got a fixed point.
Slightly stronger versions assume a weaker form of recurrence, for example the existence of periodic disk chains (Barge and Franks [1]), to obtain the existence of fixed points.
In this paper we consider homeomorphisms between compact sets in the plane and obtain a statement analogous to Theorem A in which we only make certain topological assumptions on the domain of the map. Our main theorem is the following: Theorem 1.1. Let X ⊂ R 2 be a compact, simply connected, locally connected subset of the real plane and let f : X → Y ⊂ R 2 be a homeomorphism isotopic to the identity on X. Let C be a connected component of X ∩ Y. If f has got a periodic orbit in C, then f also has got a fixed point in C.
In fact, as a corollary we will obtain a slightly stronger result: Corollary 1.2. Let X ⊂ R 2 and f : X → Y be as in Theorem 1.1 and let C be a connected component of X ∩ Y. If f has got no fixed point in C, then the orbit of every point x ∈ C eventually leaves C, i.e. there exists n = n(x) ∈ N, such that f n (x) C.
In particular, if X ∩ Y is connected and f has got no fixed points, then the non-escaping set of f is empty: {x ∈ X : f n (x) ∈ D ∀n ∈ N} = ∅.
To prove these results, in Section 2 we will first consider the case of an orientationpreserving homeomorphism f : D → E ⊂ R 2 of a Jordan domain D into the plane. In this special case, the statement will be first proved for D ∩ E connected, by suitably extending f to the real plane and applying Theorem A. We will then proceed to show that the connectedness assumption can be lifted if one formulates the result more precisely, taking into account the connected components of D ∩ E individually. Finally, in Section 3, we will deduce the general case of Theorem 1.1 by reducing the problem to the Jordan domain case.
In Section 4 we will discuss the assumptions of our results and questions about possible extensions.

Non-self maps of Jordan domains
A set D ⊂ R 2 is a Jordan domain, if it is a closed set with boundary ∂D a simple closed curve (Jordan curve). By Schoenflies' theorem, Jordan domains are precisely the planar regions homeomorphic to the (closed) disk. In this paper we assume that all Jordan curves are endowed with the counter-clockwise orientation. For a Jordan curve C and x, y ∈ C, we denote by (x, y) C (resp. [x, y] C ) the open (resp. closed) arc in C from x to y according to this orientation.
For X ⊂ R 2 , we say that x ∈ X is a fixed point for the map f : In analogy to Theorem A, we will prove the following: (1) There is a partition • I, J are non-empty, at most countable sets, We consider a Jordan domain D ⊂ R 2 and a homeomorphism f : If D ∩ E is connected, it follows that both D ∩ E and D ∪ E are Jordan domains (see Fig.1). With Proposition 2.3 in mind, the proof of the following lemma is straightforward; it can be found in the appendix. The partition of R 2 \int(D∩ E) obtained this way allows us to prove our key proposition, from which the main result of this section will follow as a corollary. Proof. Since f has got no fixed points, by Brouwer's Fixed Point Theorem we get that E D and D E. Therefore we can apply Lemma 2.4 to obtain a partition P of R 2 \ int(D ∩ E) into arcs, each connecting a point in ∂(D ∩ E) with ∞ and intersecting each of ∂D and ∂E in exactly one point.
For x ∈ R 2 \ int(D ∩ E), we denote by L x ∈ P the partition element containing the point x and let π D (x) = L x ∩ ∂D, π E (x) = L x ∩ ∂E. Denote by l D (x) and l E (x) the lengths of the arcs [π D (x), x] and [π E (x), x] as subarcs of L x , respectively.
We now construct F : R 2 → R 2 as an extension of f in such a way, that each arc L x \ D, x ∈ ∂D, is mapped into the arc L f (x) . For x ∈ D we define F(x) = f (x) and for x D we define F(x) to be the unique point y ∈ L f (π D (x)) \ E such that l E (y) = l D (x) (see Fig. 2).
The map F is an orientation-preserving homeomorphism of R 2 ; its inverse can be obtained the same way by swapping the roles of D, f and E, f −1 . Moreover, F has got no fixed points in D since , and therefore f (π D (p)) = π D (p), a contradiction. Hence F is fixed point free.
Combining Proposition 2.5 and Theorem A, we now immediately obtain Theorem 2.1.
Next, we formulate a somewhat more general form of Theorem 2.1, by lifting the restriction of D ∩ f (D) to be connected, and instead making an assertion for each individual connected component of this set. This version of the result turns out to be much more useful in its application to more general compact sets in Section 3.   Proof. We show that this more general setting can be reduced to the one in Theorem 2.1.
Each C i is a Jordan domain bounded by the union of γ i and a closed subarc of ∂E. Let {C i } be another collection of disksC i ⊂ R 2 \ D, such thatC i ∩ D = γ i and theC i are pairwise disjoint. (Note that by Schoenflies' theorem we can assume without loss of generality D to be the closed standard unit disk D = {(x, y) ∈ R 2 : x 2 + y 2 < 1}. Since the γ i are pairwise disjoint, lettingC i be the round halfdisk over ∂D centred at the midpoint of γ i gives a collection of pairwise disjoint disks as required, see Fig.3.) By Schoenflies' theorem, for each i there exists a homeomorphism g i : C i →C i and we can assume that g i is the identity on γ i . Noting that and define a homeomorphism g : E →Ẽ by gluing together the identity on C and the homeomorphisms g i : Proof. By repeating verbatim the argument in the proof of Theorem 2.6, we can construct a homeomorphismf : D →Ẽ, which coincides with f on C and such thatẼ ∩ D = C.
Observe thatf is fixed point free, so we can apply Proposition 2.5 to get an extension off to a fixed point free orientation-preserving homeomorphism F : for all n ∈ N and the forward orbit of x has an accumulation point, F n k (x) → x 0 ∈ C as k → ∞. A sufficiently small neighbourhood U of x 0 then satisfies F(U) ∩U = ∅ but clearly F n k+1 −n k (U) ∩ U ∅ for sufficiently large k, contradicting (2.1). Hence such point never escaping C does not exist.

Non-self Maps for Compact Simply Connected Planar Sets
In this section we will generalize the results of the previous section to non-self maps of compact, simply connected and locally connected sets in the real plane (also known as nonseparating Peano continua). Note that in our terminology simply connected always implies connected.
We denote the Riemann sphere byĈ. If X ⊂Ĉ is compact, connected and nonseparating, then its complement U =Ĉ\X is simply connected (an open set U ⊂Ĉ is simply connected if and only if both U andĈ \ U are connected).
The classical Riemann mapping theorem states that if U C is non-empty, simply connected and open, then there exists a biholomorphic map from U onto the open unit disk D = {z ∈ C : z < 1}, known as the Riemann map. We will make use of the following stronger result by Carathéodory (see [13,Theorem 17.14]).

Theorem B (Carathéodory's Theorem). If U
Ĉ is non-empty, simply connected and open,Ĉ \ U has at least two points, and additionally ∂U (orĈ \ U) is locally connected, then the inverse of the Riemann map, φ : D → U, extends continuously to a map from the closed unit disk D onto U.
As in the case of Jordan domains, we will need a requirement on the homeomorphisms to be orientation-preserving. To make sense of this notion for more general subsets of the plane, we cite the following result by Oversteegen and Valkenburg [15], simplified by our assumption of a nonseparating and locally connected set (see also Oversteegen and Tymchatyn [14]).
Theorem C. Let X ⊂Ĉ be compact, simply connected and locally connected, and f : X → Y ⊂Ĉ a homeomorphism. Then the following are equivalent: (1) f is isotopic to the identity on X; (2) there exists an isotopy F :Ĉ × [0, 1] →Ĉ such that F 0 = idĈ and F 1 | X = f ; (3) f extends to an orientation-preserving homeomorphism ofĈ; (4) if U =Ĉ \ X and V =Ĉ \ Y, then f induces a homeomorphismf from the prime end circle of U to the prime end circle of V which preserves the circular order.
To explain assertion (4) in the above theorem, let φ : D → U and ψ : D → V be the extended inverse Riemann maps given by Carathéodory's theorem, and denote S 1 = ∂D. Then the homeomorphismf : More details can be found in [15], for an introduction to Carathéodory's theory of prime ends the reader is referred to Milnor's book [13,Chapter 17].
Furthermore, we will make use of the following notation: For ε > 0, denote the closed ε-neighbourhood of a set X ⊂ R 2 by In the proof of our main theorem, we will consider ε-neighbourhoods of disconnected compact sets. One key fact we will need is that any two given connected components of such a set are separated by the ε-neighbourhood of the set, if ε > 0 is chosen sufficiently small. This is the following technical lemma, whose proof can be found in the appendix. Lemma 3.1. Let X be a compact subset of R n , n ∈ N. Let C = {C i } i∈I be the collection of connected components of X and assume |C| ≥ 2. Let C, C ′ ∈ C be two distinct connected components. Then for ε > 0 sufficiently small, C and C ′ lie in two distinct connected components of X ε .
We are now ready to prove our main results.
Proof of Theorem 1.1. Let C be a connected component of X ∩Y which contains a periodic orbit of f . We will prove that f has got a fixed point in C. The strategy of the proof is to construct Jordan-domain neighbourhoods of X and Y and to extend f to a homeomorphism between these neighbourhoods, so that Theorem 2.6 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 in C.
For fixed ε > 0, we first construct a closed neighbourhood N ε of X with boundary Γ = ∂N ε a Jordan curve, such that N ε \ X has a foliation {γ z } z∈Γ with the following properties: • each γ z is an arc connecting the point z ∈ Γ with a point x(z) ∈ ∂X; • γ z ∩ ∂X = {x(z)} and γ z ∩ Γ = {z} for each z ∈ Γ; • each arc γ z lies within an ε-ball centered at its basepoint x(z) ∈ ∂X: γ z ⊂ B ε (x(z)). Observe that by the last property, N ε is included in the ε-neighbourhood X ε of X.
We identify R 2 = C and C ⊂Ĉ in the usual way and denote U =Ĉ \ X. Then U satisfies the hypotheses of Theorem B, so there exists a continuous map φ : D → U (the extended inverse Riemann map), whose restriction to the open unit disk D is a conformal homeomorphism from D to U and φ(∂D) = ∂U. The map φ can be chosen such that φ(0) = ∞. Since D is compact, φ is uniformly continuous, so we can find δ = δ(ε) > 0, such that for all x, y ∈ D with x − y < δ, φ(x) − φ(y) < ε.
Assume without loss of generality that δ < 1 and set A δ = {x ∈ C : 1 − δ ≤ x ≤ 1} and N ε = φ(A δ ) ∪ X. Then N ε is a closed neighbourhood of X, whose boundary Γ = ∂N ε = φ({x ∈ D : x = 1 − δ}) is a Jordan curve. We can then construct the foliation {γ z } z∈Γ of N ε \ X by taking the images of radial lines in A δ (see Fig.4): All the required properties of N ε and its foliation {γ z } then follow because φ maps A δ \∂D bijectively and uniformly continuously onto N ε \ X; in particular the last required property follows by the choice of δ = δ(ε).
Next, since f is a homeomorphism, Y = f (X) is simply connected, compact and also locally connected (see [13,Theorem 17.15]). Hence, with V =Ĉ \ Y and φ : D → V the corresponding extended inverse Riemann mapping, we can repeat the same construction as before to obtain a closed neighbourhood M ε of Y, such that Θ = ∂M ε is a Jordan curve and {θ z } is a foliation of M ε \ X with the same properties as {γ z }.
By choosing δ > 0 small enough, such that both constructions work with this given value, we get that N ε = X ∪ φ(A δ ) and M ε = Y ∪ ψ(A δ ). Denote by {l s : s ∈ S 1 } the foliation of A δ by radial lines, such that γ z = φ(l s ) for z = φ((1 − δ) · s) and θ z ′ = ψ(l s ) for From Theorem C we know that f induces a homeomorphismf : S 1 → S 1 , such that Hence we can extend f : X → Y to F ε : N ε → M ε by mapping each arc γ z ⊂ N ε \ X to the corresponding arc θ z ′ ⊂ M ε \ X. More precisely, let H : A δ → A δ be the homeomorphism which maps the radial line segment l s linearly to the radial line segment lf (s) (i.e., in polar coordinates H = id | [1−δ,1] ×f ). Then define the homeomorphic extension of f to N ε by setting Note that for 0 < ε ′ < ε, we can repeat the above construction to obtain Jordan domains N ε ′ and M ε ′ and a homeomorphism F ε ′ : This follows directly from the main result of the previous section: the periodic point for f in C is also a periodic point for F ε ′ in K ε ′ . By Theorem 2.6, F ε ′ has a fixed point in K ε ′ .
By Claim 1 and the fact that for 0 < ε ′ < ε, F ε ′ is the restriction of F ε , F ε has got a sequence of fixed points (p k ) k∈N , such that p k ∈ N ε/k ∩ M ε/k . By passing to a convergent subsequence, we get that p k → p as k → ∞, with p ∈ K ε ∩(X ∩Y) and F ε (p k ) = F ε/k (p k ) = p k . By continuity F ε (p) = p, which proves the claim.

Claim 3. F ε has got a fixed point in C.
By Claim 2 we know that F ε (and hence F ε ′ for ε ′ ∈ (0, ε]) possesses a fixed point p ∈ K ε ∩ (X ∩ Y). If p C, then p lies in another connected component of K ε ∩ (X ∩ Y), say C ′ . We can now apply Lemma 3.1 to K ε ∩(X ∩Y) and its connected components. Since N ε ′ ⊆ X ε ′ and M ε ′ ⊆ Y ε ′ , we obtain that for ε ′ > 0 sufficiently small, p K ε ′ . By again applying Theorem 2.6 and Claim 2, we get that F ε ′ has a fixed point p ′ ∈ K ε ′ ∩ (X ∩ Y), which is also a fixed point for F ε .
We can iterate the argument and obtain a sequence of F ε -fixed points (p k ) k∈N , such that p k ∈ K 1/k ∩ (X ∩ Y). As in the proof of the previous claim, by passing to a convergent subsequence, p k → p as k → ∞, we get an F ε -fixed point p ∈ K ε ∩ (X ∩ Y). If p lies in a connected component of K ε ∩ (X ∩ Y) distinct from C, then, by Lemma 3.1, p K ε ′ for sufficiently small ε ′ . But by construction, p ∈ K ε ′ for all ε ′ ∈ (0, ε], so p ∈ C, finishing the proof of Claim 3.
Claim 3 implies that f has a fixed point in C, and Theorem 1.1 is proved.
Proof of Corollary 1.2. If C is a connected component of X ∩ Y which contains a point x such that f n (x) ∈ C for all n ∈ N, then exactly as in the proof of Theorem 1.1, we can construct an extension F of f to a closed neighbourhood N ε of X, for ε > 0 sufficiently small. Using Corollary 2.7 instead of Theorem 2.6, we get that the extension has got a fixed point. The same argument then shows that one such fixed point lies in C, finishing the proof of the corollary.

Discussion of Assumptions and Possible Extensions
Easy examples show that in our main results we cannot omit the hypothesis of a periodic orbit being completely contained in a given connected component of X ∩ f (X): take X to be a closed ε-neighbourhood of the straight line segment S = [−1, 1] × {0} and let f : X → Y be a homeomorphism which maps S to the semicircle {(x, y) : x 2 + y 2 = 1, y ≥ 0} with f (−1, 0) = (1, 0) and f (1, 0) = (−1, 0). For ε > 0 small, f has got a period-two orbit (spread over two different connected components of X ∩ f (X)) but no fixed point.
Furthermore, obvious counterexamples show that our main result, Corollary 1.2, fails when the domain X ⊂ R 2 of the homeomorphism f is not compact. Also, we are not aware of generalizations to higher dimensions; in fact, simple counterexamples to Theorem 1.1 in its current form can be constructed, when X ⊂ R n is an n-dimensional ball with n ≥ 3. Bonino [2] showed that if an orientation-preserving homeomorphism f : R 2 → R 2 possesses non-escaping points in a topological disk D bounded by a simple closed curve C, and C can be split into two arcs (Fig.5(A)), then f has got a fixed point in D. Bonino's theorem is similarly topological, but does not imply (and is not implied by) our results.
Another result related to Theorem 1.1 is due to Brown [6,Theorem 5]. Brown assumed that for the n-periodic point x ∈ X there is a connected neighbourhood W of {x} ∪ { f (x)} with f i (W) ⊂ X for i = 1, . . . , n, and showed that f then has a fixed point in X.
A somewhat different class of results relates to the question of period forcing. Gambaudo et al [9] showed that a C 1 orientation-preserving embedding of the disk into itself has periodic orbits of all periods, if it has a period-three orbit of braid type different than the rotation by angle 2π/3 (see also Kolev [11] for a topological version of this result). For f (X) X, this statement is false: a counterexample is a version of the Smale horseshoe map, where a period-two point is removed from the disc together with a narrow strip such that the remaining domain X is still simply connected (see Fig.5(B)). Then f , the restriction of the horseshoe map to X, is an orientation-preserving homeomorphism with X ∩ f (X) connected, and one can choose the removed strip narrow enough such that f still has got period-three orbits but no period-two orbit in X. We conclude with the following question: Question. Can one find conditions on X and f (X), under which every extension of f to a homeomorphism of R 2 has periodic orbits of every period passing through X, whenever f has got a period-three point in X (of braid type different than the rotation by angle 2π/3)?
The method of extending a non-self map of a compact set to a self-map of R 2 or a disk without additional fixed (or periodic) points, could possibly also be applied to this problem. Note that D∪E is also a Jordan domain and that the (a i , b i ) ∂E , i ∈ I, and (c j , d j ) ∂D , j ∈ J, together with ∂D ∩ ∂E form a partition of its boundary. Thus we obtained a collection of (pairwise disjoint) closed arcs, each connecting precisely one point of ∂(D∩ E) to precisely one point of ∂(D ∪ E). Denote the arc corresponding to z ∈ ∂(D ∪ E) by l z , and let l z = {z} whenever z ∈ ∂D ∩ ∂E.
Since D ∪ E is a Jordan domain, we can again apply Schoenflies' theorem to obtain a homeomorphism h : Let r θ be the radial line segment in R 2 \ D expressed in polar coordinates as {(r, φ) : r ≥ 1, φ = θ}. Then {r θ : θ ∈ [0, 2π)} forms a partition of R 2 \ D, which can be pulled back to a partition {h −1 (r θ ) : θ ∈ [0, 2π)} of R 2 \ int(D ∪ E). This partition consists of (pairwise disjoint) arcs m z , each connecting a point on z ∈ ∂(D ∪ E) to ∞.
Combining the above, we get that each point z ∈ ∂(D∪E) is the endpoint of two uniquely defined arcs l z and m z (l z possibly being the trivial arc {z}). Let L z ≔ l z ∪ m z , which is an arc connecting a point on ∂(D ∩ E) to ∞, for every z ∈ ∂(D ∪ E). Then {L z : z ∈ ∂(D ∪ E)} is a partition of R 2 \ int(D ∩ E) with the desired properties.
Proof of Lemma 3.1. We denote the minimal distance between any two points of two disjoint sets U, V ⊂ R n by d(U, V) = inf x∈U, y∈V x − y .
(4.1) (Note that (4.1) implies that (C ε i ) ε ∩ (C ε i+1 ) ε ∅.) Now for k ∈ N let ε k = 1/k and define We use the fact that the space of all non-empty compact subsets of a compact complete metric space, equipped with the Hausdorff distance, forms itself a compact complete metric space. Since (S k ) k∈N is a sequence of non-empty compact subsets of X ⊂ R n , it has a convergent subsequence and by passing to this subsequence we can assume S k → S as k → ∞, for some non-empty compact set S ⊆ X.
Claim. C ⊂ S , C ′ ⊂ S , and S is connected.
The inclusions follow immediately from the fact that C ⊂ S k and C ′ ⊂ S k for all k ∈ N. Further, suppose S is disconnected. Then S can be written as S 1 ∪ S 2 , where S 1 and S 2 are non-empty disjoint open sets in X, i.e., there exist disjoint open sets U 1 , U 2 ⊂ R n , such that S 1 = U 1 ∩ X and S 2 = U 2 ∩ X. Note that by compactness of S , the sets U 1 and U 2 can be chosen such that U 1 ∩ U 2 = ∅, hence Since S is contained in the open set U 1 ∪ U 2 , S k → S implies that S k ⊂ (U 1 ∪ U 2 ) for k ∈ N sufficiently large. But by (4.2) the set U 1 ∪ U 2 has a 'gap' of size ≥ d, whereas the set S k admits no such 'gaps' wider than 2/k by (4.1). For k ≥ 2/d this is a contradiction, and the claim is proved.
We get that S is a connected subset of X containing both C and C ′ , which contradicts the assumption that C and C ′ are distinct connected components of X. This finishes the proof of the lemma.