Poincaré theory for decomposable cofrontiers
Abstract
We extend Poincaré’s theory of orientationpreserving homeomorphisms from the circle to circloids with decomposable boundary. As special cases, this includes both decomposable cofrontiers and decomposable cobasin boundaries. More precisely, we show that if the rotation number on an invariant circloid of a surface homeomorphism is irrational and the boundary of is decomposable, then the dynamics are monotonically semiconjugate to the respective irrational rotation. This complements classical results by Barge and Gillette on the equivalence between rational rotation numbers and the existence of periodic orbits and yields a direct analogue to the Poincaré Classification Theorem for circle homeomorphisms. Moreover, we show that the semiconjugacy can be obtained as the composition of a monotone circle map with a ‘universal factor map’, only depending on the topological structure of the circloid. This implies, in particular, that the monotone semiconjugacy is unique up to postcomposition with a rotation.
If, in addition, is a minimal set, then the semiconjugacy is almost onetoone if and only if there exists a biaccessible point. In this case, the dynamics on are almost automorphic. Conversely, we use the AnosovKatok method to build a example where all fibres of the semiconjugacy are nontrivial.
]Poincar’e theory for decomposable cofrontiers 0] ]0
1 Introduction
Given an orientationpreserving circle homeomorphism , the Poincaré Classification Theorem states that the rotation number is welldefined and determines the qualitative dynamical behaviour of , in the sense that is rational if and only if has a periodic orbit and irrational if and only if is monotonically semiconjugate to the respective irrational rotation. This result provides the basis for a rather complete understanding of invertible dynamics on the circle. At the same time the study of cofrontiers, circloids and other classes of circlelike continua, like basin boundaries, cobasin boundaries or pseudocircles, has a long history in plane topology and continuum theory, going back to Kuratowski, Cartwright, Littlewood, Bing and others [kurafrontier, cartwrightlittlewood, Bing1951Pseudocircle]. Recently the topic has gained further momentum, since invariant circloids play a crucial role in surface dynamics. It is therefore a natural question to ask whether an analogue to Poincaré’s classical result holds for these more general continua. In this article, we extend the Poincaré Classification to circloids with decomposable boundary.^{1}^{1}1We recall that a continuum is called decomposable if it can be written as the union of two nonempty proper subcontinua. In order to define this notion, we let and and call a continuum an essential annular continuum if consists of exactly two connected components, both of which are homeomorphic to . Further, is called an essential circloid if it does not contain any strictly smaller essential annular continuum as a subset, and a cofrontier if it is a circloid with empty interior. We refer to Section 2 for further explanations and details.
In the context of the dynamics of surface homeomorphisms, circloids may appear in various situations. For instance, they separate adjacent invariant topological disks or annular domains [BargeGillette1991CofrontierRotationSets, brechneretal, WalkerPrimeEnd, Kennedy1994Pseudocircles, franks/lecalvez:2003], and any periodic point free continuum of a nonwandering surface homeomorphism is an annular continuum [koropecki2010aperiodic] that can further be decomposed into a dense union of invariant circloids and transitive annuli [jaeger:2010a]. On the twotorus, the existence of invariant circloids can often be deduced from information on the rotation set [jaeger:2009c, Davalos2013SublinearDiffusion, GuelmanKoropeckiTal2012Annularity], and invariant “foliations” consisting of circloids play an important role for the problem of linearisation [jaeger:2009b]. The respective results in topological dynamics have further applications in the theory of generic diffeomorphisms [franks/lecalvez:2003, KoropeckiNassiri2010GenericTransitivity, KoroLeCalvezNassIrrPE]. It is thus of vital interest to understand the interplay between the topological structure and the possible dynamical behaviour on such continua. However, while the relation between rational rotation numbers and periodic orbits is quite wellunderstood [cartwrightlittlewood, BargeGillette1991CofrontierRotationSets, KoroLeCalvezNassIrrPE], the more intricate question of irrational rotation factors has been left completely open so far.
The problem is complicated by the fact that the rotation number on invariant circloids is not necessarily unique. Nondegenerate rotation intervals have been shown to occur on the Birkhoff attractor [LeCalvez1988BirkhoffAttractor] and, more recently, the pseudocircle [BoronskiOprocha2014InverseLimitPseudocircle]. Such examples can be excluded by adding a mild recurrence assumption [KoroLeCalvezNassIrrPE], but even in the case of a unique rotation number a semiconjugacy does not have to exist. This was shown by Handel [handel:1982] and Herman [herman:1986], who realised the pseudocircle as a minimal set of a smooth surface diffeomorphism. In these examples, the rotation number is irrational, but the dynamics are not semiconjugate to the corresponding rotation. While the pseudocircle is the paradigm example of a circlelike continuum with highly intricate topological structure, a modification of the construction can be used to produce a variety of more ‘regular’ indecomposable continua with the same behaviour. Hence, decomposability of the circloid presents itself as the obvious minimal requirement for a possible analogue to the Poincaré classification. As the following result shows, it turns out to be sufficient as well. Recall that a monotone map is one with connected fibers.
Theorem 1.1.
Suppose is a homeomorphism homotopic to the identity with an essential invariant circloid with decomposable boundary. Then every point of has a welldefined rotation number which is independent of the point, and

is rational if and only if there is a periodic point in .

is irrational if and only if is monotonically semiconjugate to the corresponding irrational rotation by on .
Barge and Gillette showed that the rotation number on a decomposable cofrontier is always unique, and it is rational if and only if there exists a periodic orbit in the cofrontier [BargeGillette1991CofrontierRotationSets]. Theorem 1.1 complements these results to give a direct analogue to the Poincaré Classification Theorem for decomposable cofrontiers and, more generally, to circloids with decomposable boundary. Thus, dynamics with irrational rotation number on an invariant circloid with decomposable boundary are ‘linearisable’.
As it further turns out, to a great extent this linearisation does not depend on the dynamics. More precisely, there exists a ‘universal factor map’ which maps the circloid to a topological circle and semiconjugates the dynamics of any homeomorphism preserving the circloid to that of a circle homeomorphism.
Theorem 1.2.
Suppose that is an essential circloid with decomposable boundary and there exists a selfhomeomorphism of leaving invariant without periodic points in . Then there exists a continuous and onto map with the following properties.

is monotone and homotopic to the identity;

sends to ;

is injective on ;

for any homeomorphism such that there exists a homeomorphism such that and .

If is any monotone surjection, then there exists a monotone map such that . In particular, if is a homeomorphism leaving invariant and semiconjugates to an irrational rotation , then semiconjugates to (where is as in the previous item).
We give a short selfcontained proof in Section 5. It turns out, however, that the family of subcontinua of given by the fibres of coincides with a decomposition of the circloid constructed already by Kuratowski, in a purely topological context [kurafrontier]. We discuss this in Section LABEL:Kuratowski.
It is wellknown that the semiconjugacies in the Poincaré Classification Theorem are unique up to postcomposition by a rotation. As an immediate consequence of Theorem 1.2, we obtain the same statement for decomposable circloids.
Corollary 1.3.
The semiconjugacy in Theorem 1.1 is unique up to postcomposition by a rotation.
It is worth mentioning that we do not know if an indecomposable cofrontier which is invariant by a homeomorphism may be semiconjugate to an irrational rotation (however, if such an example exists the semiconjugacy cannot be monotone).
As should be expected, additional information on the topological structure of the circloid yields further information on the dynamics. We concentrate on the relation between the existence of biaccessible points and almost automorphic dynamics, whose study is a classical topic in abstract topological dynamics [veech1965almost, ellis1969lectures, auslander1988minimal]. A homeomorphism is almost automorphic if it is semiconjugate to some almost periodic homeomorphism of a space in a way that the set of points of with a unique preimage under the semiconjugation is dense in . The following statement shows how sets of this type appear in surfaces. A point of an essential circloid is called biaccessible if it is the unique intersection point of some arc with such that intersects both components of . An essential cobasin boundary is the boundary of an essential circloid, , and if has a biaccessible point belonging to , we also say that is a biaccessible point of .
Theorem 1.4.
If is an essential cobasin boundary in invariant by a homeomorphism without periodic points and there is a biaccessible recurrent point in , then is almost automorphic.
Corollary 1.5.
Let be a homeomorphism and is an essential invariant continuum such that is minimal. If has a biaccessible point, then is a decomposable cofrontier and is almost automorphic.
We note that if a planar homeomorphism is both almost periodic and minimal on an invariant continuum , then is a simple closed curve [brechneretal].
In [MMO], examples are constructed, using the AnosovKatok method, of decomposable cofrontiers arising as the boundary of Siegel disks, which admit a monotone surjection onto for which there is a subset of point inverses homeomorphic to the product of a cantor set and an interval (in particular, uncountably many fibers are nontrivial). All examples of minimal decomposable cofrontiers that we found in the literature are, to our knowledge, almost automorphic; see for instance [WalkerPrimeEnd, herman:1983]. We close with a construction that is also based on the AnosovKatok method [anosov/katok:1970] and demonstrates that this is not necessarily the case: all fibres may be nontrivial.
Theorem 1.6.
There exists a diffeomorphism leaving invariant a decomposable cofrontier such that

the rotation number on is irrational;

the dynamics on are minimal;

all the fibres of points of of the semiconjugacy given by Theorem 1.2 are nontrivial continua (i.e. not a single point).
In fact, the fibres of points of have a diameter uniformly bounded below by a positive constant (see Claim LABEL:claim:ak2) and, although we do not give a formal proof, it can be seen from the construction that all these fibres can be given a rich topological structure, reminiscent of the Knaster Buckethandle continuum.
We note that due to the nature of the construction, the rotation number of the example from Theorem 1.6 is Liouvillean. We do not know whether such an example exists with a Diophantine rotation number. Similar questions in the context of indecomposable cofrontiers have been raised by in [brechneretal] (see also [turpin]).
Acknowledgements. The authors are grateful to the anonymous referee for the suggestions that helped improve this paper. This work was initiated during the conference ‘Surfaces in São Paulo’, held in São Sebastião, 711 April 2014. TJ would like to thank the organisers and participants for creating this unique opportunity. This cooperation was supported by the BrasilianEuropean exchange program BREUDS (EU Marie Curie Action, IRSES Scheme). TJ acknowledges support of the German Research Council (Emmy Noether Grant Ja 1721/21). AK acknowledges support of CNPqBrasil and FAPERJBrasil.
2 Notation and preliminaries
We denote by the open annulus, and its universal covering map, where is a generator of the group of covering transformations.
A subset of the open annulus is called an essential annular continuum if it is compact and connected and its complement consists of exactly two connected components, both of which are unbounded. Note that in this situation one of the components is unbounded above and bounded below, whereas the other is bounded above and unbounded below, and both of them are homeomorphic to . Moreover, is the decreasing intersection of a sequence of closed annuli. We call an essential circloid if it is a minimal element with respect to inclusion amongst essential annular continua. An essential circloid with empty interior is called an essential cofrontier. The boundary of an essential circloid is called an essential cobasin boundary. It is the intersection of the boundaries of the two complementary components of the circloid and a minimal element with respect to inclusion amongst essential continua. A subset of a surface is called annular continuum (circloid/cofrontier/cobasin boundary) if it has a neighbourhood homeomorphic to such that as a subset of is an essential annular continuum (essential circloid/essential cofrontier/essential cobasin boundary) in the above sense. Note that thus an annular continuum in may be nonessential, in which case it is contained in a closed topological disk. From now on, given any annular continuum, circloid, cobasin boundary or cofrontier, we always identify its annular neighbourhood with and assume implicitly that the objects are essential in .
A closed subset is called horizontal, if there exists with , and horizontally separating if and are contained in different connected components of . It is called a horizontal strip if it separates the plane into exactly two connected components, one of them unbounded above and the other unbounded below. A horizontal strip is called minimal if it does not strictly contain a smaller horizontal strip. In this case, its boundary is called a horizontal coplane boundary and equals the intersection of the boundaries of the two complementary domains of the strip. A horizontal coplane boundary is minimal amongst horizontally separating sets. If is an essential continuum, we call the set its lift. We state the next observation as a lemma, since it will be used repeatedly. Its proof is straightforward and left to the reader.
Lemma 2.1.
The lift of an essential continuum is a minimal strip if and only if is a circloid, and the lift of an essential continuum is a coplane boundary if and only if is a cobasin boundary.
Let be a homeomorphism homotopic to the identity. Any such map lifts to a homeomorphism which commutes with the deck transformation . If is a compact invariant subset of , the rotation interval of on is defined as
(2.1) 
When is reduced to a singleton , we say has a unique rotation number in . In this case, for all . When this happens we say that has a welldefined rotation number .
Given metric spaces , a continuous map is semiconjugate to if there exists a continuous onto map such that . In this situation, we say is a factor of and is a semiconjugacy or factor map. An important case is that of monotone semiconjugacies. A continuous map is called monotone if all fibres , are connected. A set is called saturated with respect to , if implies . If is continuous, then it maps saturated open (closed) sets to open (closed) sets. As a direct consequence, we have
Lemma 2.2.
Preimages of connected sets under surjective monotone maps are connected. In particular, preimages of decomposable sets are decomposable.
A cellular continuum in a surface is one of the form where each is a closed topological disk and . This is equivalent to saying that is a continuum and has a neighborhood homeomorphic to in which is nonseparating.
A partition of a metric space into compact subsets is called an upper semicontinuous decomposition if for each open set , the union of all elements of contained in is also open. A Moore decomposition of a surface is an upper semicontinuous decomposition of into cellular continua. The following version of Moore’s theorem is contained in [daverman, Theorem 25.1] (see also Theorem 13.4 in the same book). It says essentially that the quotient space of a Moore decomposition is the same surface .
Theorem 2.3.
Given any Moore decomposition of a surface , there exists a map which satisfies the following.

is continuous and surjective;

is homotopic to the identity (and preserves orientation if is orientable);

For all , we have .
The map is called the Moore projection associated to .
Finally, we state some basic results from plane topology. We say that a subset of a topological space separates two points if the two points belong to different connected components of .
Lemma 2.4 ([Newman1992PlaneTopology, Theorem 14.3]).
If two points in the plane are separated by a closed set, then they are also separated by some connected component of that set.
Lemma 2.5 ([HockingYoung1961Topology, Theorem 228]).
In any metric space, a continuum is homeomorphic to a circle if for any pair of points of , the set is disconnected.
Lemma 2.6 ([HockingYoung1961Topology, Theorem 216]).
If is a continuum and is closed, then the closure of every connected component of intersects .
3 Minimal generators
Throughout this section, denotes a decomposable cobasin boundary and its lift. If is a continuum such that , we say that is a generator of . We say that is a minimal generator if it does not strictly contain a smaller generator. In the same way we may define generators and minimal generators for lifts of circloids. This concept has been used implicitly by Barge and Gillette in [BargeGillette1991CofrontierRotationSets]; the terminology is taken from [JaegerPasseggi2013THIrrational]. As a consequence of Zorn’s Lemma, any generator contains a minimal generator. The aim of this section is to provide a number of basic facts on minimal generators which will be crucial for the later constructions. The main objective is to derive the statements for circloids, but in order to do so first have to consider cobasin boundaries.
Lemma 3.1.
A continuum is a generator of if and only if .
Proof.
If , then is horizontally separating and by Lemma 2.1 it has to be equal to , so is a generator. To prove the converse, we first note that if is a generator then for some , since otherwise would project injectively onto contradicting the fact that is essential. If we are done; otherwise assume that is maximal with the property that , and note that is horizontally separating and so must be equal to ; in particular it contains . But for and due to the maximality of . Thus , and since is compact, cannot be contained in , so as claimed. ∎
Lemma 3.2.
Suppose that and are closed connected subsets of , with unbounded to the left and unbounded to the right. If , then is connected, and if then .
Proof.
If , then is horizontally separating, so by Lemma 2.1 it must be equal to . Assume that . Let and be the connected components of which are unbounded below and above, respectively, so . Note that cannot be horizontally separating (since this would contradict Lemma 2.1); thus and are contained in the same connected component of . Since , it follows that . Note that is simply connected, since both and are connected and unbounded. Moreover, is a closed subset in the topology of , and since separates from in it follows that separates from in . By Lemma 2.4 applied to , some connected component of separates from in . Since is closed in , we have that is closed and horizontally separating, so by Lemma 2.1 it must be equal to . This implies that , so is connected. ∎
Lemma 3.3.
There exists a minimal generator such that if and only if . Moreover, and are connected and dense in .
Proof.
Since is decomposable, there exists a decomposition into proper subcontinua. As is a cobasin boundary, both and must be inessential in , which implies that there are open topological disks and . Let be a connected component of , and . Let be a connected component of , and . The sets and project injectively onto and , respectively, so they are continua and for all , and similarly for . Since , there exists such that . The set is a generator, so it contains some minimal generator . Let be the largest integer such that , and suppose for a contradiction that . Then is closed and horizontally separating, so in particular it contains . Since is disjoint from for all , it follows that , which implies that , contradicting our choice of and .
By Lemma 3.2 we have that is connected. Note that the closure of any connected component of intersects (see Lemma 2.6), so any connected component of must contain (otherwise it would be contained in which is disjoint from ), so there is only one such component. Thus is connected. Since , it follows that is a generator and by minimality . This implies that has empty interior in the restricted topology to , and therefore is also dense in , completing the proof. ∎
Lemma 3.4.
If is any minimal generator of , then if and only if . Moreover, and are connected and dense in .
Proof.
Suppose that for some , so there exists such that . Let be as in Lemma 3.3. Replacing by for a suitable , we may assume that . This means that intersects and , where . Thus the set is closed, connected and horizontally separating, and by Lemma 2.1 it should be equal to . Thus , implying that . By minimality , contradicting the fact that with .
Knowing that , the remaining claims are proved exactly as in the last paragraph of the proof of Lemma 3.3. ∎
Given a minimal generator of , let and . With these notions, we have
Lemma 3.5.
If and are two different minimal generators of , then either or .
Proof.
Corollary 3.6.
If and are minimal generators of , then is contained in two adjacent copies of and vice versa.
A cut (of ) is a set of the form where is a minimal generator of . We denote by the family of all cuts. Note that by Lemma 3.5, cuts are pairwise disjoint.
Given a cut , we let and , so that . Further, we let and . We write if , or equivalently if . By Lemma 3.5 and its corollary, defines a total order in . We extend this notation to compare arbitrary subsets with cuts by writing if and if . If and , we simply write . For two cuts , we let and .
We note that cuts need not be connected. However, we have:
Lemma 3.7.
Given two cuts , the set is connected and its closure is .
Proof.
Let be the essential circloid such that (i.e. is the union of with all bounded connected components of ), and let be its lift. A generator of is a continuum which satisfies . In order to go over from a the decomposable cobasin boundary to the corresponding circloid , the following statements will be crucial.
Lemma 3.8.
All connected components of are topological disks with diameter bounded by a uniform constant .
Proof.
Let be a generator of , and suppose . Let be such that . Since , the latter set has diameter bounded by some constant . let . If is a connected component of with , then and we may assume replacing by for an appropriate . Thus there is a simple arc such that , , and for . If and , we have that has exactly two connected components and , the former unbounded above and the later unbounded below. The set being connected, disjoint from and contained in , must lie entirely in or . Suppose without loss of generality that . Note that and is is bounded above. If is the smallest real such that , then is disjoint from and thus contained in , and since there must exist such that . Since when , it follows that , but this is not possible since . ∎
Given a continuum , the complement consists of one unbounded component and a union of topological disks. We denote the unbounded component by and the family of disks by and let . Note that is a nonseparating continuum.
Lemma 3.9.
Suppose is a connected component of and are cuts such that . If are cuts with , then .
Proof.
Suppose for a contradiction that . Assume without loss of generality that . Since is bounded, there is such that , and we may assume and . The sets and are connected, and is also connected (by Lemma 3.7). Since is not contained in either set or , we have that . But then by Lemma 2.4 we have that , contradicting the fact that is bounded and . ∎
4 Dynamical linearisation: Proof of Theorem 1.1
Throughout this section, we assume that is a homeomorphism homotopic to the identity and is a invariant circloid with decomposable boundary. We let and denote the lifts of and by and , respectively. The next result generalizes [BargeGillette1991CofrontierRotationSets, Theorem 2.7] to circloids.
Lemma 4.1.
The rotation number exists and is independent of .
Proof.
The fact that exists for some follows from the Birkhoff Ergodic Theorem and the existence of an invariant measure for , since is a Birkhoff average for the function , where is arbitrary.
Lemma 4.2.
if and only if there exists such that .
Proof.
The ifpart is trivial. For the other implication, note that it is easy to verify that if and only if , so it suffices to assume that and show that there is a fixed point in . Fix a minimal generator of . We claim that for all . Indeed, fix a positive integer . Corollary 3.6 implies that for some . If , then which then implies and this implies that for all and , contradicting our assumption. If , we get a similar contradiction. Thus , and since and are minimal generators, must intersect both and , so as claimed. Thus is connected and bounded (again due to Corollary 3.6), and . Moreover, and , so is a nonseparating invariant continuum in and the generalization of the CartwrightLittlewood theorem due to Bell [bell] implies that contains a fixed point of . ∎
The previous lemma implies the first item from Theorem 1.1. It also follows that if is monotonically semiconjugate to an irrational rotation then is irrational. Thus to complete the proof of the theorem it remains to prove that if is irrational, then is monotonically conjugate to the corresponding irrational rotation.
For the remainder of this section we assume that is irrational, and we fix a minimal generator of . Given in , we let and denote by the cut corresponding to . Recall that there is a linear ordering on the set of cuts as defined in the previous section.
Lemma 4.3.
The mapping is strictly monotonically increasing. In particular, if . Moreover, the set is connected for all , decreasing in and increasing in .
Proof.
Suppose that , but . Then . As a consequence, all orbits in under the lift of are bounded to the right, contradicting the fact that . This shows the strict monotonicity of and the disjointness. Connectedness of is given by Lemma 3.7. ∎
As in the previous section, given and a cut , we write iff and iff . In order to extend this notion to all , note that is a union of bounded open topological disks whose boundary is contained in . Given , we denote the respective disk containing by and write iff and iff . Equivalently, iff and iff . Given a subset , we write iff and iff . Then, we define by
(4.1) 
Lemma 4.4.
The map is continuous and projects to a monotone semiconjugacy from to the irrational rotation by .
Proof.
We first show the continuity of the restriction of to . By definition, we have that
(4.2) 
Since the sets are relatively open in , this shows that preimages of open sets are open, so that is continuous.
In order to see that is continuous on all of , fix and . It suffices show to that contains a neighbourhood of . If , then by definition the whole open disk is contained in . (Note that if , then for all .) Thus, suppose that . Since is continuous, there exists such that . If , then intersects both and . Consequently, intersects and we have . However, this implies that for some with we have . Therefore Lemma 3.9 yields that for any with and thus . Altogether, we obtain , which proves the continuity of on .
In order to show the further statements, note that since by definition and , the same relations hold for and . Using these facts, it is easy to check that is a semiconjugacy from to the translation on and that commutes with the deck translation , such that projects to a semiconjugacy from to the rotation .
It remains to prove the monotonicity of , which will follow immediately from that of . We have that
(4.3) 
This can be seen as a nested intersection of continua and is therefore a continuum itself. The full fibre is obtained by adding the union of topological open disks to . However, if , then cannot intersect for any , since otherwise Lemma 3.9 would imply that for some and thus . Similarly, is disjoint from for all , and therefore . Hence, we have that is the ‘fillin’ of a continuum, and hence a continuum itself. ∎
This concludes the proof of Theorem 1.1.
5 Topological linearisation: a universal factor map
The main goal of this section is to prove Theorem 1.2. We refer the reader to Section LABEL:Kuratowski for a discussion on the relationship between the content presented here and the work of Kuratowski.
As before, we suppose is a circloid with decomposable boundary, and . We let and . Our aim is to define a Moore decomposition of such that the corresponding projection maps to a topological circle. As before, we start by decomposing , and we use the family of cuts of as the main tool.
However, cuts need not be connected, and moreover is easy to give examples where does not cover all of . In order to obtain a decomposition starting from , we define a strong partial order relation on by writing if and only if there exist uncountably many cuts such that . Similar to before, we extend this definition to arbitrary subsets by writing whenever there exist uncountably many cuts with .^{2}^{2}2We note that the requirement of uncountably many intermediate cuts is crucial for the whole construction in this section. We do not elaborate further on this, but just mention that a examples demonstrating why requiring uncountably many intermediate cuts is necessary can be produced by gluing finitely or countably many pseudoarcs together. Otherwise, a statement analogous to Lemma 5.1 does not hold. In case of onepoint sets , we write instead of . Then, given any , we define
(5.1) 
We let and call the elements fibres of . Further, we let . We note that the intersection in (5.1) can be viewed as a nested intersection of continua: by compactness, for every there exist such that . Without loss of generality we may assume that and for all , and we have . By Lemma 3.7 this is a nested intersection of continua, hence a continuum.
Lemma 5.1.
If and , then there exists such that .