Fibers and local connectedness of planar continua
aa r X i v : . [ m a t h . GN ] M a r FIBERS AND LOCAL CONNECTEDNESS OF PLANARCONTINUA
BENOˆIT LORIDANT AND JUN LUO
Abstract.
We describe non-locally connected planar continua via theconcepts of fiber and numerical scale.Given a continuum X ⊂ C and x ∈ ∂X , we show that the set of points y ∈ ∂X that cannot be separated from x by any finite set C ⊂ ∂X isa continuum. This continuum is called the modified fiber F ∗ x of X at x . If x ∈ X o , we set F ∗ x = { x } . For x ∈ X , we show that F ∗ x = { x } implies that X is locally connected at x . We also give a concrete planarcontinuum X , which is locally connected at a point x ∈ X while thefiber F ∗ x is not trivial.The scale ℓ ∗ ( X ) of non-local connectedness is then the least integer p (or ∞ if such an integer does not exist) such that for each x ∈ X thereexist k ≤ p + 1 subcontinua X = N ⊃ N ⊃ N ⊃ · · · ⊃ N k = { x } such that N i is a fiber of N i − for 1 ≤ i ≤ k . If X ⊂ C is an unshieldedcontinuum or a continuum whose complement has finitely many com-ponents, we obtain that local connectedness of X is equivalent to thestatement ℓ ∗ ( X ) = 0.We discuss the relation of our concepts to the works of Schleicher(1999) and Kiwi (2004). We further define an equivalence relation ∼ based on the fibers and show that the quotient space X/ ∼ is a locallyconnected continuum. For connected Julia sets of polynomials and moregenerally for unshielded continua, we obtain that every prime end im-pression is contained in a fiber. Finally, we apply our results to examplesfrom the literature and construct for each n ≥ X n with ℓ ∗ ( X n ) = n . Introduction and main results
Motivated by the construction of Yoccoz puzzles used in the study onlocal connectedness of quadratic Julia sets and the Mandelbrot set M , Schle-icher [11] introduces the notion of fiber for full continua (continua M ⊂ C having a connected complement C \ M ), based on “separation lines” chosenfrom particular countable dense sets of external rays that land on points of M . Kiwi [7] uses finite “cutting sets” to define a modified version of fiberfor Julia sets, even when they are not connected. Mathematics Subject Classification.
Key words and phrases.
Local connectedness, fibers, numerical scale, upper semi-continuous decomposition.
Jolivet-Loridant-Luo [5] replace Schleicher’s “separation lines” with “goodcuts”, i.e. , simple closed curves J such that J ∩ ∂M is finite and J \ M = ∅ .In this way, Schleicher’s approach is generalized to continua M ⊂ C whosecomplement C \ M has finitely many components. For such a continuum M ,the pseudo-fiber E x (of M ) at a point x ∈ M is the collection of the points y ∈ M that cannot be separated from x by a good cut; the fiber F x at x is the component of E x containing x . Here, a point y is separated from apoint x by a simple closed curve J provided that x and y belong to differ-ent components of C \ J . And x may belong to the bounded or unboundedcomponent of C \ J .Clearly, the fiber F x at x always contains x . We say that a pseudo-fiberor a fiber is trivial if it coincides with the single point set { x } .By [5, Proposition 3.6], every fiber of M is again a continuum withfinitely many complementary components. Thus the hierarchy by “fibers offibers” is well defined. Therefore, the scale ℓ ( M ) of non-local connectedness is defined as the least integer k such that for each x ∈ M there exist p ≤ k +1subcontinua M = N ⊃ N ⊃ · · · ⊃ N p = { x } such that N i is a fiber of N i − for 1 ≤ i ≤ p . If such an integer k does not exist we set ℓ ( M ) = ∞ .In this paper, we rather follow Kiwi’s approach [7] and define “modifiedfibers” for continua on the plane. The key point is: Kiwi focuses on Juliasets and uses “finite cutting sets” that consist of pre-periodic points, but weconsider arbitrary continua M on the plane (which may have interior points)and use “finite separating sets”. We refer to Example 7.1 for the differencebetween separating and cutting sets. Moreover, in Jolivet-Loridant-Luo [5],a good cut is not contained entirely in the underlying continuum M . In thecurent paper we will remove this assumption and only require that a goodcut is a simple closed curve intersecting ∂M at finitely many points. Afterthis slight modification we can establish the equivalence between the abovementioned two approaches to define fiber, using good cuts or using finiteseparating sets. See Remark 1.2 for further details.The notions and results will be presented in a way that focuses on thegeneral topological aspects, rather than in the framework of complex anal-ysis and dynamics. Definition 1.1.
Let X ⊂ C be a continuum. We will say that a point x ∈ ∂X is separated from a point y ∈ ∂X by a subset C ⊂ X if there is a separation ∂X \ C = A ∪ B with x ∈ A and y ∈ B . Here “ ∂X \ C = A ∪ B is a separation” means that A ∩ B = A ∩ B = ∅ . IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 3 • The modified pseudo-fiber E ∗ x of X at a point x in the interior X o of X is { x } ; and the modified pseudo-fiber E ∗ x of X at a point x ∈ ∂X is the set of the points y ∈ ∂X that cannot be separated from x byany finite set C ⊂ ∂X . • The modified fiber F ∗ x of X at x is the connected component of E ∗ x containing x . We say E ∗ x or F ∗ x is trivial if it consists of the point x only. ( We will show that E ∗ x = F ∗ x in Theorem 1, so the notion ofmodified pseudo-fiber is only used as a formal definition. ) • We inductively define a fiber of order k ≥ Y ⊂ X , where Y is a fiber of order k − • The local scale of non-local connectedness of X at a point x ∈ X ,denoted ℓ ∗ ( X, x ), is the least integer p such that there exist k ≤ p +1subcontinua X = N ⊃ N ⊃ N ⊃ · · · ⊃ N k = { x } such that N i is a fiber of N i − for 1 ≤ i ≤ k . If such an integer doesnot exist we set ℓ ∗ ( X, x ) = ∞ . • The (global) scale of non-local connectedness of X is ℓ ∗ ( X ) = sup { ℓ ∗ ( X, x ) : x ∈ X } . We also call ℓ ∗ ( X, x ) the local NLC-scale of X at x , and ℓ ∗ ( X ) the global NLC-scale .We firstly obtain the equality F ∗ x = E ∗ x and relate trivial fibers to localconnectedness. Here, local connectedness at a particular point does not im-ply trivial fiber. In particular, let K ⊂ [0 ,
1] be Cantor’s ternary set, let X be the union of K × [0 ,
1] with [0 , × { } . See Figure 6. Then X is locallyconnected at every x = ( t,
1) with t ∈ K , while the modified fiber F ∗ x at thispoint is the whole segment { t } × [0 , Theorem 1.
Let X ⊂ C be a continuum. Then F ∗ x = E ∗ x for every x ∈ X ;moreover, F ∗ x = { x } implies that X is locally connected at x . Secondly, we characterize modified fibers F ∗ x = E ∗ x through simple closedcurves γ that separate x from points y in X \ F ∗ x and that intersect ∂X ata finite set or an empty set.This provides an equivalent way to develop the theory of fibers, for planarcontinua, and leads to a partial converse for the second part of Theorem 1.See Remark 1.2. Theorem 2.
Let X ⊂ C be a continuum. Then F ∗ x at any point x ∈ X consists of the points y ∈ X such that every simple closed curve γ separating B. LORIDANT AND J. LUO x from y must intersect ∂X at infinitely many points. Or, equivalently, X \ F ∗ x consists of the points z ∈ X which may be separated from x by asimple closed curve γ such that γ ∩ ∂X is a finite set. This criterion can be related to Kiwi’s characterization of fibers [7, Corol-lary 2.18], as will be explained at the end of Section 4.
Remark 1.2.
We define a simple closed curve γ to be a good cut of acontinuum X ⊂ C if γ ∩ ∂X is a finite set (the empty set is also allowed).We also say that two points x, y ∈ X are separated by a good cut γ if they liein different components of C \ γ . This slightly weakens the requirements on“good cuts” in [5]. Therefore, given a continuum X ⊂ C whose complementhas finitely many components, the modified pseudo-fiber E ∗ x at any point x ∈ X is a subset of the pseudo-fiber E x at x , if E x is defined as in [5].Consequently, we can infer that local connectedness of X implies trivialityof all the fibers F ∗ x , by citing two of the four equivalent statements of [5,Theorem 2.2]: (1) X is locally connected; (2) every pseudo-fiber E x is trivial.The same result does not hold when the complement C \ X has infinitelymany components. Sierpinski’s universal curve gives a counterexample. Remark 1.3.
The two approaches, via pseudo-fibers E x and modified pseudo-fibers E ∗ x , have their own merits. The former one follows Schleicher’s ap-proach and is more closely related to the theory of puzzles in the studyof Julia sets and the Mandelbrot set; hence it may be used to analyse thestructure of such continua by cultivating the dynamics of polynomials. Thelatter approach has a potential to be extended to the study of general com-pact metric spaces; and, at the same time, it is directly connected with thefirst approach when restricted to planar continua.Thirdly, we study the topology of X by constructing an equivalencerelation ∼ on X and cultivating the quotient space X/ ∼ , which will beshown to be a locally connected continuum. This relation ∼ is based onfibers of X and every fiber F ∗ x is contained in a single equivalence class. Definition 1.4.
Let X ⊂ C be a continuum. Let X be the union of all thenontrivial fibers F ∗ x for x ∈ X and X denote the closure of X . We define x ∼ y if x = y or if x = y belong to the same component of X . Then ∼ isa closed equivalence relation on X such that, for all x ∈ X , the equivalenceclass [ x ] ∼ always contains the modified fiber F ∗ x and equals { x } if only x ∈ ( X \ X ). Consequently, every equivalence class [ x ] ∼ is a continuum,so that the natural projection π ( x ) = [ x ] ∼ is a monotone mapping, from X onto its quotient X/ ∼ . IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 5
Remark 1.5.
Actually, there is a more natural equivalence relation ≈ bydefining x ≈ y whenever there exist points x = x, x , . . . , x n = y in X suchthat x i ∈ F ∗ x i − . However, the relation ≈ may not be closed, as a subset ofthe product X × X . On the other hand, if we take the closure of ≈ we willobtain a closed relation, which is reflexive and symmetric but may not betransitive (see Example 7.3). The above Definition 1.4 solves this problem.The following theorem provides important information about the topol-ogy of X/ ∼ . Theorem 3.
Let X ⊂ C be a continuum. Then X/ ∼ is metrizable and isa locally connected continuum, possibly a single point. Remark 1.6.
The result of Theorem 3 is of fundamental significance fromthe viewpoint of topology. It also plays a crucial role in the study of complexdynamics. In particular, if J is the Julia set (assumed to be connected) ofa polynomial f ( z ) with degree n ≥ f | J : J → J induces acontinuous map f ∼ : J/ ∼→ J/ ∼ such that π ◦ f = f ∼ ◦ π . See Theorem6.1. Moreover, the modified fibers F ∗ x are closely related to impressions ofprime ends. See Theorem 6.4. Combining this with laminations on the unitcircle S ⊂ C , the system f ∼ : J/ ∼→ J/ ∼ is also a factor of the map z z n on S . However, it is not known yet whether the the decomposition { [ x ] ∼ : x ∈ X } by classes of ∼ coincide with the finest locally connectedmodel discussed in [1]. For more detailed discussions related to the dynamicsof polynomials, see for instance [1, 7] and references therein.Finally, to conclude the introduction, we propose two problems. Problem 1.7.
To estimate the scale ℓ ∗ ( X ) from above for particular con-tinua X ⊂ C such that C \ X has finitely many components, and to computethe quotient space X/ ∼ or the locally connected model introduced in [1].The Mandelbrot set or the Julia set of an infinitely renormalizable quadraticpolynomial (when this Julia set is not locally connected) provide very typi-cal choices of X . In particular, the scale ℓ ∗ ( X ) will be zero if the Mandelbrotset is locally connected, i.e. , if MLC holds. In such a case, the relation ∼ is trivial and its quotient is immediate. Remark 1.8.
Section 7 gives several examples of continua X ⊂ C . Weobtain the decomposition { [ x ] ∼ : x ∈ X } into sub-continua and representthe quotient space X/ ∼ on the plane. For those examples, the scale ℓ ∗ ( X )is easy to determine. B. LORIDANT AND J. LUO
Problem 1.9.
Given an unshielded continuum X in the plane, is it possibleto construct the “finest” upper semi-continuous decomposition of X intosub-continua that consist of fibers, in order that the resulted quotient spaceis a locally connected continuum (or has the two properties mentioned inTheorem 3)? Such a finest decomposition has no refinement that has theabove properties. If X is the Sierpinski curve, which is not unshielded, thedecomposition { [ x ] ∼ : x ∈ X } obtained in Theorem 3 does not suffice. Remark 1.10.
The main motivation for Problem 1.9 comes from [1] inwhich the authors, Blokh-Curry-Oversteegen, consider “unshielded” con-tinua X ⊂ C which coincide with the boundary of the unbounded compo-nent of C \ X . They obtain the existence of the finest monotone map ϕ from X onto a locally connected continuum on the plane, such that ϕ ( X ) is thefinest locally connected model of X and extend ϕ to a map ˆ ϕ : ˆ C → ˆ C thatmaps ∞ to ∞ , collapses only those components of C \ X whose boundary iscollapsed by ϕ , and is a homeomorphism elsewhere in ˆ C \ X [1, Theorem 1].This is of significance in the study of complex polynomials with connectedJulia set, see [1, Theorem 2]. Remark 1.11.
The equivalence classes [ x ] ∼ obtained in this paper give aconcrete upper semi-continuous decomposition of an arbitrary continuum X on the plane, with the property that the quotient space X/ ∼ is a lo-cally connected continuum. In the special case X is unshielded, the finestdecomposition in [1, Theorem 1] is finer than or equal to our decomposi-tion { [ x ] ∼ : x ∈ X } . See Theorem 6.4 for details when X is assumed to beunshielded. The above Problem 1.9 asks whether those two decompositionsactually coincide. If the answer is yes, the quotient space X/ ∼ in Theorem3 is exactly the finest locally connected model of X , which shall be in somesense “computable”. Here, an application of some interest is to study thelocally connected model of an infinitely renormalizable Julia set [4] or of theMandelbrot set, as mentioned in Problem 1.7.We arrange our paper as follows. Section 2 recalls some basic notionsand results from topology that are closely related to local connectedness.Sections 3, 4 and 5 respectively prove Theorems 1, 2 and 3. Section 6 dis-cusses basic properties of fibers, studies fibers from a viewpoint of dynamictopology (as proposed by Whyburn [14, pp.130-144]) and relates the the-ory of fibers to the theory of prime ends for unshielded continua. Finally,in Section 7, we illustrate our results through examples from the litera-ture and give an explicit sequence of path connected continua X n satisfying ℓ ∗ ( X n ) = n . IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 7 A Revisit to Local Connectedness
Definition 2.1.
A topological space X is locally connected at a point x ∈ X if for any neighborhood U of x there exists a connected neighborhood V of x such that V ⊂ U , or equivalently, if the component of U containing x is also a neighborhood of x . The space X is then called locally connected if it is locally connected at every of its points.We focus on metric spaces and their subspaces. The following character-ization can be found as the definition of locally connectedness in [14, PartA, Section XIV]. Lemma 2.2.
A metric space ( X, d ) is locally connected at x ∈ X if andonly if for any ε > there exists δ > such that any point y ∈ X with d ( x , y ) < δ is contained together with x in a connected subset of X ofdiameter less than ε . When X is compact, Lemma 2.2 is a local version of [9, p.183, Lemma17.13(d)]. For the convenience of the readers, we give here the concretestatement as a lemma. Lemma 2.3.
A compact metric space X is locally connected if and only iffor every ε > there exists δ > so that any two points of distance lessthan δ are contained in a connected subset of X of diameter less than ε . Using Lemma 2.2, we obtain a fact concerning continua of the Euclideanspace R n . Lemma 2.4.
Let X ⊂ R n be a continuum and U = S α ∈ I W α the union ofany collection { W α : α ∈ I } of components of R n \ X . If X is locally con-nected at x ∈ X , then so is X ∪ U . Consequently, if X is locally connected,then so is X ∪ U .Proof. Choose δ with properties from Lemma 2.2 with respect to x , X and ε/
2. For any y ∈ U with d ( x , y ) < δ we consider the segment [ x , y ]between x and y . If [ x , y ] ⊂ ( X ∪ U ), we are done. If not, choose the point z ∈ ([ x , y ] ∩ X ) that is closest to y . Clearly, the segment [ y, z ] is containedin X ∪ U . By the choice of δ and Lemma 2.2, we may connect z and x witha continuum A ⊂ X of diameter less than ε/
2. Therefore, the continuum B := A ∪ [ y, z ] ⊂ ( X ∪ U ) is of diameter at most ε as desired. (cid:3) In the present paper, we are mostly interested in continua on the plane,especially continua X which are on the boundary of a continuum M ⊂ C .Typical choice of such a continuum M is the filled Julia set of a rational B. LORIDANT AND J. LUO function. Several fundamental results from Whyburn’s book [14] will be veryhelpful in our study.The first result gives a fundamental fact about a continuum failing to belocally connected at one of its points. The proof can be found in [14, p.124,Corollary].
Lemma 2.5.
A continuum M which is not locally connected at a point p necessarily fails to be locally connected at all points of a nondegeneratesubcontinuum of M . The second result will be referred to as
Torhorst Theorem in thispaper (see [14, p.124, Torhorst Theorem] and [14, p.126, Lemma 2]).
Lemma 2.6.
The boundary B of each component C of the complement ofa locally connected continuum M is itself a locally connected continuum. Iffurther M has no cut point, then B is a simple closed curve. We finally recall a
Plane Separation Theorem [14, p.120, Exercise 2].
Proposition 2.7. If A is a continuum and B is a closed connected set ofthe plane with A ∩ B = T being a totally disconnected set, and with A \ T and B \ T being connected, then there exists a simple closed curve J separating A \ T and B \ T such that J ∩ ( A ∪ B ) ⊂ A ∩ B = T . Fundamental properties of fibers
The proof for Theorem 1 has two parts. We start from the equality E ∗ x = F ∗ x . Theorem 3.1.
Let X ⊂ C be a continuum. Then E ∗ x = F ∗ x for every x ∈ X .Proof. Suppose that E ∗ x \ F ∗ x contains some point x ′ . Then we can fix aseparation E ∗ x = A ∪ B with F ∗ x ⊂ A and x ′ ∈ B . Since E ∗ x is a compact set,the distance dist( A, B ) := min {| y − z | : y ∈ A, z ∈ B } is positive. Let A ∗ = (cid:26) z ∈ C : dist( z, A ) <
13 dist(
A, B ) (cid:27) and B ∗ = (cid:26) z ∈ C : dist( z, B ) <
13 dist(
A, B ) (cid:27) . Then A ∗ and B ∗ are disjoint open sets in the plane, hence K = X \ ( A ∗ ∪ B ∗ )is a compact subset of X . As E ∗ x ∩ K = ∅ , we may find for each z ∈ K afinite set C z and a separation X \ C z = U z ∪ V z IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 9 such that x ∈ U z , z ∈ V z . Here, we have U z = X \ ( C z ∪ V z ) = X \ ( C z ∪ V z )and V z = X \ ( C z ∪ U z ) = X \ ( C z ∪ U z ); so both of them are open in X .By flexibility of z ∈ K , we obtain an open cover { V z : z ∈ K } of K , whichthen has a finite subcover { V z , . . . , V z n } . Let U = U z ∩ · · · ∩ U z n , V = V z ∪ · · · ∪ V z n . Then
U, V are disjoint sets open in X such that C := X \ ( U ∪ V ) is a subsetof C z ∪ · · · ∪ C z n , hence it is also a finite set. Now, on the one hand, we havea separation X \ C = U ∪ V with x ∈ U and K ⊂ V ; on the other hand, fromthe equality K = X \ ( A ∗ ∪ B ∗ ) we can infer U ⊂ ( A ∗ ∪ B ∗ ). Combining thiswith the fact that x ′ ∈ B ∗ , we may check that A ′ := ( U ∪{ x ′ } ) ∩ A ∗ = U ∩ A ∗ and B ′ := ( U ∪ { x ′ } ) ∩ B ∗ = ( U ∩ B ∗ ) ∪ { x ′ } ⊂ B ∗ are separated in X . Let C ′ = C \ { x ′ } . Since A ′ ⊂ U and V are also separated in X , we see that X \ C ′ = U ∪ { x ′ } ∪ V = ( U ∩ A ∗ ) ∪ ( U ∩ B ∗ ) ∪ { x ′ } ∪ V = A ′ ∪ ( B ′ ∪ V )is a separation with x ∈ A ′ and x ′ ∈ ( B ′ ∪ V ). This contradicts the assump-tion that x ′ ∈ E ∗ x , because E ∗ x being the pseudo-fiber at x , none of its pointscan be separated from x by the finite set C ′ . (cid:3) Then we recover in fuller generality that triviality of the fiber at a point x in a continuum M ⊂ C implies local connectedness of M at x . Morerestricted versions of this result appear earlier: in [11] for continua in theplane with connected complement, in [7] for Julia sets of monic polynomialsor the components of such a set, and in [5] for continua in the plane whosecomplement has finitely many components. Theorem 3.2. If F ∗ x = { x } for a point x in a continuum X ⊂ C then X is locally connected at x .Proof. We will prove that if X is not locally connected at x then F ∗ x containsa non-degenerate continuum M ⊂ X .By definition, if X is not locally connected at x there exists a number r > Q x of B ( x, r ) ∩ X containing x is not aneighborhood of x in X . Here B ( x, r ) = { y : | x − y | ≤ r } . This means that there exist a sequence of points { x k } ∞ k =1 ⊂ X \ Q x suchthat lim k →∞ x k = x . Let Q k be the component of B ( x, r ) ∩ X containing x k .Then Q i ∩ { x k } ∞ k =1 is a finite set for each i ≥
1, and hence we may assume,by taking a subsequence, that Q i ∩ Q j = ∅ for i = j . Since the hyperspace of the nonempty compact subsets of X is a compactmetric space under Hausdorff metric, we may further assume that thereexists a continuum M such that lim k →∞ Q k = M under Hausdorff distance.Clearly, we have x ∈ M ⊂ Q x . The following Lemma 3.3 implies that thediameter of M is at least r . Since every point y ∈ M \ { x } cannot beseparated from x by a finite set in X , F ∗ x cannot be trivial and our proof isreadily completed. (cid:3) Lemma 3.3.
In the proof for Theorem 3.2, every component of B ( x, r ) ∩ X intersects ∂B ( x, r ) . In particular, Q k ∩ ∂B ( x, r ) = ∅ for all k ≥ .Proof. Otherwise, there would exist a component Q of B ( x, r ) ∩ X such that Q ∩ ∂B ( x, r ) = ∅ . Then, for each point y on X ∩ ∂B ( x, r ), the component Q y of X ∩ B ( x, r ) containing y is disjoint from Q . By definition of quasi-components, we may choose a separation X ∩ B ( x, r ) = U y ∪ V y with Q y ⊂ U y and Q ⊂ V y . Since every U y is open in X ∩ B ( x, r ), we have an open cover { U y : y ∈ X ∩ ∂B ( x, r ) } for X ∩ ∂B ( x, r ), which necessarily has a finite subcover, say { U y , . . . , U y t } .Let U = U y ∪ · · · ∪ U y t , V = V y ∩ · · · ∩ V y t . Then X ∩ B ( x, r ) = U ∪ V is a separation with ∂B ( x, r ) ⊂ U . Therefore, X = [( X \ B ( x, r )) ∪ U ] ∪ V is a separation, which contradicts the connectedness of X . (cid:3) Schleicher’s and Kiwi’s approaches unified
Let X be a topological space and x a point in X . The component of X containing x is the maximal connected set P ⊂ X with x ∈ P . Thequasi-component of X containing x is defined to be the set Q = { y ∈ X : no separation X = A ∪ B exists such that x ∈ A, y ∈ B } . Equivalently, the quasi-component of a point p ∈ X may be defined asthe intersection of all closed-open subsets of X containing p . Since anycomponent is contained in a quasi-component, and since quasi-componentscoincide with the components whenever X is compact [8], we can infer anequivalent definition of pseudo fiber as follows. Proposition 4.1.
Let X ⊂ C be a continuum. Two points of X are sepa-rated by a finite set C ⊂ X iff they belong to distinct quasi-components of X \ C . IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 11
The following proposition implies Theorem 2. We present it in this form,since it can be seen as a modification of Whyburn’s plane separation theorem(Proposition 2.7). Actually, the main idea of our proof is borrowed from [14,p.126] and is slightly adjusted.
Proposition 4.2.
Let C be a finite subset of a continuum X ⊂ C and x, y two points on X \ C . If there is a separation X \ C = P ∪ Q with x ∈ P and y ∈ Q then x is separated from y by a simple closed curve γ with ( γ ∩ X ) ⊂ C .Proof. We first note that every component of P intersects C . If on the con-trary a component W of P ⊂ ( P ∪ C ) is disjoint from C , then P is discon-nected and a contradiction follows. Indeed, we have P = W since P ∩ C = ∅ .And all the components of P intersecting C are disjoint from W . As C is a fi-nite set, there are finitely many such components, say W , . . . , W t . However,since a quasi-component of a compact metric space is just a component, wecan find separations P = A i ∪ B i for 1 ≤ i ≤ t such that W ⊂ A i , W i ⊂ B i .Let A = ∩ i A i and B = ∪ i B i . Then P = A ∪ B is a separation with A ∩ Q = ∅ , hence X = A ∪ ( B ∪ Q ) is a separation of X . This contradictsthe connectedness of X .Since every component of P intersects C and since C is a finite set, weknow that P has finitely many components, say P , . . . , P k . We may assumethat x ∈ P . Similarly, every component of Q ⊂ ( Q ∪ C ) intersects C and Q has finitely many components, say Q , . . . , Q l . We may assume that y ∈ Q .Let P ∗ = P ∪ · · · ∪ P k ∪ Q ∪ · · · ∪ Q l . Then X = P ∪ P ∗ , x ∈ P , y ∈ P ∗ and ( P ∩ P ∗ ) ⊂ C . Let N = { z ∈ P ; dist( z, P ∗ ) ≥ } and for each j ≥ N j = { z ∈ P : 3 − j ≤ dist( z, P ∗ ) ≤ − j +1 } . Clearly, every N j is a compact set. Therefore, we may cover N j by finitelymany open disks centered at a point in N j and with radius r j = 3 − j − , say B ( x j , r j ) , . . . , B ( x jk ( j ) , r j ).For j >
1, let us set M j = S k ( j ) i =1 B ( x ji , r j ). Then M = S j > M j isa compact set containing P . Its interior M o contains x . Moreover, P ∗ ∩ (cid:16)S j > M j (cid:17) = ∅ by definition of N j and M j , while M \ (cid:16)S j M j (cid:17) is a subsetof P ∩ P ∗ , hence we have M ∩ P ∗ = P ∩ P ∗ and y / ∈ M . Also, ∂M ∩ X isa subset of P ∩ P ∗ , hence it is a finite set. This idea is inspired from the proof of Whyburn’s plane separation theorem, seeProposition 2.7
Now M is a continuum, since P is itself a continuum and the disks B ( x ji , r j ) are centered at x ji ∈ N j . The continuum M is even locally con-nected at every point on M \ C = S j M j . Indeed, it is locally a finite unionof disks, since M j ∩ M k = ∅ as soon as | j − k | > M \ C is in one of these disks. As C is finite, it follows from Lemma 2.5that M is a locally connected continuum.Now, let U be the component of C \ M that contains y . By TorhorstTheorem, see Lemma 2.6, the boundary ∂U of U is a locally connectedcontinuum. Therefore, by Lemma 2.4, the union U ∪ ∂U is also a locallyconnected continuum. Since U is a complementary component of ∂U , theunion U ∪ ∂U even has no cut point. It follows from Torhorst Theorem thatthe boundary ∂V of any component V of C \ ( U ∪ ∂U ) is a simple closedcurve. Note that this curve separates every point of U from any point of V .Choosing V to be the component of C \ ( U ∪ ∂U ) containing x , we obtaina simple closed curve J = ∂V separating y from x .Finally, since J = ∂V ⊂ ∂U ⊂ ∂M , we see that J ∩ X is contained in thefinite set C . Consequently, J is a good cut of X separating x from y . (cid:3) This result proves Theorem 2 and is related to Kiwi’s characterizationof fibers. Restricting to connected Julia sets J ( f ) of polynomials f , Kiwi [7]had defined for ζ ∈ J ( f ) the fiber Fiber( ζ ) as the set of ξ ∈ J ( f ) such that ξ and ζ lie in the same connected component of J ( f ) \ Z for every finite set Z ⊂ J ( f ), made of periodic or preperiodic points that are not in the grandorbit of a Cremer point. Kiwi showed in [7, Corollary 2.18] that these fiberscan be characterized by using separating curves involving external rays.5. A locally connected model for the continuum X In this section, we recall a few notions and results from Kelley’s
GeneralTopology [6] and construct a proof for Theorem 3, the results of which aredivided into two parts:(1) X/ ∼ is metrizable, hence is a compact connected metric space, i.e. ,a continuum.(2) X/ ∼ is a locally connected continuum.A decomposition D of a topological space X is upper semi-continuous iffor each D ∈ D and each open set U containing D there is an open set V such that D ⊂ V ⊂ U and V is the union of members of D [6, p.99]. Givena decomposition D , we may define a projection π : X → D by setting π ( x )to be the unique member of D that contains x . Then, the quotient space D IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 13 is equipped with the largest topology such that π : X → D is continuous.We copy the result of [6, p.148, Theorem 20] as follows. Theorem 5.1.
Let X be a topological space, let D be an upper semi-continuous decomposition of X whose members are compact, and let D havethe quotient topology. Then D is, respectively, Hausdorff, regular, locallycompact, or has a countable base, provided X has the corresponding prop-erty. Urysohn’s metrization theorem [6, p.125, Theorem 16] states that a reg-ular T -space whose topology has a countable base is metrizable. Combiningthis with Theorem 5.1, we see that the first part of Theorem 3 is impliedby the following theorem, since X is a continuum on the plane and has allthe properties mentioned in Theorem 5.1. One may also refer to [10, p.40,Theorem 3.9], which states that any upper semi-continuous decompositionof a compact metric space is metrizable. Theorem 5.2.
The decomposition { [ x ] ∼ : x ∈ X } is upper semi-continuous.Proof. Given a set U open in X , we need to show that the union U ∼ ⊂ U of all the classes [ x ] ∼ ⊂ U is open in X . In other words, we need to showthat X \ U ∼ is closed in X , which implies that π ( X \ U ∼ ) = ( X/ ∼ ) \ π ( U ∼ )is closed in the quotient X/ ∼ . Here, we note that X \ U ∼ is just the unionof all the classes [ x ] ∼ that intersects X \ U .Assume that y k ∈ X \ U ∼ is a sequence converging to y , we will showthat [ y ] ∼ \ U = ∅ , hence that y ∈ X \ U ∼ . Let z k be a point in [ y k ] ∼ \ U for each k ≥
1. By coming to an appropriate subsequence, we may furtherassume that • [ y i ] ∼ ∩ [ y j ] ∼ = ∅ for i = j ; • the sequence of continua [ y k ] ∼ converges to a continuum M underHausdorff metric; • the sequence z k converges to a point z ∞ .Clearly, we have z ∞ ∈ M ; and, as X \ U is compact, we also have z ∞ ∈ X \ U .If the sequence [ y k ] ∼ is finite, then M = [ y ] ∼ , thus [ y ] ∼ \ U = ∅ . If thesequence [ y k ] ∼ is infinite, let us check that M ⊂ F ∗ y . Indeed, for any point z ∈ M and for any finite set C ⊂ X disjoint from { y, z } , all but finitely many[ y k ] ∼ are connected disjoint subsets of X \ C . It follows that there exists noseparation X \ C = A ∪ B such that y ∈ A, z ∈ B , because y and z are bothlimit points of the sequence of continua [ y k ] ∼ . Hence M ⊂ F ∗ y ⊂ [ y ] ∼ and z ∞ ∈ M ∩ ( X \ U ), indicating that [ y ] ∼ \ U = ∅ . (cid:3) Theorem 5.3.
The quotient X/ ∼ is a locally connected continuum.Proof. As X is a continuum, π ( X ) = X/ ∼ is itself a continuum. We nowprove that this quotient is locally connected. If V is an open set in X/ ∼ that contains [ x ] ∼ , as an element of X/ ∼ , then the pre-image U := π − ( V )is open in X and contains the class [ x ] ∼ as a subset. We shall prove that V contains a connected neighborhood of [ x ] ∼ . Without loss of generality, weassume that U = X . Let Q be the component of U that contains [ x ] ∼ . Bythe boundary bumping theorem [10, Theorem 5.7, p75] (see also [14, p.41,Exercise 2]), since X is connected, we have Q \ U = ∅ . Moreover, our proofwill be completed by the following claim. Claim.
The connected set π ( Q ), hence the component of V that contains[ x ] ∼ as a point, is a neighborhood of [ x ] ∼ in the quotient space X/ ∼ .Otherwise, there would exist an infinite sequence of points [ x k ] ∼ in V \ π ( Q ) such that lim k →∞ [ x k ] ∼ = [ x ] ∼ under the quotient topology. Since U = π − ( V ), every x k belongs to U . Let Q k be the component of U that contains x k . Here we have Q k ∩ Q = ∅ . And, by the above mentioned boundarybumping theorem, we also have Q k \ U = ∅ .Now, choose points y k ∈ [ x k ] ∼ for every k ≥ { y k } has alimit point y . Here, we certainly have [ y k ] ∼ = [ x k ] ∼ and [ y ] ∼ = [ x ] ∼ . Bycoming to an appropriate subsequence, we may assume that lim k →∞ y k = y and that lim k →∞ Q k = M under Hausdorff metric. Then M is a continuumwith y ∈ M ⊂ Q and M \ U = ∅ , indicating that the fiber F ∗ y contains M , hence intersects X \ U . In other words, F ∗ y * U , which contradicts theinclusions y ∈ Q ⊂ U and F ∗ y ⊂ [ y ] ∼ = [ x ] ∼ ⊂ U . (cid:3) How fibers are changed under continuous maps
In this section, we discuss how fibers are changed under continuous maps.As a special application, we may compare the dynamics of a polynomial f c ( z ) = z n + c on its Julia set J c , the expansion z z d on unit circle, andan induced map ˜ f c on the quotient J c / ∼ .Let X, Y ⊂ C be continua and x ∈ X a point. The first primary ob-servation is that f ( F ∗ x ) ⊂ F ∗ f ( x ) for any finite-to-one continuous surjection f : X → Y .Indeed, for any y = x in the fiber F ∗ x and any finite set C ⊂ Y that isdisjoint from { f ( x ) , f ( y ) } , we can see that f − ( C ) is a finite set disjointfrom { x, y } . Since y ∈ F ∗ x there exists no separation X \ f − ( C ) = A ∪ B IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 15 with x ∈ A, y ∈ B ; therefore, there exists no separation Y \ C = P ∪ Q with f ( x ) ∈ P, f ( y ) ∈ Q . This certifies that f ( y ) ∈ F ∗ f ( x ) .By the above inclusion f ( F ∗ x ) ⊂ F ∗ f ( x ) we further have f ( X ) ⊂ Y . Here X is the union of all the nontrivial fibers F ∗ x in X , and Y the union ofthose in Y . It follows that f ([ x ] ∼ ) ⊂ [ f ( x )] ∼ . Therefore, the correspondence[ x ] ∼ ˜ f −−→ [ f ( x )] ∼ gives a well defined map ˜ f : X/ ∼ → Y / ∼ that satisfies thefollowing commutative diagram, in which each downward arrow ↓ indicatesthe natural projection π from a space onto its quotient. X f −−−−−−−−−→ Y ↓ ↓ X/ ∼ ˜ f −−−−−−−−−→ Y / ∼ Given an open set U ⊂ Y / ∼ , we can use the definition of quotient topologyto infer that V := ˜ f − ( U ) is open in X/ ∼ whenever π − ( V ) is open in X .On the other hand, the above diagram ensures that π − ( V ) = f − ( π − ( U )),which is an open set of X , by continuity of f and π .The above arguments lead us to a useful result for the study of dynamicson Julia sets. Theorem 6.1.
Let
X, Y ⊂ C be continua. Let the relation ∼ be defined asin Theorem 3. If f : X → Y is continuous, surjective and finite-to-one then ˜ f ([ x ] ∼ ) := [ f ( x )] ∼ defines a continuous map with π ◦ f = ˜ f ◦ π . Remark 6.2.
Every polynomial f c ( z ) = z n + c restricted to its Julia set J c satisfies the conditions of Theorem 6.1, if we assume that J c is connected;so the restricted system f c : J c → J c has a factor system ˜ f c : J c / ∼→ J c / ∼ ,whose underlying space is a locally connected continuum.Let X ⊂ C be an unshielded continuum and U ∞ the unbounded com-ponent of C \ X . Here, X is unshielded provided that X = ∂U ∞ . Let D := { z ∈ ˆ C : | z | ≤ } be the unit closed disk. By Riemann MappingTheorem, there exists a conformal isomorphism Φ : ˆ C \ D → U ∞ that fixes ∞ and has positive derivative at ∞ . The prime end theory [2, 13] builds acorrespondence between an angle θ ∈ S := ∂ D and a continuum Imp ( θ ) := n w ∈ X : ∃ z n ∈ D with z n → e i θ , lim n →∞ Φ( z n ) = w o We call
Imp ( θ ) the impression of θ . By [3, p.173, Theorem 9.4], we may fixa simple open arc R θ in C \ D landing at the point e i θ such that Φ( R θ ) ∩ X = Imp ( θ ).We will connect impressions to fibers. Before that, we obtain a usefullemma concerning good cuts of an unshielded continuum X on the plane. Here a good cut of X is a simple closed curve that intersects X at a finitesubset (see Remark 1.2). Lemma 6.3.
Let X ⊂ C be an unshielded continuum and U ∞ the unboundedcomponent of C \ X . Let x and y be two points on X separated by a goodcut of X . Then we can find a good cut separating x from y that intersects U ∞ at an open arc.Proof. Since each of the two components of C \ γ intersects { x, y } , we have γ ∩ U ∞ = ∅ . Since γ ∩ X is a finite set, the difference γ \ X has finitelymany components. Let γ , . . . , γ k be the components of γ \ X that lie in U ∞ . Let α i = Φ − ( γ i ) be the pre-images of γ i under Φ. Then every α i is asimple open arc in { z : | z | > } whose end points a i , b i are located on theunit circle; and all those open arcs α , . . . , α k are pairwise disjoint.If k >
2, rename the arcs α , . . . , α k so that we can find an open arc β ⊂ ( C \ D ) disjoint from S ki =1 α i that connects a point a on α to a point b on α . Then γ ∪ Φ( β ) is a Θ-curve separating x from y (see [14, PartB, Section VI] for a definition of Θ-curve). Let J and J denote the twocomponents of γ \ Φ( β ) = γ \ { Φ( a ) , Φ( b ) } . Then J ∪ Φ( β ) and J ∪ Φ( β ) areboth good cuts of X . One of them, denoted by γ ′ , separates x from y [14,Θ-curve theorem, p.123]. By construction, this new good cut intersects U ∞ at k ′ open arcs for some 1 k ′ k −
1. For relative locations of J , J andΦ( β ) in ˆ C , we refer to Figure 1 in which γ is represented as a circular circle,although a general good cut is usually not a circular circle. If k ′ >
2, weΦ( β ) J J γ ∋ Φ( a ) Φ( b ) ∈ γ Figure 1.
The Θ-curve together with the arcs J , J , and Φ( β ).may use the same argument on γ ′ and obtain a good cut γ ′′ , that separates x from y and that intersects U ∞ at k ′′ open arcs for some 1 k ′′ k − k − x from y that intersects U ∞ at a single open arc. (cid:3) Theorem 6.4.
Let X ⊂ C be an unshielded continuum. Then every im-pression Imp ( θ ) is contained in a fiber F ∗ w for some w ∈ Imp ( θ ) . IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 17
Proof.
Suppose that a point y = x on Imp ( θ ) is separated from x in X bya finite set. By Proposition 4.2, we can find a good cut γ separating x from y . By Lemma 6.3, we may assume that γ ∩ U ∞ is an open arc γ . Let a and b be the two end points of α = Φ − ( γ ), an open arc in C \ D .Fix an open arc R θ in C \ D landing at the point e i θ such that Φ( R θ ) ∩ X = Imp ( θ ). We note that e i θ ∈ { a, b } . Otherwise, there is a number r > R θ ∩ { z : | z | < r } lies in the component of( C \ D ) \ ( { a, b } ∪ α ) (difference of C \ D and { a, b } ∪ α )whose closure contains e i θ . From this we see that Φ( R θ ∩ { z : | z | < r } ) isdisjoint from γ and is entirely contained in one of the two components of C \ γ , which contain x and y respectively. Therefore,Φ( R θ ∩ { z : | z | < r } )hence its subset Imp ( θ ) cannot contain x and y at the same time. Thiscontradicts the assumption that x, y ∈ Imp ( θ ).Now we will lose no generality by assuming that e i θ = a . Then Φ( R θ )intersects γ infinitely many times, since Φ( R θ ) \ Φ( R θ ) contains { x, y } .This implies that a is the landing point of R θ ⊂ ( C \ D ).Let w = lim z → a Φ | α ( z ). Then { x, y, w } ⊂ Imp ( θ ), and the proof will becompleted if we can verify that Imp ( θ ) ⊂ F ∗ w .Suppose there is a point w ∈ Imp ( θ ) that is not in F ∗ w . By Lemma6.3 we may find a good cut γ ′ separating w from w that intersects U ∞ at an open arc γ ′ . Let α ′ = Φ − ( γ ′ ). Let I be the component of S \ α ′ that contains a . Since w / ∈ γ ′ , the closure α ′ does not contain the point a . Therefore, R θ ∩ { z : | z | < r } is disjoint from α ′ for some r >
1. Forsuch an r , the image Φ( R θ ∩ { z : | z | < r } ) is disjoint from γ ′ . On theother hand, the good cut γ ′ separates w from w . Therefore, the closure ofΦ( R θ ∩ { z : | z | < r } ) hence its subset Imp ( θ ) does not contain the twopoints w and w at the same time. This is a contradiction. (cid:3) Remark 6.5.
Let J c be the connected Julia set of a polynomial. The equiva-lence classes [ x ] ∼ obtained in this paper determine an upper semi-continuousdecomposition of J c , such that the quotient space is a locally connected con-tinuum. Theorem 6.4 says that the impression of any prime end is entirelycontained in a single class [ x ] ∼ . Therefore, the finest decomposition men-tioned in [1, Theorem 1] is finer than { [ x ] ∼ : x ∈ J c } . Currently it is notclear whether these two decompositions just coincide. This is proposed asan open question in Problem 1.9. Facts and Examples
In this section, we give several examples to demonstrate the differencebetween (1) separating and cutting sets, (2) the fiber F ∗ x and the class [ x ] ∼ ,(3) a continuum X ⊂ C and the quotient space X/ ∼ for specific choicesof X . We also construct an infinite sequence of continua which have scalesof any k ≥ ∞ , although the quotient of each of thosecontinua is always homeomorphic with the unit interval [0 , Example 7.1 ( Separating Sets and Cutting sets).
For a set M ⊂ C , aset C ⊂ M is said to separate or to be a separating set between two points a, b ⊂ M if there is a separation M \ C = P ∪ Q satisfying a ∈ P, b ∈ Q ;and a subset C ⊂ M is called a cutting set between two points a, b ∈ M if { a, b } ⊂ ( X \ C ) and if the component of X \ C containing a does notcontain b [8, p.188, § L be the segment between the points (2 ,
1) and (2 ,
0) on the plane, Q the one between ( − ,
0) and c = (0 , ), and P the broken line connecting(2 ,
0) to ( − ,
0) through (0 , − x , x ) f −−→ cbQ a L P Q (2 , , , − − , L L P Figure 2.
The continuum X and its quotient as a Hawaiianearring minus an open rectangle.( x , x ) and ( x , x ) g −−→ ( x , x ). For any k ≥
1, let L k +1 = g ( L k ) and Q k +1 = g ( Q k ); let P k +1 = f ( P k ). Let B k = L k ∪ P k ∪ Q k . Then { B k : k ≥ } isa sequence of broken lines converging to the segment B between a = (0 , b = (0 , N = ( S k B k ) S B . Then N is a continuum, which isnot locally connected at each point of B . Moreover, the singleton { c } isa cutting set, but not a separating set, between the points a and b . Theonly nontrivial fiber is B = { } × [0 ,
1] = F ∗ x for each x ∈ B . So we have ℓ ∗ ( N ) = 1. Also, it follows that [ x ] ∼ = B for all x ∈ B and [ x ] ∼ = { x } otherwise. In particular, the broken lines B k are still arcs in the quotientspace but, under the metric of quotient space, their diameters converge to IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 19 zero. Consequently, the quotient N/ ∼ is topologically the difference of aHawaiian earring with a full open rectangle. See the right part of Figure 2.In other words, the quotient space N/ ∼ is homeomorphic with the quotient X/ ∼ of Example 7.2. Example 7.2 ( The Witch’s Broom).
Let X be the witch’s broom [10,p.84, Figure 5.22]. See Figure 3. More precisely, let A := [ , × { } ; let Figure 3.
An intuitive depiction of the witch’s broom andits quotient space. A k be the segment connecting (1 ,
0) to ( , − k ) for k ≥
0. Then A = [ k ≥ A k is a continuum (an infinite broom) which is locally connected everywherebut at the points on [ , × { } . Let g ( x ) = x be a similarity contractionon R . Let X = { (0 , } ∪ A ∪ f ( A ) ∪ f ( A ) ∪ · · · ∪ f n ( A ) ∪ · · · · · · . The continuum X is called the Witch’s Broom . Consider the fibers of X ,we have F ∗ x = { x } for each x in X ∩ { ( x , x ) : x > } and for x = (0 , F ∗ (1 , = [ , × { } , F ∗ (2 − k , = [2 − k − , − k +1 ] ×{ } ( k ≥ F ∗ ( x , = [2 − k , − k +1 ] × { } (2 − k < x < − k +1 , k ≥ . Consequently, [ x ] ∼ = { x } for each x in X ∩ { ( x , x ) : x > } , while[ x ] ∼ = [0 , × { } for x ∈ [0 , × { } . See the right part of Figure 3 for adepiction of the quotient X/ ∼ . Example 7.3 ( Witch’s Double Broom).
Let X be the witch’s broom.We call the union Y of X with a translated copy X + ( − ,
0) the witch’sdouble broom (see Figure 4). Define x ≈ y if there exist points x = x,x , . . . , x n = y in Y such that x i ∈ F ∗ x i − . Then ≈ is an equivalence and isnot closed. Its closure ≈ ∗ is not transitive, since we have ( − , ≈ ∗ (0 , , ≈ ∗ (1 , − ,
0) is not related to (1 ,
0) under ≈ ∗ . ( − ,
0) (0 ,
0) (1 , Figure 4.
Relative locations of the points ( ± ,
0) and (0 , Example 7.4 ( Cantor’s Teepee).
Let X be Cantor’s Teepee [12, p.145].See Figure 5. Then the fiber F ∗ p = X ; and for every other point x , F ∗ x isexactly the line segment on X that crosses x and p . Therefore, ℓ ∗ ( X ) = 1.Moreover, [ x ] ∼ = X for every x , hence the quotient is a single point. In thiscase, we also say that X is collapsed to a point. p Figure 5.
A simple representation of Cantor’s Teepee.
Example 7.5 ( Cantor’s Comb).
Let
K ⊂ [0 ,
1] be Cantor’s ternary set.Let X be the union of K × [0 ,
1] with [0 , × { } . See Figure 6. We call X the Cantor comb. Then the fiber F ∗ x = { x } for every point on X that is off K × [0 , x on K × [0 , F ∗ x is exactly thevertical line segment on K × [0 ,
1] that contains x . Therefore, ℓ ∗ ( X ) = 1.Moreover, [ x ] ∼ = F ∗ x for every x , hence the quotient is homeomorphic to[0 , X is locally connected at every point lying on thecommon part of [0 , ×{ } and K × [0 , Figure 6.
Cantor’s Comb, its nontrivial fibers, and the quo-tient X/ ∼ . IBERS AND LOCAL CONNECTEDNESS OF PLANAR CONTINUA 21
Example 7.6 ( More Combs).
We use Cantor’s ternary set
K ⊂ [0 , { X k : k ≥ } , such that the scale ℓ ∗ ( X k ) = k for all k ≥
1. We also determine the fibers and compute thequotient spaces X k / ∼ . Let X be the union of X ′ = ( K + 1) × [0 ,
2] with[1 , × { } . Here K + 1 := { x + 1 : x ∈ K} . Then X is homeomorphicwith Cantor’s Comb defined in Example 7.5. We have ℓ ∗ ( X ) = 1 andthat X / ∼ is homeomorphic with [0 , X be the union of X with[0 , × ( K + 1). See Figure 7. Then the fiber of X at the point (1 , ∈ X Figure 7.
A simple depiction of X , the largest fiber, andthe quotient X / ∼ .is F ∗ (1 , = X ∩ { ( x , x ) : x ≤ } , which will be referred to as the “largestfiber”, since it is the fiber with the largest scale in X . See the central partof Figure 7. The other fibers are either a single point or a segment, of theform { ( x , x ) : 0 ≤ x ≤ } for some x ∈ K + 1. Therefore, we have ℓ ∗ ( X ) = 2 and can check that the quotient X / ∼ is homeomorphic with[0 , X be the union of X with X n(cid:16) x , x (cid:17) : ( x , x ) ∈ X o . Then the largest fiber of X is exactly F ∗ (1 , = X ∩ { ( x , x ) : x ≤ } ,which is homeomorphic with X . Therefore, ℓ ∗ ( X ) = 3; moreover, X / ∼ is also homeomorphic with [0 , X = X ∪ X . Then the largest fiber of X is F ∗ (1 , = X ∩ { ( x , x ) : x ≤ } ,which is homeomorphic with X . Similarly, we can infer that ℓ ∗ ( X ) = 4and that X / ∼ is homeomorphic with [0 , X k for k ≥ X k +2 = X S X k defines a path-connected continuum for all k ≥ X k +1 . Therefore, we have ℓ ∗ ( X k ) = k ; moreover, the quotient space X k / ∼ is always homeomorphic Figure 8.
A depiction of X , X , the largest fibers, and thequotients X / ∼ and X / ∼ .to the interval [0 , X ∞ = { (0 , } ∪ ∞ [ k =2 X k ! is a path connected continuum and that its largest fiber is homeomorphicto X ∞ itself. Therefore, X ∞ has a scale ℓ ∗ ( X ∞ ) = ∞ , and its quotient ishomeomorphic to [0 , Acknowledgements
The authors are grateful to the referee for very helpful remarks, especiallythose about a gap in the proof for Theorem 3.1 and an improved proof forLemma 2.4. The first author was supported by the Agence Nationale dela Recherche (ANR) and the Austrian Science Fund (FWF) through theproject
Fractals and Numeration
ANR-FWF I1136 and the FWF Project22 855. The second author was supported by the Chinese National NaturalScience Foundation Projects 10971233 and 11171123.
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