A Core Decomposition of Compact Sets in the Plane

Transcription

1 A Core Decomposition of Compact Sets in the Plane Benoît Loridant 1, Jun Luo 2, and Yi Yang 3 1 Montanuniversität Leoben, Franz Josefstrasse 18, Leoben 8700, Austria 2,3 School of Mathematics, Sun Yat-Sen University, Guangzhou , China 3 Corresponding author: yangyi32@mail2.sysu.edu.cn July 14, 2017 Abstract A compact metric space is called a generalized Peano space if all its components are locally connected and if for any constant C > 0 all but finitely many of the components are of diameter less than C. Given a compact set C, there usually exist several upper semi-continuous decompositions of into subcontinua such that the quotient space, equipped with the quotient topology, is a generalized Peano space. We show that one of these decompositions is finer than all the others and call it the core decomposition of with Peano quotient. For specific choices of, this core decomposition coincides with two models obtained recently, namely the locally connected models for unshielded planar continua (like connected Julia sets of polynomials) and the finitely Suslinian models for unshielded planar compact sets (like disconnected Julia sets of polynomials). We further answer several questions posed by Curry in In particular, we can exclude the existence of a rational function whose Julia set is connected and does not have a finest locally connected model. eywords. Locally connected, finitely Suslinian, core decomposition. This author was supported by the Agence Nationale de la Recherche (ANR) and the Austrian Science Fund (FWF) through the project FAN ANR-FWF I1136. This author was supported by the Chinese National Natural Science Foundation Projects and

2 1 Introduction and main results In this paper, a compact metric space is called a compactum and a connected compactum is called a continuum. If, L are two compacta, a continuous onto map π : L such that the preimage of every point in L is connected is called monotone [21]. We are interested in compacta and continua in the plane or in the Riemann sphere. Given a compactum C, an upper semi-continuous decomposition D of is a partition of such that for every open set B the union of all D D with D B is open in (see [13]). Let π be the natural projection sending x to the unique element of D containing x. Then a set A D is said to be open in D if and only if π 1 (A) is open in. This defines the quotient topology on D. An upper semi-continuous decomposition of is monotone if each of its elements is a subcontinuum of. In this case, π is a monotone map. Finally, let D and D be two monotone decompositions of a compactum C, with projections π and π, and suppose that D and D both satisfy a topological property (T). We say that D is finer than D with respect to (T) if there is a map g : D D such that π = g π. If a monotone decomposition of a compactum is finer than every other monotone decomposition of with respect to (T), then it is called the core decomposition of with respect to (T) and it will be denoted by D T or simply D, if (T) is fixed. Clearly, the core decomposition D is unique, if it exists. Recently, core decompositions with respect to two specific topological properties were studied for the special class of compacta C that are unshielded, that is, = W for the unbounded component W of C \. More generally, when dealing with subsets of the Riemann sphere, a compactum Ĉ is called unshielded if = W for some component W of Ĉ \. In particular, if a rational function R : Ĉ Ĉ has a completely invariant Fatou component, then its Julia set is an unshielded compactum (see [1, Theorem (i)]). First, Blokh-Curry-Oversteegen prove in [4, Theorem 1] that the core decomposition D LC an unshielded continuum with respect to the property of being locally connected always exists. A special case is when is the connected Julia set of a polynomial. In our paper, we will solve the existence of D LC for all continua C, without assuming that is unshielded. In particular, our core decomposition will apply to the study of connected Julia sets of rational functions on the extended complex plane Ĉ, thus negatively answers [6, Question 5.2]. However, if we only consider upper semi-continuous decompositions then there might be two decompositions D 1, D 2 of an unshielded continuum C which are both Peano continua under quotient topology, such that the only decomposition finer than D 1 and D 2 is the decomposition {{z} : z } into singletons. Actually, let C [0, 1] C be Cantor s ternary set. Let be the union of of 2

3 {x + iy : x C, y [0, 1]} with {x + i : x [0, 1]} and call it the Cantor Comb. See the following figure for an approximation of. Let p 1 be the restriction to of the projection x + iy x. O Then D 1 = Figure 1: A rough approximation of the Cantor Comb. { p 1 1 (x) : x [0, 1] } coincides with D LC, the core decomposition of with respect to local connectedness. Let D 2 be the union of all the translates C y := {x + iy : x C} of with 0 y 1 and all the single point sets {z = x + i} with x / C. Then D 2 is an upper semi-continuous decomposition of, which is not monotone and which is a Hawaiian earing under quotient topology. Clearly, the only decomposition finer than both D 1 and D 2 is the decomposition {{x} : x } into singletons. Second, given an unshielded compactum C, the result of [2, Theorem 4] indicates the existence of the core decomposition of with respect to the property of being finitely Suslinian, denoted by D F S. Here we recall that a compactum is finitely Suslinian if every collection of pairwise disjoint subcontinua whose diameters are bounded away from zero is finite. Since every finitely Suslinian continuum is locally connected, we see that [4, Theorem 1] is a special case of [2, Theorem 4]. We may wonder about the existence of the core decomposition D F S of an arbitrary compactum C. However, there are examples of continua C failing to have such a core decomposition (see [2, Example 14] and Section 2 of this paper). We will replace the property of being finitely Suslinian by the property of being a generalized Peano space. This class of compacta will be defined below. For every compactum C, we will prove the existence of the core decomposition D P S Peano space. We will briefly call D P S with respect to the property of being a generalized the core decomposition of with Peano quotient. Since a finitely Suslinian compactum is also a generalized Peano space, the decomposition D P S than D F S is finer for any compactum C, when the latter exists. If the compactum C is unshielded, we will prove in Section 2 that our core decomposition D P S finitely Suslinian model D F S, given by [2, Theorem 4]. The core decomposition D LC coincides with the finest we will obtain just coincides with DP S, when C is a continuum. It is unknown whether such a core decomposition exists for a general continuum or 3

4 compactum, e.g., when can not embedded into the plane. Our work is motivated by recent studies and possible applications in the field of complex dynamics, but we will rather focus on the topological part. The concept of generalized Peano space has its origin in an ancient result by Schönflies. It also has motivations from some recent works by Blokh, Oversteegen and their colleagues. We will show that this property can be used advantageously in discussing core decompositions, besides the properties of being locally connected or finitely Suslinian. Schönflies result reads as follows. Theorem. [13, p.515, 61, II, Theorem 10]. If is a locally connected compactum in the plane and if the sequence R 1, R 2,... of components of C \ is infinite, then the sequence of their diameters converges to zero. The above theorem gives a necessary condition for planar compacta to be locally connected. This condition is also necessary for planar compacta to be finitely Suslinian, as we will prove in Theorem 4.1. However, in both cases, the condition is not sufficient. For instance, Sierpinski s universal curve is not finitely Suslinian but its complement has infinitely many components whose diameters converge to zero. Also, the closed topologist s sine curve is not locally connected but its complement has a single component. This motivates us to introduce the following condition, which happens to be necessarily fulfilled by every planar compactum, if is assumed to be finitely Suslinian or locally connected. Schönflies Condition. A compactum in the plane fulfills the Schönflies condition if for the region U bounded by any two parallel lines L 1 and L 2, the difference U \ has at most finitely many components intersecting both L 1 and L 2. Theorem 1. If a compactum in the plane is locally connected or finitely Suslinian then it satisfies the Schönflies condition. We will prove an equivalent formulation of the Schönflies condition in Lemma 3.3: a compactum C satisfies the Schönflies condition if and only if for the region U bounded by any two parallel lines L 1 and L 2, the intersection U has at most finitely many components that intersect both L 1 and L 2. Moreover, we will show that the above Schönflies condition entirely characterizes the local connectedness for continua in the plane. Theorem 2. A continuum in the plane is locally connected if and only if it satisfies the Schönflies condition. Continua like Sierpinski s universal curve indicate that the above theorem does not hold if we replace locally connected with finitely Suslinian. 4

5 The main purpose of this paper is not to characterize the finitely Suslinian compacta, but to find appropriate candidates for the core decomposition, instead of the finitely Suslinian property. Such a core decomposition of planar compacta will have interesting applications to the study on Julia sets of rational functions. See for instance the open questions proposed at the end of [6]. Combined with earlier models developed by Blokh-Curry-Oversteegen [2, 4], the results of Theorems 1 and 2 provide some evidence that planar compacta satisfying the Schönflies condition seem to be a reasonable model for the above mentioned core decompositions. Therefore, we give a nontrivial characterization of the Schönflies condition as follows. Theorem 3. A compactum C satisfies the Schönflies condition if and only if it has the following two properties. (1) Every component of is locally connected. (2) For every C > 0, all but finitely many components of are of diameter less than C. A compact metric space satisfying the above two properties in Theorem 3 will be called a generalized Peano space, simply Peano space. In particular, a Peano space is a Peano continuum if it is connected. Note that the term Peano space has been used as a synonym for Peano continuum in the literature (see [8, p.199] or [9, p.117]). In our paper, a connected Peano space means a Peano continuum, while a Peano space might be disconnected. We are now in the position to introduce our strategy to prove the existence of D P S, the core decomposition with Peano quotient, for every compactum in the plane. Let us define a relation R on any given compactum C as follows. Given two disjoint simple closed curves J 1 and J 2, we denote by U(J 1, J 2 ) the component of Ĉ \ (J 1 J 2 ) bounded by J 1 J 2. This is an annulus in the extended complex plane Ĉ. We say that two points x, y are related under R provided that there exist two disjoint simple closed curves J 1 x and J 2 y such that U(J 1, J 2 ) contains an infinite sequence of components Q k intersecting both J 1 and J 2, whose limit lim k Q k under Hausdorff distance contains {x, y}. Definition 4. Let C be a compactum and R the relation defined above. We denote by R be the collection of all the closed equivalence relations on containing R (as subsets of ). Moreover, we denote by the intersection of all the equivalence relations of R. It is also an element of R, and we call it the minimal equivalence containing R, or the Schönflies equivalence on, for short. One can check that the equivalence class [x] under the Schönflies equivalence is a continuum for every x (see Proposition 5.1). Moreover, if in Theorem 3 is assumed to be unshielded, 5

6 then it is finitely Suslinian if and only if it satisfies the Schönflies condition (see Theorem 2.1). Also, if the compactum in Definition 4 is unshielded, then coincides with the relation developed in [2], given by the finest finitely Suslinian model (see Section 2). From now on, we denote by D the collection of equivalence classes [x] := {z : z x}. Thus D is the decomposition induced by the Schönflies equivalence on. It is standard to verify that D is necessarily a compact, Hausdorff and secondly countable space under quotient topology [11, p.148, Theorem 20]. Therefore, it is metrizable by Urysohn s metrization theorem [11, p.125, Theorem 16]. We will prove that it satisfies the two properties of Theorem 3, thus is a Peano space. Theorem 5. Under quotient topology D is a Peano space. After this we will show that D is finest in the following sense. Theorem 6. Let be the Schönflies equivalence on a compactum C and π(x) = [x] the natural projection from to D. If f : Y is monotone map onto a Peano space Y, then there is an onto map g : D Y with f = g π. By Theorems 5 and 6, we can conclude that D equals the core decomposition D P S with Peano quotient. of Theorem 7. Every compactum C has a core decomposition D P S with respect to the property of being a Peano space. It coincides with the decomposition D induced by the Schönflies equivalence on. Remark 8. Theorem 7 answers [6, Question 5.2] and partially answers [6, Question 5.2]. In the first part of [6, Question 5.2], Curry asks: for what useful topological properties P does there exist a finest decomposition of every Julia set J(R) (of a rational function R) satisfying P? By Theorem 7, the property of being a Peano space is such a property. Moreover, in the last part of [6, Question 5.2], Curry asks: which of these (properties) is the appropriate analogue for the finest locally connected model? Since the core decomposition D P S in Theorem 7 generalizes the earlier finest models obtained in [2, 4], the answer is again the property of being a Peano space. Moreover, in the middle part of [6, Question 5.2], Curry asks: is the decomposition (satisfying the right property P) dynamic? This interesting question provides a reasonable angle to apply the core decomposition obtained in Theorem 7 to the study of complex dynamics, in particular, to the study on dynamics of a rational function restricted to its Julia set. Remark 9. For any compactum, the decomposition {Q : Q is a component of } always induces a Peano quotient, whose components are single points. Therefore, an important problem 6

7 is to determine whether the core decomposition D P S of a given compactum C induces a quotient space having a non-degenerate component. In such a case, we say that D P S is a nondegenerate core decomposition; otherwise, we say that it is a degenerate core decomposition. Clearly, the core decomposition of every indecomposable continuum C is degenerate, since D P S = {}. The studies of Blokh-Curry-Oversteegen [2, 4], whose models are generalized by the core decomposition introduced in this paper, already provide very interesting results on the existence of non-degenerate core decompositions. For instance, by [4, Theorem 27], if a continuum X has a well-slicing family, then the image of X under the natural projection π : D P S is a non-degenerate continuum, hence has a non-degenerate core decomposition. If is the Julia set of a polynomial, then it is stated in [2, Corollary 24] that has a nondegenerate core decomposition D P S if and only if has a periodic component Q which, as a plane continuum, has a non-degenerate core decomposition DQ P S. In other words, to compute the core decomposition D P S we just need to compute the core decomposition DP S Q for all the periodic components Q of. If the above Julia set is connected and is finitely irreducible, the result of [6, Theorem 4.1] indicates that the core decomposition D P S D P S satisfies either DP S = {} or = {{x} : x }. Actually, in the former case is an indecomposable continuum and in the latter case it is homeomorphic to [0, 1]. Finally, if X is an unshielded continuum and Y X is a subcontinuum, Blokh-Oversteegen-Timorin [5] obtained recently a sufficient condition for the core decomposition D P S Y of Y to embed canonically into that of X. As an application to complex dynamics, the authors also considered the special case that X is the connected Julia set of a renormalizable polynomial P and Y is the so-called small Julia set, for a polynomiallike map obtained as a restriction of some iterate P n with n > 1. Combining these results with the core decomposition obtained in our paper, one may investigate problems like the local connectedness of Julia set of infinitely renormalizable polynomials. We arrange our paper as follows. Section 2 briefly recalls facts on local connectedness, laminations in complex dynamics and core decompositions. We provide an argument based on Theorems 3 and 5 that the core decompositions D P S and DF S of an unshielded compactum C are equal. Section 3 gives preliminary lemmas needed in the proofs of the main theorems. Section 4 proves Theorems 1 to 3. Sections 5 and 6 respectively prove Theorems 5 and 6. Acknowledgments. We are grateful to the referee for pointing out significant and confusing typos and for an interesting question that led to Remark 9. 7

8 2 Local Connectedness, Lamination, and Core Decomposition The investigation of local connectedness dates back to the nineteenth century. Cantor proved that the unit interval and the unit square have the same cardinality. In other words, there exists a bijection h : [0, 1] [0, 1] 2, and this map h can not be continuous. Peano and some of his contemporaries further obtained continuous surjections from [0, 1] onto planar domains like squares and triangles. The range of a continuous map from [0, 1] into a metric space is therefore often called a Peano continuum. Peano continua were then fully characterized via the notion of local connectedness: indeed, Hahn and Mazurkiewicz showed that a continuum is a Peano continuum if and only if it is locally connected. Among the Peano continua of the plane, the boundary of a bounded simply connected domain U provides a special case. By the Riemann Mapping Theorem, there is a conformal isomorphism from the unit open disk D = { z < 1} onto U. Furthermore, Carathéodory s theorem states that this conformal mapping has a continuous extension to the closed disk D if and only if the boundary U is locally connected. Considering U as a domain in the extended complex plane Ĉ, we may assume, after the action of a Möbius map, that U. Then X = C \ U is a full continuum, i.e., it has a connected complement U. Moreover, there is a conformal isomorphism Φ from D = { z > 1} { } onto U, fixing Ĉ and having a real derivative at. In the study of quadratic dynamics, examples of the above map Φ are (1) Böttcher maps for hyperbolic polynomials z z 2 + c with c lying in a hyperbolic component of the Mandelbrot set M and (2) the conformal isomorphism sending D onto Ĉ \ M. For the map Φ in (1), the boundary of Φ(D ) is the Julia set J c of z z 2 + c, which is known to be locally connected. In this case, J c is the image of the unit circle D = D under a continuous map (called Carathéodory s loop), hence may be considered as the quotient space of an equivalence relation on D. This equivalence relation is a lamination in Thurston s sense [20]. Douady [7] proposed a pinched disc model describing full locally connected continua in the plane. Extending the lamination in a natural way to a closed equivalence relation L on the closed unit disk D, he obtains that c is homeomorphic with the quotient D/L, where c is the filled Julia set of the polynomial z z 2 + c (J c = c ). The pinched disc model works even if the full continuum is not locally connected. The map Φ in (2) provides a typical example, in which the boundary of Φ(D ) coincides with that of the Mandelbrot set M. Denote by R θ the image of {re 2πθi : r > 1} under Φ for θ [0, 1]. R θ ( is called the external ray at θ. If lim re 2πθi) is a point on M, denoted as c θ, we say that r 1 Φ R θ lands at c θ. It is known that all external rays R θ with rational θ lands. Douady therefore 8

9 { } [7] defines an equivalence relation on e 2πθi : θ Q [0, 1] by setting θ Q M θ if and only if c θ = c θ. As a subset of D D, the closure of Q M (denoted M) turns out to be an equivalence relation on D (see [7, Theorem 3] for fundamental properties of M ). Let us now recall the main ideas of Blokh-Curry-Oversteegen [4] concerning locally connected models for unshielded continua in the plane. Let Ĉ be an unshielded continuum with = U, where U is the unbounded component of C \. Let Φ be a conformal mapping that sends D to U and fixes. For any θ [0, 1], the impression at e 2πθi, defined by Imp(θ) = { } lim Φ(z i) : {z i } D, lim z i = e 2πθi, i i is a subcontinuum of. By [4, Lemma 13] there is a minimal closed equivalence relation I on such that every equivalence class is made up of impressions and is a subcontinuum of. By [4, Lemma 16], if R is an arbitrary closed equivalence on such that the quotient space /R is a locally connected continuum then I is contained in R (as subsets of ). The first part of [4, Lemma 17] obtains that the quotient /I is a locally connected continuum, called the locally connected model of. Now we may define a closed equivalence relation on D by requiring that θ θ if and only if Imp(θ) and Imp(θ ) lie in the same equivalence class [x] I. Then, by the second part of [4, Lemma 17], the equivalence is a lamination such that the induced quotient D/ is homeomorphic to /I. In particular, when is the Julia set of a polynomial f of degree d 2 without irrationally neutral cycles, iwi [12] investigates the structure of the classes [x] I and shows that every [x] I coincides with the fiber at x [12, Definition 2.5] defined by Fiber(x) = {y : no finite set separates y from x in }. Here, a finite set C separates two points of if these points are in distinct components of \ C. We refer to [12, Corollary 3.14] and [12, Proposition 3.15] for important properties of Fiber(x), and to Schleicher s earlier works [17, 18, 19] for another approach in defining fibers. To generalize the above model, Blokh, Curry and Oversteegen [2] define an equivalence on an unshielded compactum C to be the minimal closed equivalence such that every limit continuum is contained in a single class [x] := {z : z x}. Recall that a limit continuum is the limit lim N k under Hausdorff distance of an infinite sequence of pairwise k disjoint subcontinua N k. The quotient space D F S = {[x] : x } is necessarily a compact metrizable space [16, p.38, Theorem 3.9]. The authors of [2] further check that it is finitely Suslinian [2, Lemma 13]. Every element d of the above decomposition D F S, as a subset of C, possesses the following 9

10 property: the union of all the bounded components of C \ d does not intersect. The authors of [2] then use Moore s theorem to prove that D F S with finitely Suslinian quotient [2, Theorem 19]. In other words, D F S of with respect to the finitely Suslinian property. is the finest monotone decomposition of is the core decomposition Let now C be a any compactum. Let D = {[x] : x } with defined as above. On the other side, let be the Schönflies equivalence on, defined in Definition 4 as the minimal closed equivalence relation containing the relation R. We write D = {[x] : x }. We want to compare these two decompositions. The definition of R indicates that if (z 1, z 2 ) R then there is a limit continuum containing both z 1 and z 2. This in turn indicates that is contained in as subsets of, hence the decomposition D always refines D. These two decompositions turn out to be equal provide that is unshielded. Note that in this case D = DF S is the core decomposition of with respect to the finitely Suslinian property [2, Theorem 19]. Actually, the unshielded assumption of implies that the bounded components of C \ d for every d D are all disjoint from. Let d be the union of d with the bounded components of C \ d. Then D C := {d : d D } {z} : z / d d D is a monotone decomposition of C, such that d 1 d 2 = for any d 1 d 2 D. By Moore s Theorem, the quotient D C is homeomorphic to the plane and the natural projection Π : C D C sends to a planar compactum. Since every d is disjoint from the unbounded component W of C \, the image Π(W) is a region in the plane D C whose boundary contains Π(). That is to say, Π() is also an unshielded compactum in the plane D C. On the other hand, for any x, y it is direct to check that Π(x) = Π(y) if and only if π(x) = π(y). Therefore, the quotient D is homeomorphic to Π() hence may be embedded into the plane as an unshielded compactum. Theorem 5 of this paper says that D is also a Peano space. By the following theorem, such a planar compactum is finitely Suslinian. Consequently, the core decomposition decomposition D F S is finer than D, and we have D F S = D. Theorem 2.1. If an unshielded compactum C is a Peano space then it is finitely Suslinian. Proof. By Theorem 3 and the definition of Peano space, we only need to consider the case when is an unshielded continuum. Recall that a continuum X is regular at a point x X if for every neighborhood V x of x there exists a neighborhood U x of x whose boundary U x = U x X \ U x is a finite set [22, p.19]. A regular continuum is just one that is regular at each of its points. Here it is standard to check that a regular continuum is finitely Suslinian. Therefore, our proof will 10

11 be completed if only we can verify that, a locally connected unshielded planar continuum, is a regular continuum. We will use the notions of pseudo fiber and fiber for planar continua, recently introduced in [10], from which a numerical scale is developed that measures the extent to which such a continuum is locally connected. More precisely, for any point x, the pseudo fiber E x at x consists of the points y such that there does not exist a simple closed curve γ with γ X a finite set, called a good cut, such that x and y lie in different component of C \ γ; the fiber F x at x is the component of E x that contains x. By [10, Proposition 4.2], for the locally connected unshielded continuum C every pseudo fiber E x equals the single point set {x}. Therefore, given any x and any open set U x, we can choose good cut γ y such that x and y lie in different component of C \ γ y. Let U y, V y be the component of C \ γ y with x U y and y V y. Then {V y : y \ U} is an open cover of the compact set \ U. Fix a finite sub-cover {V y1,..., V yn }. Then n U x := U yi i=1 is open in, contains x, and is contained in U. Recall that, for 1 i n, the intersection B i := U yi V yi is contained in γ yi hence is also a finite set. Since the boundary of U x in is defined to be the intersection U x ( \ U x ) and is a subset of n ( ) U x V yi, i=1 which is in turn a subset of i B i and hence is also a finite set. This verifies that is regular at x. Consequently, from flexibility of x we can infer that is a regular continuum. Note that another proof of this theorem can be found in [3, Lemma 2.7]. We mention that if the compactum C is not assumed to be unshielded, then the core decomposition of with respect to the finitely Suslinian property may not exist. Consider for instance the locally connected continuum C of [2, Example 14]. It admits two monotone decompositions D 1, D 2 such that the quotients are finitely Suslinian. However, the only partition finer than both D 1 and D 2 is the trivial decomposition {{x} : x }. Therefore, the core decomposition of with respect to the finitely Suslinian property does not exist, while the trivial decomposition {{x} : x } is the core decomposition of with respect to the property of being a Peano space. We end up this section with an example of a continuum C having two properties. Firstly, the core decomposition D has an element d such that at least two components of C \ d intersect ; secondly, the resulted quotient space D can not be embedded into the plane. 11

12 Example 2.2. Let the compactum C be the union of the closure of the unit disk D = { (1 {z C : z < 1} and the spiral curve L = + e t ) e 2πit : t 0}. By routine works one may check that the core decomposition of with respect to the property of being a Peano space is exactly given by D = {{x} : x (D L)} { D}. Clearly, the quotient space is the one-point union of a sphere with a segment, thus can not be embedded into the plane. 3 Some Useful Lemmas The lemmas in this section give a couple of results that are used in latter sections. Lemma 3.1 is from [14, Lemma 2.1] and will be used in proving Lemmas 3.2 and 3.3. Lemma 3.1. Suppose that A [0, 1) [0, 1] and B (0, 1] [0, 1] are disjoint closed sets. Then there exists a path in [0, 1] 2 \ (A B) starting from a point in (0, 1) {0} and leading to a point in (0, 1) {1}. Before stating the next lemma, we recall the following definitions and facts. For X C, we say that X = A B (A, B ) is a separation of X if A B = A B =. Remember that, if x 0 is a point in X then the component of X containing x 0 is the maximal connected set P X with x 0 P. The quasi-component of X containing x 0 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 containing a point p X may be defined as the intersection of all closed-open subsets of X containing p. Any component is contained in a quasicomponent, and quasi-components coincide with the components whenever X is compact [13]. If X is compact we denote by X be the union of X with all the bounded components of C \ X. We call X the topological hull of X, following Blokh-Curry-Oversteegen [2]. Lemma 3.2. Let C be compactum and x 0 /. Then x 0 lies in the unbounded component of C \ provided that it does not lie in the topological hull P for any component P of. Proof. Fix a large enough circular disk D r with radius r > 0 whose interior contains. For each component P of, since x 0 is assumed to be in the unbounded component of C \ P, we may choose a path α P disjoint from P which joins x 0 to a fixed point x 1 D r. Let δ be a 12

13 number smaller than the distance dist(p, α P ) := { z 1 z 2 : z 1 P, z 2 α P }. As P is also a quasi-component of, we may choose a separation = A y,p B y,p with P A y,p and y B y,p for any point y with dist(y, P) δ. By compactness of {y : dist(y, P) δ}, there are finitely many points y 1,..., y l with dist(y i, P) δ such that B y1,p, B y2,p,..., B yl,p are open under the induced topology of and form a cover of {y : dist(y, P) δ}. Clearly, = A P B P is also a separation with α P A P =, where l A P = A yi,p and l B P = B yi,p. i=1 i=1 Since every A P is both open and closed under the induced topology of, we see that there are finitely many components P 1,..., P m with ( m ) = A Pi. i=1 Rename α Pi as α i for 1 i m. Let A 1 = A P1. And, for 2 i m, let i 1 i 1 A i = A Pi \ = A Pi \. A j j=1 A Pj j=1 Then A 1,..., A m are disjoint compact sets such that m i=1 A i. Moreover, every α i is an arc satisfying α i A i =. To finish our proof, we will show that there is a path α disjoint from that joins x 0 to x 1. Recall that the paths α 1, α 2 : [0, 1] C, with common initial point α i (0) = x 0 and common endpoint α i (1) = x 1, are homotopic relative to {0, 1} under the straight line homotopy F : [0, 1] 2 C defined by F(t, s) = tα 1 (s) + (1 t)α 2 (s) for any t, s [0, 1]. The disjointness of A 1, A 2 indicates that F 1 (A 1 ), F 1 (A 2 ) are disjoint compact subsets of the unit square. By Lemma 3.1 there is a path β in [0, 1] 2 \ ( F 1 (A 1 ) F 1 (A 2 ) ) starting from a point in {0} (0, 1) and leading to a point in {1} (0, 1). Then F(β) is a path disjoint from A 1 A 2 and joins x 0 to x 1. Repeating the above argument on the paths F(β) and α 3, we can find a path disjoint from A 1 A 2 A 3 that joins x 0 to x 1. Inductively, we can find a path α disjoint from m i=1 A i, hence disjoint from, that joins x 0 to x 1. Lemma 3.3. Let C be a compact set and U the region bounded by two parallel lines L 1 and L 2. Let U i be the component of C \ (L 1 L 2 ) with U i = L i. We have the following: (1) If L 1 L 2 and no connected subset of intersects both L 1 and L 2, then there is a separation = A 1 A 2 with A 1, A 2 compact sets such that (U i ) A i. 13

14 (2) If U \ has at least m 2 components intersecting both L 1 and L 2, then U has at least m 1 components intersecting both L 1 and L 2. (3) If U has at least m 2 components intersecting both L 1 and L 2, then U \ has at least m components intersecting both L 1 and L 2. Remark 3.4. The results of Lemma 3.3 still hold if we replace C by Ĉ, L 1, L 2 by two disjoint simple closed curves J 1, J 2 Ĉ and U by the region W bounded by J 1 J 2. For Part (1), the argument is still valid. For Part (2), we may remove an open arc α W, which joins a point in J 1 to a point in J 2 and has a closure disjoint from ; then the difference W \ α contains W and has the same topology of U. For part (3), one may use the pattern of polar brick wall tiling, as indicated in the figure below, to find such an open arc α. r=1 R=2 Figure 2: A Polar Brick Wall Tiling, in which every two tiles either are disjoint or intersect at a non-degenerate arc on the boundary. Proof for Lemma 3.3. We first prove Item (1). Every component of the compact set U is also a quasi-component. Therefore, for any a (L 1 ) and any b (L 2 ), there exists a separation U = W a,b V a,b with a W a,b and b V a,b such that the sets W a,b, V a,b are closed in C and relatively open in U. In particular, the collection {V a,b : b (L 2 )} is an open cover of the compact set L 2 in U, so there exists a finite subcover V a,b1,..., V a,bk. Let k k W a = W a,bi and V a = V a,bi. i=1 i=1 Then U = W a V a is a separation such that a W a and (L 2 ) V a. By flexibility of a L 1, the collection {W a : a (L 1 )} is an open cover of L 1 in U and has a finite subcover W a1,..., W al. Let l l W = W ai and V = V ai. i=1 i=1 14

15 Then U = W V is a separation such that (L 1 ) W and (L 2 ) V. If we set A 1 = W (U 1 ) and A 2 = V (U 2 ), then = A 1 A 2 is a separation with (U 1 ) A 1 and (U 2 ) A 2. By construction, A 1 and A 2 are closed subsets of C. This proves Item (1). The proof of Item (2) reads as follows. Let R 1,..., R m be m components of U \ intersecting both L 1 and L 2. For each i {1,..., m}, let α i be a simple arc in the component R i of U \ with endpoints a i L 1, b i L 2 and α i \ {a i, b i } U R i. Choose a line L perpendicular to L 1 such that all of the arcs {α i } 1 i m are in the same component of C \ L. We can assume that a 1 is the nearest point to L among the collection {a i } 1 i m, and a j is the nearest point to L among the set of points {a i } j i m, then the arcs α i are renamed according to the distance of their endpoints on L 1 to the line L. Since the arcs α i are disjoint, we can use Jordan curve theorem to infer that the distance from b j to L is smaller than that from b j+1 to L for 1 j m 1. Let β i be the arc on L 1 with endpoints a i, a i+1 for 1 i m 1. Let γ i be the arc on L 2 with endpoints b i, b i+1 for 1 i m 1. Then Γ i = α i β i γ i α i+1 is a simple closed curve for 1 i m 1. Let W i U be the bounded component of C \ Γ i. By a theorem of Schönflies [15, p.68,theorem 6], there is a homeomorphism h i : W i [0, 1] 2 such that h i (a i ) = (0, 0), h i (a i+1 ) = (0, 1), h i (b i ) = (1, 0), h i (b i+1 ) = (1, 1). ) Since α i and α i+1 lie in distinct components of U \, the compact set h i (W i intersects each of h i (β i ) and h i (γ i ) and is disjoint from h i (α i α i+1 ) = {0, 1} [0, 1]. See left part of Figure 3 for relative locations of h i (α i ), h i (α i+1 ), h i (β i ) and h i (γ i ). h i (a i+1 ) h i (α i+1 ) h i (b i+1 ) h i (α i+1 ) h i (β i ) h i (γ i ) U1 L 1 L 2 U2 h i (a i ) h h i (α i ) i (b i ) h i (α i ) Figure 3: Relative locations of the points h i (a i ), h i (b i ), h i (a i+1 ), h i (b i+1 ) in [0, 1] 2. ) We claim that h i (W i has a component P i that intersects both h i (β i ) and h(γ i ). This ( ) will verify Item (2), since P i is a closed subset of [0, 1] 2, hence N i := h 1 i (P i ) W i is a sub-continuum of and intersects both L 1 and L 2. W i W i+1 = α i+1 =. 15 Note that N i N i+1 =, since

16 ) Suppose on the contrary that h i (W i has no component intersecting both h i (β i ) and h(γ i ). We will use Item (1) to induce a contradiction. ) To this end, we put = h i (W i. Then, let L 1 be the line through h i(a i ), h i (a i+1 ) and L 2 the line through h i(b i ), h i (b i+1 ). Moreover, let U i be the component of C \ (L 1 L 2 ) with Ui = L i. See right part of Figure 3. By Item (1), we are able to find a separation = A 1 A 2 into compact sets with (U i ) A i. It follows that A 1 [0, 1) [0, 1] and A 2 (0, 1] [0, 1]. In particular, A 1 and A 2 satisfy the conditions in Lemma 3.1, from which we can infer the existence of a path P in [0, 1] 2 \ (A 1 A 2 ) starting at a point in (0, 1) {0} h i (α i ) and leading to a point in (0, 1) {1} h i (α i+1 ). The inverse h 1 i (P) is then a path in W i \ ( U \ ), which connects a point in α i R i to a point in α i+1 R i+1. This is impossible, since R i, R i+1 are distinct components of U \. Finally we prove part (3). Let Q 1,..., Q m be m components of U intersecting both L 1 and L 2. Clearly ǫ = 1 3 min {dist(q i, Q j ) : 1 i < j m} is a positive number. For 1 i m, let Q i (ǫ) be the open ǫ-neighborhood of Q i. Then for every point x in (U ) \ Q i (ǫ) the quasi-component of U containing x is disjoint from the one containing Q i, so that there is a separation U = C x D x with Q i C x and x D x. Under the induced topology of U, the collection { } D x : x (U ) \ Q i (ǫ) is an open cover of the compact set (U ) \ Q i (ǫ), which has a finite sub-cover { } D xk : x 1,, x n (U ) \ Q i (ǫ). Then U = C i D i is a separation with Q i C i Q i (ǫ) for 1 i m, where C i = n C xk and D i = k=1 Moreover, U = C D is also a separation, where n D xk. k=1 m C = C i and m D = D i. i=1 i=1 Denote by d the distance between L 1 and L 2. Let δ = min{dist(c, D), ǫ}. Fix an integer N 1 > 0 with d 2N 1 < δ 4 and divide U into 2N 1 equal strips by 2N 1 1 lines parallel to L 1 such that the width of each strip is d 2N 1. In the rest part of our proof we assume that L 1 is the horizontal axis and L 2 is on the upper half plane. 16

17 Let T 1 = {B k : k Z} be a tiling of the lowest strip by squares of side length T 1 is a cover and the squares have disjoint interiors. Then T 2n 1 := { ( ) } (2n 2)d B k +, 0 : k Z 4N 1 is a tiling of the (2n 1)-th strip for 2 n N 1, and T 2n := { ( (2n 1)d B k +, 4N 1 d 4N 1 is a tiling of the (2n)-th strip for 1 n N 1. Moreover, T := ) } : k Z 2N 1 i=1 d 2N 1, such that T i is a tiling of the whole strip U that represents a brick wall pattern, such that two squares either are disjoint or intersect at a non-degenerate segment. For 1 i m, let C i be the union of all the squares in T intersecting C i. Let R i be the component of C i containing Q i. Clearly, we have Q i C i C i Q i(ǫ). By Torhorst Theorem [13, p.512, 61, II, Theorem 4], the unbounded component of C \ R i is bounded by a simple closed curve J i. Clearly, the curves J 1,..., J m are pairwise disjoint. Choose a i (J i L 1 ) and b i (J i L 2 ) with dist(a i, L) = dist(j i L 1, L) and dist(b i, L) = dist(j i L 2, L). Then J i \ {a i, b i } is made up of two open arcs; one of them must be contained in U and will be denoted as A i. Let H be the union of L 1, L 2 and all the arcs A 1,..., A m. Fix a permutation i 1 i 2 i m of 1, 2,..., m such that the distance from a ik to L is smaller than that from a ik+1 to L for 1 k m 1. Then C\H has exactly m 1 components W 1, W 2,..., W m 1 and the boundary of W k is A ik A ik+1 α k β k for 1 k m, where α k L 1 and β k L 2 are minimal arcs containing { a ik, a ik+1 } and { bik, b ik+1 }, respectively. Since J ik J ik+1 =, from the choice of the points a ik, b ik and the arc A ik, we can infer that J ik \ A ik is contained in the closure W ik, which necessarily contains Q ik. Since Q ik intersects both L 1 and L 2, the arc A ik is separated from all the arcs A il with l > k by Q ik in U. That is to say, all the arcs A 1, A 2,..., A m lie in different components of U \, indicating that U \ has m components which intersect both L 1 and L 2. Let R be the closed relation on a compact set C, firstly mentioned before Definition 4. The following lemma provides an equivalent approach to define R. In the sequel, the bounded component of a simple closed curve Γ C is denoted as Int(Γ) and called the interior of Γ. Lemma 3.5. Given a compact set C and n 3 disjoint simple closed curves Γ 1,..., Γ n C such that Γ 2,..., Γ n Int(Γ 1 ) and Int(Γ j ) Int(Γ j ) = for i j 2. Let W be the annulus bounded by Γ 1, Γ 2. Let W be the only region with W = n k=1 Γ k. If the intersection 17

18 W has infinitely many components P k each of which intersects both Γ 1 and Γ 2, such that lim k P k = P under Hausdorff distance, then (z 1, z 2 ) R for any z 1 (Γ 1 P ) and z 2 (Γ 2 P ). Proof. We only consider the case n = 3, to which the other cases may be reduced. Fix a circular disk D whose interior contains. By Lemma 3.2, there are two mutually exclusive possibilities: (1) the interior Int(Γ 3 ) is contained in P for some component P of W or (2) a point x 0 Γ 3 may be connected to a point x 1 on the circle D by a path α disjoint from W. In the former case, every P k other than P is disjoint from P ; each of those components is a component of W ( P ) hence also a component of W. This guarantees that (z 1, z 2 ) R for any z 1 (Γ 1 P ) and z 2 (Γ 2 P ). In the latter case, let y be the first point on α that also lies in Γ 1 Γ 2 and α 0 α the irreducible sub-path with end points x 0, y. There are two subcases, y Γ 2 or y Γ 1, and we just consider the subcase y Γ 2 since the same argument applies to the other subcase. We may slightly thicken α 0 and find two disjoint arcs α, α close enough to α 0, each of which does not intersect W and joins a point on Γ 3 to a point on Γ 2, such that α, α are contained in W except for their end points. Then the unbounded component of C \ (Γ 2 Γ 3 α α ) is bounded by a simple closed curve Γ 2 contained in Int(Γ 2), such that the region W bounded by Γ 1, Γ 2 is an annulus. Clearly, we have W = W, and thus every P k is also a component of W. Consequently, we have (z 1, z 2 ) R for any z 1 (Γ 1 P ) and z 2 (Γ 2 P ) (Γ 2 P ). 4 Proofs for Theorems 1 to 3 Firstly, we copy the ideas of Schönflies result [13, p.515, 61, II, Theorem 10] and obtain a necessary condition for a planar compactum to be finitely Suslinian. Theorem 4.1. Given a finitely Suslinian compactum C. If the sequence R 1, R 2,... of components of C \ is infinite then the sequence of their diameters converges to zero. Proof. Suppose conversely that there exists ǫ > 0 and infinitely many integers i 1 < i 2 < such that the diameter δ(r in ) > 3ǫ. For each component R in, choose an arc α in R in with diameter larger than 3ǫ. We may assume that α in converges to α 0 under Hausdorff distance. 18

19 Here we have δ(α 0 ) 3ǫ. Choose two points p, q α 0 with p q = 3ǫ. Then, we can fix two points p 1, p 2 in the interior of the segment pq with p 1 p 2 = 2ǫ. Let L i be the line through p i which is perpendicular to pq. Let U be the region bounded by L 1 L 2. Since lim α i n n = α 0, there exists an integer N such that α in intersects both L 1 and L 2 for all n > N. By part (2) of Lemma 3.3, there exist infinitely many components of U which intersect both L 1 and L 2. This contradicts the condition that is finitely Suslinian. Secondly, we prove Theorem 1 as follows. Proof for Theorem 1. The part for locally connected compacta is a direct corollary of Schönflies result [13, p.515, 61, II, Theorem 10]. So we only consider the part for finitely Suslinian compacta. Suppose on the contrary that there exist two parallel lines L 1, L 2 such that the difference U \ has infinitely many components R 1, R 2,... intersecting each of L 1 and L 2. Here U is the only component of C \ (L 1 L 2 ) bounded by L 1 L 2. By part (2) of Lemma 3.3, U has infinitely many components intersecting each of L 1 and L 2. This contradicts the assumption that is finitely Suslinian. Then we continue to prove Theorem 2. Proof for Theorem 2. We just show that a continuum C satisfying the Schönflies condition is locally connected. Suppose on the contrary that is not locally connected at a point x 0. By definition of local connectedness [13, p.227, 49, I, Definition], there would exist a closed square V centered at x 0 such that the component P 0 of V containing x 0 is not a neighborhood of x 0 with respect to the induced topology on V. In other words, there exists a sequence {x k } k=1 in (V ) \ P 0 with lim k x k = x 0 such that the components of V containing x k, denoted P k, are pairwise disjoint. Recall that the hyperspace of all closed nonempty subsets of V is a compact metric space under Hausdorff distance. Coming to an appropriate subsequence, if necessary, we may assume that P k converges to P in Hausdorff distance. From this, we see that P is a sub-continuum of P 0 and that the diameter of P k, denoted δ(p k ), converges to δ(p ). By connectedness of, each P k must intersect V hence P intersects V. Since P k P under Hausdorff distance, we can pick some point y 0 ( V P ) and points y k ( V P k ) for all k 1 such that y 0 := lim k y k. 19

20 Since V consists of four segments and contains the infinite set of points {y k },we may fix a line L 1 crossing infinitely many y k, which necessarily contains y 0. Then, fix a line L 2 parallel to L 1 which separates x 0 from y 0, so that x 0 and y 0 lie in different components of C \ L 2. Let U be the strip bounded by L 1 and L 2. Obviously, there exists an integer N such that P n intersects both L 1 and L 2 for n N. Without loss of generality, we may assume that every P k intersects both L 1 and L 2. It follows that for all k 1 the intersection U P k = V U P k has a component Q k intersecting both L 1 and L 2. Otherwise, by part (1) of Lemma 3.3 there is a separation U P k = A 1 A 2, where A i is the union of all the components of U P k intersecting L i. Let U 1, U 2 be the two components of C \ U, with L i = U i. Then P k = [A 1 (U 1 P k )] [B 1 (U 2 P k )] is a separation. This contradicts connectedness of P k Therefore, by part (3) of Lemma 3.3, our proof will be completed if only we can show that for all but two integers k 1 the continuum Q k is also a component of U. To this end, for all k 1 we may fix two points a k (L 1 Q k ) and b k (L 2 Q k ). Then, we fix a line L perpendicular to L 1 and disjoint from. See Figure 4. Now, we call a continuum Q k a nearest L 1 L L j,1 L k,1 L l,1 a j a k a l Q j Q k Q l b j b k b l L 2 L j,2 L k,2 L l,2 Figure 4: Relative locations of a k, b k and Q j, Q k, Q l. component (respectively a furthest component) if dist (a k, L) < max {dist (a j, L) : j k} (dist (a k, L) > max {dist (a j, L) : j k}). Clearly, there exist at most one nearest component and at most one furthest component. We claim that all the other Q k is also a component of U. Actually, if U i denotes the component of C \ U with U i = L i we can choose for all k 1 two rays L k,1 U 1 and L k,2 U 2 parallel to L such that a k L k,1 and b k L k,2. The above Figure 4 gives a simplified depiction for relative locations of L, L i and a j, a k, a l. If Q k is neither nearest nor furthest there exist two components Q j, Q l with dist (a j, L) < dist (a k, L) < dist (a l, L). In this case, we can use an appropriate brick wall tiling of U and the Jordan curve theorem to infer that Q k \ (L 1 L 2 ) is contained in a bounded component W of 20

21 C \ M, where M = a j a l b j b l Q j Q l is a continuum which lies entirely in V. Therefore, (W M) is a subset of V ; and we are able to choose a separation W = A B with Q k A and (Q j Q l ) A =. From this we can infer that Q k is also a component of A, which is disjoint from B 1 := (U \ W). That is to say, the intersection U is divided into two disjoint compact subsets, A and B B 1. Combining this with the fact that Q k is a component of A, we already verify that Q k is also a component of U. Finally, we prove Theorem 3. Proof of Theorem 3. Suppose that (1) and (2) hold and assume that does not satisfy the Schönflies condition. Then there exists a region U bounded by two parallel lines L 1 and L 2 such that U has infinitely many components {N k } which intersect both L 1 and L 2. Due to (2), there exists infinitely many {N ki } in one component of which is denoted by P, and these N ki are also in different components of U P. That is to say, P does not satisfy the Schönflies condition. By Theorem 2, we reach a contradiction to the local connectedness of P. This verifies the if part. To prove the only if part we assume that satisfies the Schönflies condition and verify conditions (1) and (2) as follows. Given any component P of and the region U bounded by any two parallel lines L 1 and L 2, it is routine to check that every component of U P is also a component of U. Thus U P has finitely many components intersecting both L 1 and L 2. By Theorem 2, P is locally connected. If (2) is not true, so that there exist an infinite sequence of sub-continua {N k } lying in distinct components of, denoted Q k, such that their diameters δ(n k ) are greater than a positive constant C, then we can choose a subsequence {N kn } converging to a continuum N X under Hausdorff distance. Clearly, the diameter δ(n ) C. So we can choose two points x, y N with x y = C and two parallel lines L 1, L 2 perpendicular to the line crossing x, y and intersecting the interior of the segment xy. Now we can see that all but finitely many N k Q k must intersect L 1 and L 2 at the same time. This implies that for all but finitely many integers k 1, there exists a component P k of U Q k intersecting both L 1 and L 2. Here U is the region bounded by L 1, L 2. For those integers k, the continuum P k is also a component of U, which contradicts the assumption that satisfies the Schönflies condition. 21