Approximations by disjoint continua and a positive entropy conjecture
aa r X i v : . [ m a t h . GN ] J u l APPROXIMATIONS BY DISJOINT CONTINUA ANDA POSITIVE ENTROPY CONJECTURE
DAVID S. LIPHAM
Abstract.
E.D. Tymchatyn constructed a hereditarily locally connected continuum whichcan be approximated by a sequence of mutually disjoint arcs. We show the example re-opens a conjecture of G.T. Seidler and H. Kato about continua which admit positiveentropy homeomorphisms. We prove that every indecomposable semicontinuum can beapproximated by a sequence of disjoint subcontinua, and no composant of an indecompos-able continuum can be embedded into a Suslinian continuum. We also prove that if Y is ahereditarily unicoherent Suslinian continuum, then there exists ε > such that every two ε -dense subcontinua of Y intersect. Introduction
In 1990, G.T. Seidler proved that every homeomorphism on a regular curve has zero topo-logical entropy [14, Theorem 2.3]. He conjectured:
Every homeomorphism on a rational curvehas zero topological entropy [14, Conjecture 3.4]. In 1993, H. Kato asked a related question: If f : X → X is a homeomorphism of a continuum X , and the topological entropy of f ispositive, is X non-Suslinian? [5, Question 1]. A positive answer to the latter implies theformer because every rational continuum is Suslinian (see Section 2 for definitions).In 2016, a positive answer to Kato’s question was announced [11, Corollary 27]. The proofin [11] relies on [11, Theorem 17], which is stronger than: Proposition 1 ([12, Theorem 30]) . If ( Y, d ) is a continuum and { X n } ∞ n =0 is a collection ofmutually disjoint subcontinua of Y such that d H ( X n , Y ) := sup y ∈ Y d ( y, X n ) → as n → ∞ , then Y is non-Suslinian. Unfortunately, Proposition 1 is false by [15, Example 3]. The example, constructed by E.D.Tymchatyn in 1983, is a hereditarily locally connected continuum which is the closure of afirst category ray and is therefore the limit of a sequence of disjoint arcs in Hausdorff distance d H . It is well-known that hereditarily locally connected continua are rational and Suslinian.Seidler’s conjecture thus remains an open problem. We remark that [6, Theorem 2.8] is alsocontingent on [11, Corollary 27].The hypothesis of Proposition 1 defines what it means for a continuum to be approximatedby a sequence of disjoint subcontinua. Approximations of continua from within were originallystudied by J. Krazinkiewicz and P. Minc in [7]. They proved: If Y is a hereditarily unicoherentplane continuum which contains disjoint ε -dense subcontinua for each ε > , then Y containsan indecomposable continuum [7, Theorem 1] . S. Curry later proved that if the continuum Y is tree-like then it can be written as the union of two indecomposable subcontinua [3, Mathematics Subject Classification.
Key words and phrases. continuum, rational, Suslinian, entropy, indecomposable.
Theorem 5]. These results do not extend to non-planar continua, as there exists a (hereditarilydecomposable) dendroid which is approximated by a sequence of disjoint subcontinua [7,Example 2]. We can, however, reach the weaker conclusion that Y is non-Suslinian. Theorem 1. If Y is a hereditarily unicoherent continuum which contains disjoint ε -densesubcontinua for every ε > , then Y is non-Suslinian. So Proposition 1 is true for all hereditarily unicoherent continua, including all tree-likecontinua.Next, we will investigate the role of continuum-wise connected spaces, or semicontinua , inapproximations. We show that each continuum Y which densely contains an indecomposablesemicontinuum X can be approximated by a sequence of disjoint continua (in the sense ofProposition 1). This will be a consequence of Theorem 2. Further, if X is homeomorphicto a composant of an indecomposable continuum, then Y is non-Suslinian. An even strongerresult is stated in Theorem 3. Theorem 2. If ( X, d ) is an indecomposable semicontinuum, then there is a sequence ofpairwise disjoint continua K , K , K , . . . ( X such that K n → X in the Vietoris topology.In particular, d ( x, K n ) → for each x ∈ X . Theorem 3. If X is homeomorphic to a composant of an indecomposable continuum, then X cannot be embedded into a Suslinian continuum. Moreover, every compactification of X contains c = | R | pairwise disjoint dense semicontinua. Theorem 2 extends [9, Corollary 1.2]. In Theorem 3, “Suslinian” cannot be replaced with“hereditarily decomposable”, as there exists a hereditarily decomposable plane continuumwhich homeomorphically contains composants of the bucket-handle continuum. See [13, Sec-tion 5] and [9, Section 1.1]. We also remark that Theorem 3 is false for indecomposable con-nected sets in general; there exists an indecomposable connected subset of the plane whichcan be embedded into a Suslinian continuum [8, Examples 2 and 4].2.
Preliminaries
All spaces under consideration are separable and metrizable.A continuum is a compact connected metrizable space with more than one point. An arc isa continuum homeomorphic to [0 , . A connected set X is decomposable if X can be writtenas the union of two proper closed connected subsets. Otherwise, X is indecomposable .An indecomposable semicontinuum is a continuum-wise connected space which cannot bewritten as the union of two proper closed connected subsets. A composant of a continuum isthe union of all proper subcontinua that contain a given point. Observe that each composantof an indecomposable continuum is an indecomposable semicontinuum. The class of spaceswhich are homeomorphic to composants of indecomposable continua includes all singulardense meager composants ; see [9].A continuum Y is: • hereditarily unicoherent if H ∩ K is connected for every two subcontinua H and K ; • regular if Y has a basis of open sets with finite boundaries; • hereditarily locally connected if every subcontinuum of Y is locally connected; • rational if Y has a basis of open sets with countable boundaries; • Suslinian if Y contains no uncountable collection of pairwise disjoint subcontinua [10];and • hereditarily decomposable if every subcontinuum of Y is decomposable. PPROXIMATIONS BY DISJOINT CONTINUA 3
For continua, it is well-known that:regular ⇒ hereditarily locally connected ⇒ rational ⇒ Suslinian ⇒ hereditarily decomposable ⇒ one-dimensional . One-dimensional continua are frequently called curves .For any topological space X we let X denote the set of non-empty closed subsets of X .A sequence ( A n ) ∈ [2 X ] ω converges to X in the Vietoris topology provided for every finitecollection of non-empty open sets U , . . . , U k ⊂ X there exists N < ω such that A n ∈ h U , . . . , U k i := { A ∈ X : A ∩ U i = ∅ for each i ≤ k } for all n ≥ N . If ( X, d ) is compact, then A n → X in the Vietoris topology if and only if lim n →∞ d H ( A n , X ) = 0 . Here d H ( A n , X ) = sup x ∈ X d ( x, A n ) is the Hausdorff distance between A n and X . A subset E of X is said to be ε -dense if d H ( E, X ) < ε , i.e. if E intersects everyball of radius ε in X . 3. Proofs
Proof of Theorem 1.
Let Y be a hereditarily unicoherent Suslinian continuum. We willfind ε > such that every two ε -dense subcontinua of Y intersect.Note that Y is decomposable, so there exist proper subcontinua H and K of Y such that Y = H ∪ K . Let W be an open subset of Y such that H ∩ K ⊂ W and W = Y . Eachconnected component of Y \ W contains a non-degenerate continuum by [4, Lemma 6.1.25],so the Suslinian property of Y implies that the set of connected components of Y \ W iscountable. By Baire’s theorem there is a component C of Y \ W such that C has non-emptyinterior in Y . Let ε > such that H \ K , K \ H , and C each contain open balls of radius ε .Let E and E be any two ε -dense subcontinua of Y . By hereditary unicoherence of Y , M := ( C ∪ E ∪ E ) ∩ ( H ∩ K ) is connected. Note that M = ( E ∩ H ∩ K ) ∪ ( E ∩ H ∩ K ) , where each set in that union isnon-empty and closed. Therefore E ∩ E = ∅ . (cid:4) We now prepare to prove Theorem 2. Following [1, Definition 4.5], if X is a semicontinuum, K ⊂ X , and U is a finite collection of open subsets of X , then we say K disrupts U if nocontinuum in X \ K intersects each member of U . Lemma 1. If X is an indecomposable semicontinuum, then no finite collection of non-emptyopen subsets of X is disrupted by (the union of ) finitely-many proper continua K , K , . . . , K n − ( X. Proof.
Let X be an indecomposable semicontinuum. Let K , K , . . . , K n − ( X be continua.Suppose for a contradiction that K := S { K i : i < n } disrupts a finite collection of non-empty open sets. Let l be the least positive integer with the property that some collection ofnon-empty open sets of size l is disrupted by K . That is, l = min {|U| : U is a collection of non-empty open subsets of X , and K disrupts U} . Since K is nowhere dense, l ≥ . Let V = { V , V , . . . , V l − } be a collection of non-emptyopen sets such that K disrupts V . By minimality and finiteness of l , N := [ { M ⊂ X \ K : M is a continuum and M ∩ V j = ∅ for each j ≥ } contains a dense subset of V . D.S. LIPHAM
We claim that every constituent M ⊂ N is contained in a semicontinuum S ⊂ N suchthat S intersects some K i . To see this, fix p ∈ M and q ∈ V . Since X is a semicontinuum,there is a continuum L ⊂ X such that { p, q } ⊂ L . The assumption K disrupts V implies ( M ∪ L ) ∩ K = ∅ , whence L ∩ K = ∅ . Boundary bumping [4, Lemma 6.1.25] in L now showsthat for each i < ω there is a continuum L i ⊂ L \ K such that p ∈ L i and d ( L i , K ) < − i . Thesemicontinuum S := S { M ∪ L i : i < ω } is contained in N , and S ∩ K = ∅ by compactnessof K . We conclude that N ′ := [ { S : S is a maximal semicontinuum in N } has at most n connected components. As V ⊂ N ′ , this implies some component C of N ′ isdense in a non-empty open subset of V . Then C is a closed connected subset of X \ V withnon-empty interior. This violates indecomposability of X . (cid:3) Proof of Theorem 2.
Let { U i : i < ω } be a basis for X consisting of non-empty opensets. Put U n = h U , . . . , U n − i . Let K ( X be any continuum. Assuming mutually disjoint K , . . . , K n − have been defined so that K i ∈ U i for each i < n , by Lemma 1 there exists K n ∈ U n such that K n ∩ K i = ∅ for each i < n . The sequence ( K n ) is as desired. (cid:4) Proof of Theorem 3.
Suppose X is homeomorphic to a composant of indecomposablecontinuum I . Let Y = γX be any compactification of X with associated embedding γ : X ֒ → γX . Let ι : X ֒ → I be a homeomorphic embedding such that ι [ X ] is a composant of I . Let Z be the closure of the diagonal {h ι ( x ) , γ ( x ) i : x ∈ X } in the product I × Y . More precisely,define ξ : X ֒ → I × Y by ξ ( x ) = h ι ( x ) , γ ( x ) i and put Z = ξ [ X ] . By the proof of [9, Theorem1.1], Z is an indecomposable continuum and ξ [ X ] is a composant of Z . By Lavrentiev’sTheorem [4, Theorem 4.3.21], the homeomorphism π Y ↾ ξ [ X ] extends to a homeomorphismbetween G δ -sets Z ′ ⊂ Z (with ξ [ X ] ⊂ Z ′ ) and Y ′ ⊂ Y . By [2, Theorem 9], Z ′ contains c composants of Z . Thus Y ′ contains c pairwise disjoint semicontinua which are dense in Y . (cid:4) Question A ray is a homomorphic image of the interval [0 , ∞ ) . If h : [0 , ∞ ) → X is a homeomor-phism, then X is a ray which limits onto itself if h ([ n, ∞ )) is dense in X for every n < ω .This is equivalent to saying X is first category in the sense of Baire. If Y is a one-dimensionalnon-separating plane continuum which is the closure of a ray that limits onto itself, then Y is indecomposable [3, Theorem 8]. Question 1. If Y is a continuum in the plane which contains first category ray (limiting ontoitself ), then is Y non-Suslinian? References [1] D. Anderson, The shore point existence problem is equivalent to the non-block point existence problem.Topology Appl. 262 (2019), 1–10.[2] H. Cook, On subsets of indecomposable continua, Colloquium Mathematicae 13.1 (1964) 37–43.[3] S. Curry, One-dimensional nonseparating plane continua with disjoint ε -dense subcontinua. TopologyAppl. 39 (1991), no. 2, 145–151.[4] R. Engelking, General Topology, Revised and completed edition Sigma Series in Pure Mathematics 6,Heldermann Verlag, Berlin, 1989.[5] H. Kato, Continuum-Wise Expansive Homeomorphisms. Canadian Journal of Mathematics, 45(3),(1993) 576–598.[6] H. Kato, Monotone maps of G-like continua with positive topological entropy yield indecomposability.Proc. Amer. Math. Soc. 147 (2019), no. 10, 4363–4370.[7] J. Krasinkiewicz and P. Minc, Approximations of continua from within, Bull. Acad. Polon. Sci. Sér. Sci.Math. Astronom. Phys. 25 (1977), no. 3, 283–289. PPROXIMATIONS BY DISJOINT CONTINUA 5 [8] D.S. Lipham, Dispersion points and rational curves, Proc. Amer. Math. Soc.148 (2020), 2671–2682.[9] D.S. Lipham, Singularities of meager composants and filament composants, Topology Appl., Volume260 (2019) 104–115.[10] A. Lelek, On the topology of curves II, Fund. Math. 70 (1971), 131–138.[11] C. Mouron, Mixing sets, positive entropy homeomorphisms and non-Suslinian continua. Ergodic TheoryDynam. Systems 36 (2016), no. 7, 2246–2257.[12] C. Mouron, The topology of continua that are approximated by disjoint subcontinua, Topology Appl.,Volume 156 (2009) 558–576.[13] C. Mouron and N. Ordoñez, Meager composants of continua, Topology Appl., Volume 210 (2016) 292–310.[14] G.T. Seidler, The topological entropy of homeomorphisms on one dimensional continua, Proc. Amer.Math. Soc. 108 (1990) 1025–1030.[15] E.D. Tymchatyn, Some rational continua. Rocky Mountain J. Math. 13 no. 2 (1983) 309–320.
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