Finitely Suslinian models for planar compacta with applications to Julia sets

Blokh, Alexander; Curry, Clinton P.; Oversteegen, Lex G.

doi:10.1090/s0002-9939-2012-11607-7

Cited by 11 publications

(36 citation statements)

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“…The studies of Blokh-Curry-Oversteegen [3,2], 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 [2,Theorem 27], if a continuum X ⊂ K has a "well-slicing family", then the image of X under the natural projection π : K → D P C K is a non-degenerate continuum, hence K has a non-degenerate core decomposition.…”

Section: Definitionmentioning

confidence: 84%

“…Actually, let C ⊂ [0, 1] ⊂ C be Cantor's ternary set. Let K be the union of {x + iy : x ∈ C, y ∈ [0, 1]} with {x + i : x ∈ [0, 1]} and call it the In 2013, Blokh-Curry-Oversteegen obtained for any unshielded compactum K ⊂ C the core decomposition D F S K with respect to the property of being finitely Suslinian [3,Theorem 4]. Here a compactum is finitely Suslinian provided that every collection of pairwise disjoint subcontinua whose diameters are bounded away from zero is finite.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Here a compactum is finitely Suslinian provided that 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 [2,Theorem 1] is a special case of [3,Theorem 4]. We may wonder about the existence of the core decomposition D F S K of an arbitrary compactum K ⊂ C. However, there are examples of continua K ⊂ C failing to have such a core decomposition (see [3,Example 14] and Section 2 of this paper).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Since every finitely Suslinian continuum is locally connected, we see that [2,Theorem 1] is a special case of [3,Theorem 4]. We may wonder about the existence of the core decomposition D F S K of an arbitrary compactum K ⊂ C. However, there are examples of continua K ⊂ C failing to have such a core decomposition (see [3,Example 14] and Section 2 of this paper). We will replace the property of being finitely Suslinian by the property of being a Peano compactum.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Moreover, in the last part of [6,Question 5.4], Curry asks: which of these (properties) is the appropriate analogue for the finest locally connected model? Since the core decomposition D P C K in Theorem 7 generalizes the earlier finest decompositions obtained in [2,3], the property of "being a Peano compactum" is a reasonable candidate. What remains is to verify that the decomposition D P C K is "dynamic", in the sense that the rational function R sends every element of D P C K into an element of D P C K .…”

Section: Definitionmentioning

confidence: 87%

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A core decomposition of compact sets in the plane

Loridant

Luo

Yang

2019

Advances in Mathematics

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A Peano continuum means a locally connected continuum. A compact metric space is called a Peano compactum if all its components are Peano continua and if for any constant C > 0 all but finitely many of its components are of diameter less than C. Given a compact set K ⊂ C, there usually exist several upper semi-continuous decompositions of K into subcontinua such that the quotient space, equipped with the quotient topology, is a Peano compactum. We prove that one of these decompositions is finer than all the others and call it the core decomposition of K with Peano quotient. This core decomposition gives rise to a metrizable quotient space, called the Peano model of K, which is shown to be determined by the topology of K and hence independent of the embedding of K into C. We also construct a concrete continuum K ⊂ R 3 such that the core decomposition of K with Peano quotient does not exist. For specific choices of K ⊂ C, the above mentioned core decomposition coincides with two models obtained recently, namely the locally connected model for unshielded planar continua (like connected Julia sets of polynomials) and the finitely Suslinian model for unshielded planar compact sets (like polynomial Julia sets that may not be connected). The study of such a core decomposition provides partial answers to several questions posed by Curry in 2010. These questions are motivated by other works, including those by Curry and his coauthors, that aim at understanding the dynamics of a rational map f :Ĉ →Ĉ restricted to its Julia set.In this paper, a compact metric space is called a compactum and a connected compactum is called a continuum. If K, L are two compacta, a continuous onto map π : K → L such that the preimage of every point in L is connected is called monotone [19]. We are interested in compacta in the complex plane C or in the Riemann sphereĈ. Given a compactum K ⊂ C, an upper semi-continuous decomposition D of K is a partition of K such that for every open set B ⊂ K the union of all d ∈ D with d ⊂ B is open in K (see [11]). Let π be the natural projection sending x ∈ K to the unique element of D that contains x. Then a set A ⊂ D is said to be open in D if and only if π −1 (A) is open in K. This defines the quotient topology on D. If all the elements of such a decomposition D are compact the equivalence on K corresponding to D is a closed subset of K × K and the quotient topology on D is metrizable [9, p.148, Theorem 20]. Equipped with an appropriate metric compatible with the quotient topology, the quotient space D is again a compactum.An upper semi-continuous decomposition of a compactum K ⊂ C is monotone if each of its elements is a subcontinuum of K. In this case, the mapping π described above is a monotone map. Let D and D ′ be two monotone decompositions of a compactum K ⊂ 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 com...

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Section: Definitionmentioning

confidence: 84%

Section: Introduction and Main Resultsmentioning

confidence: 99%