Irreducible Julia sets of rational functions

Curry, Clinton P.

doi:10.1080/10236190903244857

Cited by 8 publications

(13 citation statements)

References 9 publications

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“…our core decomposition will apply to the study of connected Julia sets of rational functions on the extended complex planeĈ. Such a result is very helpful in searching for an answer to [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 K ⊂ 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 ∈ K} into singletons.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…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 . This is what Curry asks, in [6,Question 5.2] and in the middle part of [6,Question 5.4]. We expect further work on the decomposition D K in this direction, especially towards applications to the dynamical study of a rational function restricted to its Julia set (see for instance [21]).…”

Section: Definitionmentioning

confidence: 93%

“…Our Theorem 7 indicates that the property of "being a Peano compactum" is such a property. 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.…”

Section: Definitionmentioning

confidence: 99%

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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: Introduction and Main Resultsmentioning

confidence: 98%

Section: Definitionmentioning

confidence: 93%

Section: Definitionmentioning

confidence: 99%

See 1 more Smart Citation

A core decomposition of compact sets in the plane

Loridant

Luo

Yang

2019

Advances in Mathematics

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“…We find a partial generalization for it, keeping items (ii) and (iii) untouched for the moment. [4,5,9,17]. It is noteworthy that these models also date back to the 1980's, when Thurston and Douady and their colleagues started applying Carathéodory's Continuity Theorem to the study of polynomial Julia sets, which are assumed to be connected and locally connected.…”

Section: Property S and The Property Of Being A Peano Compactummentioning

confidence: 99%

“…These results provide direct motivations for the current definition of Peano compactum. They are responses to [9,Question 5.4]. This question asks for reasonable choices of topological properties (P), such as the combination of the above (3.1) and (3.2), so that all rational Julia sets have a core decomposition with respect to (P).…”

Section: Appendix: From Peano Continuum To Peano Compactummentioning

confidence: 99%

On continuous extension of conformal homeomorphisms of infinitely connected planar domains

Luo¹,

Yao²

2020

Preprint

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Let ϕ : D → Ω be a homeomorphism from a circle domain D onto a domain Ω ⊂ Ĉ. We obtain necessary and sufficient conditions (1) for ϕ to have a continuous extension to the closure D and (2) for such an extension to be injective. Further assume that ϕ is conformal and that ∂Ω has at most countably many non-degenerate components {Pn} whose diameters have a finite sum n diam(Pn) < ∞. When the point components of ∂D or those of ∂Ω form a set of σ-finite linear measure, we can show that ϕ continuously extends to D if and only if all the components of ∂Ω are locally connected. This generalizes Carathéodory's Continuity Theorem, that concerns the case when D is the open unit disk z ∈ Ĉ : |z| < 1 , and allows us to derive a new generalization of the Osgood-Taylor-Caratheodry Theorem.

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A note on core decomposition of Mandelbrot set

Yang

Yao

2020

Chaos, Solitons & Fractals

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Irreducible Julia sets of rational functions

Cited by 8 publications

References 9 publications

A core decomposition of compact sets in the plane

A core decomposition of compact sets in the plane

On continuous extension of conformal homeomorphisms of infinitely connected planar domains

A note on core decomposition of Mandelbrot set

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