Density of repelling fixed points in the Julia set of a rational or entire semigroup

Stankewitz, Rich

doi:10.1080/10236190903203929

Cited by 12 publications

(16 citation statements)

References 24 publications

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“…This implies then that g −1 ∈ G for each g ∈ G and thus G = G −1 . Hence E(G) which is backward invariant under G must also be forward invariant under G. (4) is shown in [19].…”

Section: Propositionmentioning

confidence: 98%

Backward iteration algorithms for Julia sets of Möbius semigroups

Stankewitz¹,

Sumi²

2016

DCDS-A

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“…This implies then that g −1 ∈ G for each g ∈ G and thus G = G −1 . Hence E(G) which is backward invariant under G must also be forward invariant under G. (4) is shown in [19].…”

Section: Propositionmentioning

confidence: 98%

Backward iteration algorithms for Julia sets of Möbius semigroups

Stankewitz¹,

Sumi²

2016

DCDS-A

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“…If ♯J(G) ≥ 3, then J(G) is the smallest set in {∅ = K ⊂Ĉ | K is compact, ∀g ∈ G, g(K) ⊂ K}. For more details on these properties of rational semigroups, see [11,17,10,22]. For the dynamics of postcritically bounded polynomial semigroups, see [29,30,19].…”

Section: Toolsmentioning

confidence: 99%

“…[22,23]). [17] is a very nice (and short) article for an introduction to the dynamics of rational semigroups. For other research on rational semigroups, see [37,18,19,35,36], and [21]- [33].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets

Sumi

2013

Journal of the London Mathematical Society

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We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with bounded planar postcritical set. Using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that if the Julia set of such a semigroup is disconnected, then there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation which are parameterized by the Cantor set, densely inside the Julia set of the semigroup. Moreover, we give a sufficient condition for a fiberwise Julia set Jγ to satisfy that Jγ is a Jordan curve but not a quasicircle, the unbounded component ofĈ \ Jγ is a John domain and the bounded component of C \ Jγ is not a John domain. We show that under certain conditions, a random Julia set is almost surely a Jordan curve, but not a quasicircle. Many new phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are found and systematically investigated. Definition 1.2. For each rational map g :Ĉ →Ĉ, we set CV (g) := {all critical values of g :Ĉ → C}. Moreover, for each polynomial map g :Ĉ →Ĉ, we set CV * (g) := CV (g) \ {∞}. For a rational semigroup G, we set P (G) := g∈G

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“…It is possible to take 1 as a leaf-vertex, and consider the subsemigroup of such dynamical Belyi polynomials having the same Julia set. (If two polynomials have different Julia sets, then their forward orbit is dense in the plane by [60].) Then this subsemigroup acts on the inverse images of 1 like the action of the power maps on the roots of unity.…”

Section: Semigroup Homomorphisms Of Belyi-extending Mapsmentioning

confidence: 99%