Completely invariant Julia sets of polynomial semigroups

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if A and B are two connected components of the Julia set, then one of A and B surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the cardinality of the set of all connected components of the Julia set of such a semigroup is given. By using this, we give a criterion for the Julia set to be connected. Moreover, we show that for any n ∈ N ∪ {ℵ0}, there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to n. Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials are found and systematically investigated.

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“…Lemma 3.1-2). For other research on rational semigroups, see [19,20,21,47,22,24,44,43,45,46], and [27]- [40].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets

Sumi¹

2011

Discrete &Amp; Continuous Dynamical Systems - A

136

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“…Each m ∈ M has the form m After its inception in the paper [16] of Hinkkanen and Martin, the study of rational semigroups has grown in the past decade (see, for example, [13,29,30,36,39] as well as the references therein). These works generally focus, however, on semigroups that contain at least one non-Möbius map.…”

Section: Rational Semigroupsmentioning

confidence: 99%

“…We should mention that there exists a unique stationary Borel measure µ for this process whose support is exactly A and that an ergodic theorem holds for µ-almost all orbits. See [1,9,17] or [28] for details and precise statements of all the above results.…”

Section: Contracting Iterated Function Systemsmentioning

confidence: 99%

Complex dynamics of Möbius semigroups

Fried

Marotta

Stankewitz

2012

Ergod. Th. Dynam. Sys.

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AbstractWe study the dynamics of semigroups of Möbius transformations on the Riemann sphere, especially their Julia sets and attractors. This theory relates to the dynamics of rational functions, rational semigroups, and Möbius groups and we compare and contrast these theories. We particularly examine Caruso’s family of Möbius semigroups, based on a random dynamics variant of the Fibonacci sequence.

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“…The sets N (G) and J(G) are, however, not necessarily completely invariant under the elements of G. This is in contrast to the case of single function dynamics. For a discussion on completely invariant Julia sets the reader is referred to [19], [20] and [18]. Definition 2.2.…”

Section: Proposition 22 Ifmentioning

confidence: 99%

Uniformly perfect sets, rational semigroups, Kleinian groups and IFS’s

Stankewitz

2000

Proc. Amer. Math. Soc.

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We show that the Julia set of a non-elementary rational semigroup G is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of G. This also proves that the limit set of a non-elementary Möbius group is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of the group and this implies that the limit set of a finitely generated non-elementary Kleinian group is uniformly perfect.

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Completely invariant Julia sets of polynomial semigroups

Cited by 26 publications

References 2 publications

Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets

Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets

Complex dynamics of Möbius semigroups

Uniformly perfect sets, rational semigroups, Kleinian groups and IFS’s

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