Cantor Julia sets in a family of even elliptic functions

Koss, Lorelei

doi:10.1080/10236190903217796

Cited by 7 publications

(12 citation statements)

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“…The first examples of meromorphic functions with Cantor Julia sets appeared in [5], and examples of elliptic functions with Cantor Julia sets appeared in [14] and [19]. Here, we present criteria which guarantee that the Julia set of g Ω is Cantor.…”

Section: Cantor Julia Setsmentioning

confidence: 99%

“…Using Theorem 2.3, ℘ Ω 1 maps R to R ∪ {∞}, and Proposition 2.4 implies that ℘ Ω 1 (ω 1 /2) = 1. We will use the following lemma, which was proved in [19]. We prove that g Ω on the lattice Ω = Let Ω 1 = [ω 1 , εω 1 ] be the standard triangular lattice, and let k = ω 1 /2.…”

Section: Fundamental Dichotomy For Elliptic Functions 3935mentioning

confidence: 99%

“…In [19], the author investigated elliptic functions of the form f = 1/℘ on real rectangular lattices, and examples with Cantor Julia sets were found within every equivalence class of real rectangular lattice shape. However, there are many functions within this family whose Julia sets are not well understood.…”

Section: Theorem 11 If the Finite Critical Point Of A Quadratic Polmentioning

confidence: 99%

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A fundamental dichotomy for Julia sets of a family of elliptic functions

Koss

2009

Proc. Amer. Math. Soc.

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Abstract. We investigate topological properties of Julia sets of iterated elliptic functions of the form g = 1/℘, where ℘ is the Weierstrass elliptic function, on triangular lattices. These functions can be parametrized by C − {0}, and they all have a superattracting fixed point at zero and three other distinct critical values. We prove that the Julia set of g is either Cantor or connected, and we obtain examples of each type.

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Section: Cantor Julia Setsmentioning

confidence: 99%

Section: Fundamental Dichotomy For Elliptic Functions 3935mentioning

confidence: 99%

Section: Theorem 11 If the Finite Critical Point Of A Quadratic Polmentioning

confidence: 99%

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A fundamental dichotomy for Julia sets of a family of elliptic functions

Koss

2009

Proc. Amer. Math. Soc.

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“…Recall that J(f ) is a Cantor Julia set if J(f ) is a compact, totally disconnected, perfect subset of Ĉ. Up to now, double toral bands, which often yield Cantor Julia sets, have been studied the most ( [16], [20], [21], [13]), and there are examples of elliptic functions with double toral bands where J(f ) is not Cantor [22]. It has also been shown for ℘ Λ , that if Λ is a square lattice or triangular lattice then J(℘ Λ ) is connected [8,16].…”

Section: Introductionmentioning

confidence: 99%

Single and double toral band Fatou components in meromorphic dynamics

Hawkins

Koss

2023

Conform. Geom. Dyn.

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We analyze the existence and types of unbounded Fatou components for elliptic functions and other meromorphic functions with doubly periodic Julia sets. We show that apart from Herman rings and Siegel disks, all types of dynamics can occur in these domains, which are called toral bands. We show that toral bands are not necessarily periodic, and we give results about the number of distinct residue classes of critical points in each toral band.

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“…There are also some criterions to obtain the Cantor Julia sets for rational maps (see [Mil93] and [Yin92]). The first examples of transcendental meromorphic functions with Cantor Julia sets was produced by Devaney and Keen in [DK88] (see also [Kos10] for different examples).…”

Section: Introductionmentioning

confidence: 99%