Connectivity properties of Julia sets of Weierstrass elliptic functions

Hawkins, Jane; Koss, Lorelei

doi:10.1016/j.topol.2004.08.018

Cited by 21 publications

(45 citation statements)

References 15 publications

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“…Connectivity properties of Julia sets of elliptic functions were first studied by the author and Hawkins in [14], but a complete classification is known for only two families, and both of these families always have connected Julia sets. The Julia set of the Weierstrass elliptic ℘-function on any triangular lattice is always connected L. KOSS [14]; this family of functions contains examples where the Fatou set is nonempty as well as examples where the Julia set is the entire sphere.…”

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

confidence: 99%

“…The Julia set of the Weierstrass elliptic ℘-function on any triangular lattice is always connected L. KOSS [14]; this family of functions contains examples where the Fatou set is nonempty as well as examples where the Julia set is the entire sphere. Additional topological properties of Julia sets in this family were studied in [16].…”

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

confidence: 99%

“…In [11], Hawkins proved that the Weierstrass elliptic function on any rhombic square lattice always has Julia set equal to the entire sphere and hence is connected. Isolated examples of other elliptic functions with connected Julia sets were constructed in [14] as well.…”

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

confidence: 99%

“…Examples of elliptic functions with Cantor Julia sets first appeared in [14]. 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.…”

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

confidence: 99%

“…Results on how the lattice shape influences the dynamics of the Weierstrass elliptic function can be found in [10,11,12,13,14,15,16].…”

Section: Proposition 24 ([6])mentioning

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

confidence: 99%

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

confidence: 99%