On the degree of convergence of lemniscates in finite connected domains

Laiyi, Zhu

doi:10.1016/j.jat.2004.02.007

Cited by 1 publication

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“…In particular, Theorem 1.5 implies Theorem 1.2 for uniformly perfect sets, since the compact sets E s shrink to E as s decreases to 0. We also mention that results similar to Theorem 1.5 were obtained in [1] and [11], but for the rate of approximation by polynomial lemniscates in Hilbert's Lemniscate Theorem.…”

Section: Introductionsupporting

confidence: 61%

Fekete polynomials and shapes of Julia sets

Lindsey

Younsi

2019

Trans. Amer. Math. Soc.

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We prove that a nonempty, proper subset S of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if S is bounded andĈ \ int(S) is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation in terms of geometric and potential theoretic quantities.Theorem 1.1 (Lindsey [14]). Let E ⊂ C be any closed Jordan domain. Then for any > 0, there exists a polynomial P such that d(E, K(P )) < , d(∂E, J (P )) < .Here K(P ) := {z ∈ C : P m (z) ∞ as m → ∞} is the filled Julia set of P , J (P ) := ∂K(P ) is the Julia set of P and d is the Hausdorff distance.Note that Theorem 1.1 remains valid if E is any nonempty connected compact set with connected complement, by a simple approximation process.Approximating compact sets by fractals has proven over the years to be a fruitful technique in the study of important problems in complex analysis, such as the universal dimension spectrum for harmonic measure (cf. the work of Carelson-Jones [7], Binder-Makarov-Smirnov [2], and the references therein). Other related works include [3], where it was shown that any connected compact set in the plane can be approximated by dendrite Julia sets, and [13], containing applications to computer graphics.An important feature of the approach in [14] which seems to be absent from other works is that it is constructive and can easily be implemented to obtain explicit images of Julia sets representing various shapes.In this paper, we prove the following generalization of Theorem 1.1.

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Section: Introductionsupporting

confidence: 61%