Computing the Laurent Series of the Map Ψ: C − D → C − M

Bielefeld, Ben; Fisher, Yuval; Vonhaeseler, F.

doi:10.1006/aama.1993.1002

Cited by 8 publications

(24 citation statements)

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“…Taking advantage of the one million coefficients, we incidentally find a new upper bound for the area. Recall that the area is given by A ∞ := lim N →∞ A N , namely Gronwall's area formula, where [20]; the first 8000 coefficients were used in [2]. In [18], A 240000 was firstly obtained.…”

Section: Error Estimatementioning

confidence: 99%

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Family of Invariant Cantor Sets as Orbits of Differential Equations Ii: Julia Sets

Chen

Kawahira

et al. 2011

Int. J. Bifurcation Chaos

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Abstract. The Julia set of the quadratic map fµ(z) = µz(1 − z) for µ not belonging to the Mandelbrot set is hyperbolic, thus varies continuously. It follows that a continuous curve in the exterior of the Mandelbrot set induces a continuous family of Julia sets. The focus of this article is to show that this family can be obtained explicitly by solving the initial value problem of a system of infinitely coupled differential equations. A key point is that the required initial values can be obtained from the antiintegrable limit µ → ∞. The system of infinitely coupled differential equations reduces to a finitely coupled one if we are only concerned with some invariant finite subset of the Julia set. Therefore, it can be employed to find periodic orbits as well. We conduct numerical approximations to the Julia sets when parameter µ is located at the Misiurewicz points with external angle 1/2, 1/6, or 5/12. We approximate these Julia sets by their invariant finite subsets that are integrated along the reciprocal of corresponding external rays of the Mandelbrot set starting from the anti-integrable limit µ = ∞. When µ is at the Misiurewicz point of angle 1/128, a 98-period orbit of prescribed itinerary obtained by this method is presented, without having to find a root of a 2 98 -degree polynomial. The Julia sets (or their subsets) obtained are independent of integral curves, but in order to make sure that the integral curves are contained in the exterior of the Mandelbrot set, we use the external rays of the Mandelbrot set as integral curves. Two ways of obtaining the external rays are discussed, one based on the series expansion (the Jungreis-Ewing-Schober algorithm), the other based on Newton's method (the OTIS algorithm). We establish tables comparing the values of some Misiurewicz points of small denominators obtained by these two algorithms with the theoretical values.

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Section: Error Estimatementioning

confidence: 99%

“…which determines coefficient Γ n,m for n ≥ 0 and m ≥ 2 n+1 − 1 in terms of Γ j,k with j ≥ n + 1 and k ≤ m. According to [2,18,20], all Γ n,m are real numbers, and all b m are dyadic rationals. Some calculations by hand using (5.…”

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confidence: 99%

Family of Invariant Cantor Sets as Orbits of Differential Equations Ii: Julia Sets

Chen

Kawahira

et al. 2011

Int. J. Bifurcation Chaos

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“…p n (ψ(z)) = z 2 n + o(1) as z → ∞, a fact that they used to prove (4). Jungreis [12] proved earlier that b 2 n+1 = 0 for n ≥ 1 (see also [3,8,14]). Bielefeld, Fisher, and Haeseler [3] proved that no constants and K exist so that |b m |< K/m 1+ for all m.…”

Section: Introductionmentioning

confidence: 90%

New approximations for the area of the Mandelbrot set

Bittner

Cheong

Gates

et al. 2017

Involve

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Abstract. Due to its fractal nature, much about the area of the Mandelbrot set M remains to be understood. While a series formula has been derived by Ewing and Schober to calculate the area of M by considering its complement inside the Riemann sphere, to date the exact value of this area remains unknown. This paper presents new improved upper bounds for the area based on a parallel computing algorithm and for the 2-adic valuation of the series coefficients in terms of the sum-of-digits function.

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“…• Γ 0,m = b m−1 for m ≥ 1. Moreover, they obtained the following very useful backward recursion formula which determines coefficient Γ n,m for n ≥ 0 and m ≥ 2 n+1 − 1 in terms of Γ j,k with j ≥ n + 1 and k ≤ m. According to [2,18,20] Some coefficients are a 2 = 1/2, a 3 = 1/8, a 4 = 1/4, a 5 = 15/128, a 6 = 0, and a 7 = 81/1024. 's method).…”

Section: Jungreis-ewing-schober (Jes) Algorithm (Series Expansion)mentioning

confidence: 99%