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JuliaSetBoettcher[c,z]

gives the Böttcher coordinate of z with respect to the quadratic Julia set Jc.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Neat Examples  
See Also
Related Guides
History
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JuliaSetBoettcher

JuliaSetBoettcher[c,z]

gives the Böttcher coordinate of z with respect to the quadratic Julia set Jc.

Details and Options

  • The Julia set of a function f is the closure of the set of all repelling fixed points of f.
  • The Böttcher function is given by and satisfies the conjugacy .
  • With the option MaxIterations->m, the sequence will be iterated at most m times to approximate .
  • The default setting is MaxIterations->100.

Examples

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Basic Examples  (2)

Get the Böttcher coordinate of for the Julia set given by :

Wolfram Language code: JuliaSetBoettcher[1 + I, 2. - I]

Since is in the Julia set given by , the result is undefined:

Wolfram Language code: N[JuliaSetBoettcher[I / 2, -0.1]]

Scope  (2)

JuliaSetBoettcher threads itself element-wise over lists:

Wolfram Language code: JuliaSetBoettcher[1 + I, {2I , I + 1, I + 2}]
Wolfram Language code: N /@ %

JuliaSetBoettcher works on all kinds of numbers:

Wolfram Language code: JuliaSetBoettcher[1 + I, #]& /@ {-2, -2.9, 3 / 4, 0.36, 3 I, 1 + I, 1 / Pi, N[1 / Pi, 30]}
Wolfram Language code: N[%]

Generalizations & Extensions  (1)

Find a series approximation for the Böttcher function for the Julia set for :

Wolfram Language code: Series[JuliaSetBoettcher[.1, z], {z, Infinity, 5}]

Neat Examples  (3)

Use the absolute value to display "equipotential lines" for a Cantor-like Julia set:

Wolfram Language code: ArrayPlot[Table[Sin[100 Abs@JuliaSetBoettcher[1 + I, x + I * y]], {y, -1.4, 1.4, .01}, {x, -1.6, 1.6, .01}]]

Use the argument to display "field lines" for a connected Julia set:

Wolfram Language code: ArrayPlot[Table[Sin[200 Arg@JuliaSetBoettcher[.3 - .2I, x + I * y, MaxIterations -> 119]], {y, -2.0, 2.0, .02}, {x, -1.6, 1.5, .02}]]

Use the argument to display the "external rays" with angle :

Wolfram Language code: ArrayPlot[Table[(If[#1 == 0. + 0. I, .5, Boole[N[#2 - .004] < Arg[#1] < N[#2 + .004]]]&)[JuliaSetBoettcher[.3 - .3 I, x + I y, MaxIterations -> 38], -0.8], {y, -2, 2, .03}, {x, -2.6, 2.5, .03}]]
Wolfram Research (2014), JuliaSetBoettcher, Wolfram Language function, https://reference.wolfram.com/language/ref/JuliaSetBoettcher.html.

Text

Wolfram Research (2014), JuliaSetBoettcher, Wolfram Language function, https://reference.wolfram.com/language/ref/JuliaSetBoettcher.html.

CMS

Wolfram Language. 2014. "JuliaSetBoettcher." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JuliaSetBoettcher.html.

APA

Wolfram Language. (2014). JuliaSetBoettcher. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JuliaSetBoettcher.html

BibTeX

@misc{reference.wolfram_2026_juliasetboettcher, author="Wolfram Research", title="{JuliaSetBoettcher}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/JuliaSetBoettcher.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_juliasetboettcher, organization={Wolfram Research}, title={JuliaSetBoettcher}, year={2014}, url={https://reference.wolfram.com/language/ref/JuliaSetBoettcher.html}, note=[Accessed: 12-June-2026]}