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 :

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

Scope  (2)

JuliaSetBoettcher threads itself element-wise over lists:

JuliaSetBoettcher works on all kinds of numbers:

Generalizations & Extensions  (1)

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

Neat Examples  (3)

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

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

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

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_2025_juliasetboettcher, author="Wolfram Research", title="{JuliaSetBoettcher}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/JuliaSetBoettcher.html}", note=[Accessed: 21-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_juliasetboettcher, organization={Wolfram Research}, title={JuliaSetBoettcher}, year={2014}, url={https://reference.wolfram.com/language/ref/JuliaSetBoettcher.html}, note=[Accessed: 21-February-2025 ]}