# JuliaSetIterationCount

JuliaSetIterationCount[f,z,p]

returns the number of iterations, beginning with the complex number , of the function needed to determine whether p is in the Julia set of f.

returns the number of iterations, beginning with the complex number , of the function needed to determine whether p is in the Julia set of .

JuliaSetIterationCount[f,z,{p1,p2,}]

returns a list of the number of iterations required to determine whether each member of {p1,p2,} is in the Julia set of f.

JuliaSetIterationCount[c,{p1,p2,}]

returns a list of the number of iterations required to determine whether each member of {p1,p2,} is in the Julia set of .

# Details and Options

• The Julia set of a function f is the closure of the set of all repelling fixed points of f.
• JuliaSetIterationCount uses the same "OrbitDetection" algorithm as JuliaSetPlot.
• With , where n is a positive integer, the function will be iterated at most n times to determine if z lies outside of the Julia set. If z is not found to lie outside the Julia set, JuliaSetIterationCount returns n+1. The default setting is MaxIterations->1000.
• With , each iteration is internally calculated to n digits of precision. Without this option, the amount of precision used is determined based on the precision of p and the value of MaxIterations.

# Examples

open allclose all

## Basic Examples(2)

Four iterations are needed to determine that is not in the Julia set of :

Calculate the iterations for a list of numbers:

## Scope(6)

Find the iteration count of for the Julia set of where :

Find the iteration count of 0 for the Julia set of a polynomial:

Find the iteration count of 0 for the Julia set of a rational function:

Find the iteration counts for a list of numbers:

Find the iteration counts for an array of numbers:

JuliaSetIterationCount works on all kinds of numbers:

## Options(2)

### MaxIterations(1)

MaxIterations must be increased if the number of iterations needed exceeds 1000:

### WorkingPrecision(1)

Increasing WorkingPrecision can increase accuracy, at the cost of more time:

Setting WorkingPrecision too low can result in the iteration entering a false loop:

## Properties & Relations(3)

ArrayPlot applied to is essentially JuliaSetPlot[c]:

The red dot in the middle comes from taking the Log after JuliaSetIterationCount returns 0:

JuliaSetIterationCount can accept lists, which is faster than applying it to each member of a list:

## Possible Issues(1)

The precision of a point may affect the result:

## Neat Examples(2)

Display the iteration count as height:

Create a three-dimensional image by varying a parameter:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2022_juliasetiterationcount, author="Wolfram Research", title="{JuliaSetIterationCount}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/JuliaSetIterationCount.html}", note=[Accessed: 06-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_juliasetiterationcount, organization={Wolfram Research}, title={JuliaSetIterationCount}, year={2014}, url={https://reference.wolfram.com/language/ref/JuliaSetIterationCount.html}, note=[Accessed: 06-June-2023 ]}