JuliaSetPoints

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`JuliaSetPoints`

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JuliaSetPoints[f,z]

returns a list of coordinates approximating the real and imaginary parts of the complex numbers in the Julia set of the rational function f of the variable z.

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JuliaSetPoints[c]

returns a list of coordinates of points approximating the Julia set of the function .

Details and Options

The Julia set of a function f is the closure of the set of all repelling fixed points of f.
JuliaSetPoints uses the same "InverseIteration" algorithm as JuliaSetPlot.
JuliaSetPoints has the options:
"ClosenessTolerance" 0.004 minimum distance between points

"Bound" 6 radius around the origin in which to search
For polynomial functions, "Bound" is automatically determined to ensure the entire Julia set is captured.

Examples

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Basic Examples (2)Summary of the most common use cases

Find points of the Julia set of :

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-nzi1mj

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0cpudo4genslna-bmhim6

Out[2]=2

Find points of the Julia set of :

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-sg0li5

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0cpudo4genslna-w6agjd

Out[2]=2

Scope (2)Survey of the scope of standard use cases

JuliaSetPoints[c] generates the Julia set of a function of the form :

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-3z9h7p

Out[1]=1

JuliaSetPoints[f,z] generates the Julia set of polynomials or more general rational functions:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-d60e6r

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0cpudo4genslna-ndet5d

Out[2]=2

Options (2)Common values & functionality for each option

"ClosenessTolerance" (1)

Increase "ClosenessTolerance" to make a quick, low-resolution picture of a Julia set:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-20ttz1

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0cpudo4genslna-x60uy8

Out[2]=2

Decrease "ClosenessTolerance" to make a high-resolution picture of a small part of a Julia set:

In[3]:=3

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https://wolfram.com/xid/0cpudo4genslna-vg0xeu

Out[3]=3

"Bound" (1)

For some rational functions, increasing "Bound" can find more points:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-80kqrt

Out[1]=1

Properties & Relations (2)Properties of the function, and connections to other functions

JuliaSetPlot[c] generates essentially a ListPlot of the result of JuliaSetPoints[c]:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-h3eg8n

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0cpudo4genslna-4eu27

Out[2]=2

JuliaSetPoints[c] is the same as JuliaSetPoints[z^2+c,z]:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-bepum6

Out[1]=1

Possible Issues (1)Common pitfalls and unexpected behavior

If the value of the "Bound" option is too low for a rational function, no points may be returned:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-rm7kfb

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0cpudo4genslna-jsiz8t

Out[2]=2

Some very large Julia sets can take a long time to compute with this method:

In[3]:=3

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https://wolfram.com/xid/0cpudo4genslna-8ah7n9

Out[3]=3

Interactive Examples (1)Examples with interactive outputs

Explore the Julia sets for which the parameter c is on the unit circle:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-qhez88

Out[1]=1

Neat Examples (3)Surprising or curious use cases

Stack successively finer approximations to a Julia set:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-cku1d6

Out[1]=1

Add a dimension by varying a parameter:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-b1qk4r

Out[1]=1

Visualize the Julia sets given by points on part of the unit circle:

In[1]:=1

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https://wolfram.com/xid/0cpudo4genslna-vawpo0

Out[1]=1

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