Talk:Julia set

	Mathematics portal This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics
Mid	This article has been rated as Mid-priority on the project's priority scale.

Systems: Chaos theory Mid‑importance

	Systems science portal This article is within the scope of WikiProject Systems, which collaborates on articles related to systems and systems science.SystemsWikipedia:WikiProject SystemsTemplate:WikiProject SystemsSystems
Mid	This article has been rated as Mid-importance on the project's importance scale.
	This article is within the field of Chaos theory.

Archives

1

This page has archives. Sections older than 365 days may be automatically archived by when more than 10 sections are present.

The GIF of Julia sets does not display well

The preview of the GIF on the right cannot display. Could it be a problem with my browser?

Thanks, --Shiyu Ji (talk) 02:22, 16 April 2016 (UTC)[reply]

Huge video file

The Julia_circling.ogg video is very pretty but is 70MB. Is this really a suitable size to have on the wiki infrastructure? 79.75.125.119 (talk) 03:51, 8 January 2017 (UTC)[reply]

External links modified

Hello fellow Wikipedians,

I have just modified 2 external links on Julia set. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 04:26, 29 April 2017 (UTC)[reply]

Example Julia set renders aren’t Julia sets

A Julia set is defined as boundary of the set of points such that $\lim _{n\to \infty }z_{n}=f(z_{n-1})\neq \infty$ . Notably, $f(x)$ is a rational function, i.e. the result of the division of two polynomials $P(x)$ and $Q(x)$ , where a polynomial “is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents” (qtd. from Polynomial).

Unfortunately, about half of the pictures in the Examples of Julia sets section aren’t actually Julia sets (such as $e^{z}$ , $e^{z^{3}}$ , ${\sqrt {\sinh z^{2}}}$ ), and none of them are non-polynomial rational functions.

Is there a name for something like a Julia set where $f(x)$ is a non-rational function? If so, we should move these images there.
These images should be removed from the Julia set page — they’re confusing, incorrect, and unnecessary.

B637275 (talk) 20:58, 17 May 2017 (UTC)[reply]

Update: I have removed the images from the gallery. If anyone has a problem with this — and frankly, this should be pretty uncontroversial — please raise it here.

B637275 (talk) 23:11, 18 May 2017 (UTC)[reply]

As stated in the Generalizations section (which needs expansion and references), the definition of Julia sets can be applied to maps besides rational functions. I'm not aware of any special name for Julia sets of non-rational functions; I've only seen them called "Julia sets". Popular fractal software ignores the mathematical restrictions and allows rendering of any function, treating points that are bounded under iteration as members of the "filled Julia set", which is likely where these pictures came from. Since this article is about mathematical Julia sets, not fractal software, I don't disagree with removing the images. But perhaps they would be appropriate when the Generalizations section is expanded.

Rick Sidwell (talk) 17:41, 19 May 2017 (UTC)[reply]

Non-standard formal definitions

The definitions used on this page are quite strange. Here are the definitions I am familiar with:

(1) J(f) is simply the set of points where iterating f from the given point stays bounded.

(1b) I learned today that the standard definition limits f to be a rational polynomial. That seems to be true, but also a widely discarded restriction, depending on the context.

(2) I had never heard of a Fatou set before visiting this article, but upon a little web searching, it is a kind of Julia set, not the complement of a Julia set.

Lexspoon (talk) 12:46, 21 June 2017 (UTC)[reply]

I am not sure how to deal with all the misapprehensions here, but to take the last point: the usage "Fatou set" to denote the complement of the Julia set has become standard in recent decades. Admittedly, it isn't quite as ancient. Writing in 1990, Beardon^[1] says

Although the use of the term "Julia set" is standard, the use of "Fatou set" was suggested as late as 1984 (in ^[2]). It seems appropriate, but the reader should be aware of the common alternatives, namely the stable set, and the set of normality.

Chris Thompson (talk) 11:37, 27 September 2017 (UTC)[reply]

Fair enough, but I believe my larger point stands despite this. I am not in the right research circles where Fatou sets come up for me, and that's just an example. Meramorphic function would be another example. For the goals of Wikipedia, it seems better to start with definitions that can be understood with the least background by the widest set of people, and then to also include definitions that are interesting for some other reason, e.g. being baseline definitions in certain research circles.

I would propose a few steps to improve this page.

Combine this page with "filled-in Julia set", similar to how Natural number discuses the set 0,1,2... and also the set 1,2,3... The two Julia set pages aren't using the distinction to accomplish two usefully distinct articles. They just have overalapping sets of useful information. A combined page could use "Julia set" for the boundary and "filled-in Julia set" for the boundary plus interior, though if so, I really think it deserves a callout that many groups of people, sometimes million-plus groups of people, "Julia set" by itself would mean the filled-in version.
Start with the definition that takes the least background but is still adequate. Surely that must be the one about the set of non-escaping points under iteration.
Drop "Julia laces" and "Fatou dusts" from the intro. "Fatou dust" is a kind of Julia set, but the wording of the intro makes it sound like Fatou=dust and Julia=lace. This is just confusing and should be scrapped from the intro. For reference, Robert Munafo writes: "A Julia set that is a Cantor set is called a Fatou dust.".
However, it would be a really good subsection to discuss the two categories of Julia sets--dust versus fully connected.
Drop "behave similarly" from the intro. It is good to try and explain things with simple clear intuition, but I don't think this is really a good starting point since there are so many ways that similar could be defined.

On the question of definitions, I have looked it up today and see the following. I think it would help if Wikipedia used what I'm seeing on other sites, which also matches my own experience in various circles that talk about Julia sets regularly.

I first used Google's page rank, as a way to be objective but maybe to skew for popularity over correctness. Here are the first several results from a search on "julia set meaning" that I tried today.

The Oxford English Dictionary uses what I expected: "a set of complex numbers which do not converge to any limit when a given mapping is repeatedly applied to them."
Complex Analysis: A visual and Interactive Introduction, by Ponce Campuzano, agrees: "Similarly as we did for the Mandelbrot set, we obtain a sequence of complex numbers zn with n=0,1,2,.... Again, the points zn are said to form the orbit of z0, and the Julia set is defined as follows: If the orbit zn fails to escape to infinity, the initial z0 is said to belong to the filled-in Julia Set.".
This Wikipedia article is the third result when I do the search.
Jim Belk, a lecturer in mathematics, gives the following in his class notes. "The filled Julia set for f is the set $\{p_{1}\in C|the\;orbit\;of\;p_{1}\;is\;bounded\}$ . This agrees with the above two.
Math World uses this: "The "filled-in" Julia set J_R is the set of points z which do not approach infinity after R(z) is repeatedly applied (corresponding to a strange attractor). The true Julia set J is the boundary of the filled-in set (the set of "exceptional points"). ". This agrees with the above except for the distinction of filled versus boundary.
A YouTube video. I have not watched it.
An Introduction to Julia and Fatou Sets, by Scott Sutherland. This article uses "Julia set" as the boundary, and it uses a "normal family" definition similar but a little different from what's on this page now.

Second, I pulled a few sources that seem interesting to me.

Benoit Mandlebrot, widely hailed for reviving interest in Julia's work. "Iteration associates to each mf a f* Julia set: the boundary of the domain that fails to iterate to infinity".
Robert L. Devaney, a popular explainer of fractals for the public. In "Illuminating the Mandelbrot Set", written in 2019, he writes: "By definition, the filled Julia set of Pc is the set of all points in the complex plane whose orbits are bounded under iteration of Pc".

Lexspoon (talk) 17:15, 25 May 2025 (UTC)[reply]

References

^ Beardon, Alan F. (1991). Iteration of Rational Functions. Springer. p. 50. ISBN 0-387-95151-2.
^ Blanchard, P. (1984). "Complex analytic dynamics on the Riemann sphere". Bulletin of the American Mathematical Society. 11: 85–141.

Mistakes in the definition of the Fatou set

Currently, there are mistakes in the definition of the Fatou set in the section "Formal definition". First, number of Fatou domains can be infinite. Second, the Julia set of a rational map can coincide with the whole Riemann sphere, in which case there are no Fatou domains (and so their union is not dense in the plane). Also, the definition presented is not formal at all and leaves lots of space for interpretations. I think, this section should be rewritten carefully. I might do it later. Artem149598 (talk) 12:32, 2 August 2017 (UTC)[reply]

External links modified

Hello fellow Wikipedians,

I have just modified 2 external links on Julia set. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 11:15, 2 December 2017 (UTC)[reply]

Pseudocode examples

The current pseudocode examples are wrong and do not produce graphical representations of Julia sets.

I hereby propose to improve the first example with the following changes:

Before the loop set x = x0 and y = y0
Explain the meaning of x and y (real and imaginary part)
Set the loop to:

  while (x*x + y*y < 4  AND  iteration < max_iteration) 
  {
    xtemp = x*x - y*y
    y = 2*x*y + cy 
    x = xtemp + cx
    
    iteration = iteration + 1
  }

Where cx and cy are the real and imaginary part of c.

In the second example I propose the following changes:

Before the loop set x = x0 and y = y0 and do variable renaming analogous to above
Set y = (x*x + y*y) ^ (n / 2) * sin(n * atan2(y, x)) + cy
Set xtmp = (x*x + y*y) ^ (n / 2) * cos(n * atan2(y, x)) + cx
Remove type specifier float
Remove semicolons

Reasons

Without renaming the variables in Example 1 the code would not work.
Without changing c to cx, cy the code would not work.
Variables with zero-index indicates start values and should not be changed.
Type specifers must not occur inconsistently in pseudocode and in this case the type specifier is unnecessary.
Semicolons should not occur inconsistently.

Daknuett (talk) 18:36, 30 December 2017 (UTC) Daniel[reply]

This pseudocode sometimes does not plot the entire Julia set when c is large. This is because there is a mistake in the section "Quadratic polynomials". Contrary to what is said there, points in the filled Julia set do not always have modulus <= 2. (As a specific example, if c = -6, then z = 3 is a fixed point of z^2 + c [because 3^2 - 6 = 3], so 3 is in the filled Julia set.) I have corrected the error in "Quadratic polynomials" and the pseudocode. This unfortunately makes things a bit more complicated, but I think it is important for the article to be correct. DaveWitteMorris (talk) 17:21, 2 October 2019 (UTC)[reply]

Contradictory definitions

In section "Examples", the Julia set of $f(z)=z^{2}$ is described as unit circle, but evalutaing $J(f_{c})=\left\{z\in \mathbb {C} :\forall n\in \mathbb {N} ,|f_{c}^{n}(z)|\leq 2\right\}$ as found under "Quadratic polynomials", we would obtain the closed unit disk.--Hagman (talk) 13:40, 25 July 2019 (UTC)[reply]

The disk is the filled Julia set, not the Julia set. I fixed the error in "Quadratic polynomials". DaveWitteMorris (talk) 02:36, 2 October 2019 (UTC)[reply]

Julia set renders coordinate system inconsistencies

In section Quadratic polynomials, the pictures of julia sets seem to use a mathematical coordinate with imaginary - north. Except for Julia set for fc, c = 0.8i, which seems to be - south. This results in renders being inconsistent with each other and the text given. As normally the the - south coordinate is used to my limited understanding, all the other renders apart from Julia set for fc, c = 0.8i might be incorrect between the images and formula given for those images.

EDIT. After checking all the formulas against the wolfram alpha fractal plotter, it seem that the render for c = 0.8i is the only one flipped.

In addition, the Julia set for fc, c = 0.285 + 0.0–01i seems a bit weird written like that, I assume it's actually rendered as c = 0.285 + 0.01iMyrrysart (talk) 12:49, 28 May 2025 (UTC)[reply]

EDIT. Went ahead and checked all the renders, found another oddity among the set, now all the numbers should be the same as what would result in the render itself. — Preceding unsigned comment added by Myrrysart (talk • contribs) 15:09, 28 May 2025 (UTC)[reply]

[1] Beardon, Alan F. (1991). Iteration of Rational Functions. Springer. p. 50. ISBN 0-387-95151-2.

[2] Blanchard, P. (1984). "Complex analytic dynamics on the Riemann sphere". Bulletin of the American Mathematical Society. 11: 85–141.

[1]

[2]