Talk:Julia set

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Needs a definition of "regular"

The article uses the term "regular" in several places, with no link nor internal explanation.

Also, should "perturbation" really link to "Perturbation theory?" I think the term here is being used loosely, and Perturbation theory is about using slight changes to solve specific problems. Same word, but not a useful link.

Thomaso (talk) 15:31, 27 February 2016 (UTC)[reply]

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]

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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]

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]

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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]

[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]