Are these permissible ways of writing Mandelbrot's Equation?

[emergentecon]

When asked about his work, Mandelbrot wrote his equation as such: z -> z^2 + c

Is it permissible to also write it as:
z = z^2 + c
and / or
f(z) = z^2 + c

 

[Simon Bridge]

No - well, you can if you really want to but neither of them is correct in that the relations do not describe the same process.

$z\to z^2+c: z,c\in\mathbb C$ refers to an iteration. Notice that it is not an equation. Equations have an "=" sign.
You could write $z_{n+1}=z_{n}^2+c$ to relate the next term in the iteration with the previous one ... but it kinda misses the point.
Think: what is the purpose of the iterations?

$z=z^2+c$ would only evaluate true for one or two values of z - think what the "=" sign means.

$f(z)=z^2+c$ would be treating z as a variable when, crucially, Mandelbrot's $z$ is a specific test point.
The idea is to test it to see if it is a member of the set.

 

[emergentecon]

No - well, you can if you really want to but neither of them is correct in that they relations do not describe the same process.

$z\to z^2+c: z,c\in\mathbb C$ refers to an iteration. Notice that it is not an equation. Equations have an "=" sign.
You could write $z_{n+1}=z_{n}^2+c$ to relate the next term in the iteration with the previous one ... but it kinda misses the point.
Think: what is the purpose of the iterations?

$z=z^2+c$ would only evaluate true for one or two values of z - think what the "=" sign means.

$f(z)=z^2+c$ would be treating z as a variable when, crucially, Mandelbrot's $z$ is a specific test point.
The idea is to test it to see if it is a member of the set.

Ok thank you very much for the explanation . . .

 

[emergentecon]

I'm curious though, why on this site, they make use of the f(z) approach in discussing the mandelbrot set? http://www.math.cornell.edu/~lipa/mec/lesson5.html

 

[Simon Bridge]

Because they really really want to? Maybe they left something out or are just abusing the notation - its quite a chatty page.
Lets see - reading - they are not actually using $f(z)=z^2+c$ as an equivalent to writing $z\to z^2+c$.

They are basically putting: $z\to f(z)$ or $z_{n+1}=f(z_n)$ ... that use is implicit in the context. When you use f(z) like this it is called a "map".

 

[emergentecon]

For instance, from here: http://mathworld.wolfram.com/MandelbrotSet.html

The term Mandelbrot set can also be applied to generalizations of "the" Mandelbrot set in which the function

is replaced by some other function. In the above plot, [PLAIN]http://mathworld.wolfram.com/images/equations/MandelbrotSet/Inline92.gif, [PLAIN]http://mathworld.wolfram.com/images/equations/MandelbrotSet/Inline93.gif, and http://mathworld.wolfram.com/images/equations/MandelbrotSet/Inline94.gif is allowed to vary in the complex plane. Note that completely different sets (that are not Mandelbrot sets) can be obtained for choices of

that do not lie in the fractal attractor. So, for example, in the above set, picking

inside the unit disk but outside the red basins gives a set of completely different-looking images.

So the function which generates the mandelbrot set, is the function

iterated over complex numbers?

 

[emergentecon]

Because they really really want to? Maybe they left something out or are just abusing the notation - its quite a chatty page.
Lets see - reading - they are not actually using $f(z)=z^2+c$ as an equivalent to writing $z\to z^2+c$.

They are basically putting: $z\to f(z)$ or $z_{n+1}=f(z_n)$ ... that use is implicit in the context. When you use f(z) like this it is called a "map".

True, and in fact, they refer to it as a map from what I see . . .

 

[Simon Bridge]

... same with the mathworld one.
Don't worry, look hard enough and you will find someone with some kind of authority making the error.
That does not make it correct - it just makes them being unconventional (at best) or wrong (at worst).
Make sure you also read and understand the su

In general:
You can if you really really want to...
You can use any definitions you like, but if you depart from the conventions, you have to explain your definitions in the accompanying text.
You can say: Let = be the addition operator and + be the equality; then 1=1+2 would be a valid mathematical statement evaluating "true".

As the Old Man of the Mountain was known to say: "Everything is permissible."