Formal definition of "inifinite detail" regarding factals such as the Mandelbrot Set

Question

The Mandelbrot Set is typically described as having infinite detail, e.g.:

Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnification

(Wikipedia)

Similar terms are used to characterise fractals, e.g.:

Fine or detailed structure at arbitrarily small scales.

(Wikipedia)

It's fairly apparent from looking at the Mandelbrot Set that is has this property, but is there a formal or mathematical definition of "infinite detail"?

Secondary question - is there a definition that also captures that fact that the set isn't a comparatively simple repetitive structure like the Cantor Set or the Koch Snowflake, that also have infinite detail but in a much less interesting fashion.

Answer 1

Detail at all scales

The notion of "has detail at all scales" is one that can be described mathematically in a number of different ways. My own intuition is related to some notion of "roughness". Broadly speaking, a function $f:\mathbb{R}\to\mathbb{R}$ is smooth at some point $a$ if it has a derivative at $a$.^[1] If $f$ has a derivative at $a$, then the function will start to "look like" a line when you zoom in on it enough—this idea is made rigorous in terms of local linear approximations. Smooth functions are the antithesis of fracals—as you zoom in, there is no detail to see; the function just looks like a line.

By contrast, a function which is continuous but nowhere differentiable is like "infinitely rough". No matter how much you zoom in on such a function, you will continue to see more detail See, for example, a Weierstrass function

or the blancmange curve

This notion of smoothness can be generalized to higher dimensions. For example, we might regard the boundary of the Mandelbrot set as a "curve" in $\mathbb{R}^2$. This curve is continuous, but not differentiable.

Fractality

Rigorous mathematical attempts to define the term "fractal" stray away from these kinds of wibbly-wobbly hand-wavey notions. Most notions of fractality have something to do with the dimension of the object being described. Mandelbrot proposed that any set with Hausdorff dimension smaller than its topological dimension should be considered fractal. This doesn't capture every set which we might want to call fractal, but it gives the right flavor.

Because this definition misses certain sets which probably ought to be considered fractal (and includes some sets which probably shouldn't be considered fractal), Mandelbrot later rejected this definition. At this moment in history, there is still no widely accepted definition of what a "fractal" is—most authors choose to leave the term undefined, or give "local" definitions (i.e. definitions specific to whatever they are writing).

In most cases, the definition of "fractal" set typically has something to do with the dimension or dimensions of that set. Mandelbrot proposed that sets with interesting Hausdorff dimension should be considered fractal, while my PhD advisor suggested that sets which have non-real complex dimensions should be considered fractal.

But, again, there is no widely accepted or universal definition.

Non-self-similar fractals

In some sense, a "generic" set is fractal. Sets which are not fractal are special and rare. There is some vocabulary related to attempts to classify sets. For example, there are non-fractal sets, self-similar fractal sets, self-affine fractal sets, etc. The Mandelbrot set is not truly generic, in the sense that it can be thought of as the attractor of a non-linear iterated function system—it has more structure than a truly generic set.

If I want to distinguish the Mandelbrot set from sets which are truly self-similar, I might describe it as "non self-similar". Describing it as such indicates that it is a wilder, or more pathological, or more "interesting" object than something like the Cantor set. On the other hand, if I want to tell a reader that the Mandelbrot set is more structured than an arbitrary set in the complex plane, I might describe it as "statistically self-similar". It is not strictly self-similar, in the sense that it is not composed of identical copies of itself, but it does have some kind of self-similarties, in the sense that one can see little "mini-brots" if one zooms in on certain areas of the set.

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[1] Technically, the term "smooth" usually means "has derivatives of all orders at every point in its domain". The definition I have adopted here is not entirely unheard of, but is a little non-standard. Perhaps a better term would be smooth enough at $a$.

Answer 2

I will limit myself to IFS in $\Bbb R^2$. We speak of "self-similarity". We have the property:

Let $n\in \Bbb N, n>1$. Let $f_1, ..., f_n$ $n$ contractions of $\Bbb R^2$. Let $\mathcal A:=IFS(\{f_1,...,f_n\})$. Then, $\mathcal A=f_1(\mathcal A)\cup...\cup f_n(\mathcal A).$

Otherwise written, $\mathcal A$ is the union of $n$ copies of itself, with for $k\in [\![a,n]\!], f_k(\mathcal A)=f_k(f_1(\mathcal A)\cup...\cup f_n(\mathcal A)),$ etc. Here they are, the "infinite details".

We can then define "self-similarity" as :

Definition : a compact $\mathcal B$ of $\Bbb R^2$ such that there exists $p>1$ contractions $g_1,...,g_p$ such that $\mathcal B=g_1(\mathcal B)\cup...\cup g_p(\mathcal B)$.

With this definition, we can speak about self similarity at least for IFS.

It is easy to establish that a square is self-similar. One must therefore be careful when asserting that a compact is not self-similar.