I made this post a long time ago (see my profile picture), but only posting it now. Everything below is details, but the question itself boils down to: is this method already known and/or worth pursuing for either analysis or visualization of sequences?
Also some pretty pictures are at the bottom of the post.
I was trying to think up some way to quantitatively compare results for a certain time map with diferent initial conditions and I noticed that for a fixed number of steps $N$ the points in phase space start clustering as the initial condition approaches a periodic case. So it seemed possible to use their relative density as a measure of 'almost-periodicity'.
Finally, I came up with the following procedure.
Take a discrete time map where $r_n$ is a vector in (Euclidean) space:
$$r_{n+1}=f(r_n)$$
Here initial conditions is denoted as $r_1$. Place every $r_n$ as a point in phase space. After a fixed number of steps $N$ define the following values:
$$L_x=\max{x_n}-\min{x_n}$$ $$L_y=\max{y_n}-\min{y_n}$$ $$L_z=\max{z_n}-\min{z_n}$$
and so on in case of $r_n$ being multidimensional.
Define the neighborhood of $r_n$ as a box with dimensions {$2\varepsilon_x, 2\varepsilon_y, 2\varepsilon_z$}, centered at $r_n$ where:
$$\varepsilon_x=\frac{L_x}{N}$$
For every point count the number of points in its neighborhood (including itself), call it $K_n$.
Finally introduce scale invariant density of points as:
$$S_N=\frac{1}{N} \sum_{n=1}^N K_n$$
Defined like that $S_N$ has a number of interesting properties. First, consider some cases (I will be using one dimensional maps as examples).
Identity map:
$$x_{n+1}=x_n$$
$$S_N=N$$
For every other map I have considered $S_N$ grows slower than $N$.
A 2-periodic map:
$$x_{n}=(-1)^n$$
$$S_N= \begin{cases} \frac{N}{2}, & N=2k \\ \frac{(N-1)^2}{2N}+1, & N=2k+1 \end{cases}$$
$$S_N= \frac{N}{2}, \quad N >> 1$$
In the same way, any p-periodic map has:
$$S_N= \frac{N}{p}, \quad N >> 1$$
$$x_{n}=n$$
$$S_N=1$$
Only other example of constant $S_N$ appears to be a constant is random map uniformly distributed on an interval, in which case we obtain for large $N$:
$$<S_N> \approx 3.00$$
In most cases I haven't been able to find an exact expression, but the experiments show everything clear enough, here's $S_n$ for $N \leq 200$ for $x_n=n^2$ and $x_n=2^{n}$.
For $x_n=2^{-n}$ it's exactly the same, which is why I call $S_N$ scale invariant.
In other words, $S_N$ (or really, it's $N$ derivative) appears to be measuring the speed of convergence of a sequence (or divergence to infinity). I'm pretty sure a measure like that has already been introduced somewhere, so my question is:
Which common methods of discrete time maps analysis are related to $S_N$ and how?
Can this method be improved upon for either analysis or visualization of various sequences?
As an illustration, here is $S_N$ for $N=100$ depending on the parameters for the logistic map:
I think it shows a lot of useful information, such as the areas where it becomes chaotic (characterized by small values of $S_N$) as well as its fractal nature when approaching chaotic regions.
As another example, here's a low resolution Mandelbrot set (where I set up a cut off value for $z_n$ so it doesn't get above $10^5$ in absolute value, otherwise I'd risk overflow in computation).
Note that we see some interesting structure inside the set, while most of the common algorithms (like escape time) leave it all the same color.
And this is a related Burning ship fractal, which again, features a lot of structure that other algorithms don't show:
I believe your post breaks up into two separate mathematical topics. On the one hand, you are interested in the approximation of discrete dynamical systems by limiting continuous curves. On the other hand, you are interested in how your fractal simulations differ from more common approaches. And in both cases, you are seeking answers rooted in literature references.
Regarding the first topic, I will start by mentioning one idea behind ergodic theory, which is that the rigorous study of chaos can be quite challenging for deterministic systems, but becomes tractable for certain random systems. The slogan of an ergodic dynamical system is that "time averages equal space averages", which can be interpreted as saying that chaotic behavior smooths out when averaged over time in the right way. At a conceptual level, this is not so different than the continuous approximations of the discrete quantities you are plotting.
Due to this analogy, I think it is beneficial to phrase some of what you are doing in the language of probability theory. Your scale invariant density of points is aiming towards the concept of an empirical distribution - which is deterministic, being determined by the first $n$ stages of your dynamical system. And as we know from the central limit theorem, if your empirical distributions were being generated from points that were "nudged" with a little bit of random noise, then even as the points bounced around chaotically, their aggregate behavior would tend towards a nice continuous limit. So without delving further into the details of your calculations (I believe there are some inconsistencies / ambiguities in your definition and examples of $S_N$), I would be inclined to say that when phrased in a suitably precise way, you will discover that your approach is similar to the circle of ideas I have just mentioned. There is some very beautiful mathematics in the literature surrounding the topics I have mentioned, built upon by many researchers over the past century, and it pays to delve deeply (starting from the wiki pages and going to the books they cite) into these topics if you are interested in developing your methods further. The precise way in which a discrete dynamical system approaches a continuous one is specifically the subject of study in hydrodynamic limits, although be warned that the level of sophistication of the sources cited at that page is quite high and will require familiarity with rigorous mathematical analysis, partial differential equations and probability theory.
Regarding your second topic, my feeling is that the artifacts you are seeing that do not normally appear in pictures of these fractals amount to numerical errors, which likely arise due to a similar mechanism as a Moire pattern, and that link contains further references aimed at a broad audience to explain the concept and how it arises.