I was thinking about this when I saw a screensaver appear on my machine which produced a similar-looking plot to the ones here. From this site:
https://plus.maths.org/content/extracting-beauty-chaos
we have a graph of the "Lyapunov exponent" of a discrete-time map, in particular here
$$f(x) = Q(P(x))$$
with
$$P(x) = (4p)x(1 - x)$$ $$Q(x) = (4q)x(1 - x)$$
and the graph shown plots the Lyapunov exponent as a color value for each value of the parameters $(p, q)$ (reddish-orange colors being positive values and white-bluish blare colors being negative values).
The part that I'm interested in is that the graph appears to be like it is a stylized plot of a projection of some 3-dimensional curved shape, whose surface has been shaded with some kind of shader under some kind of lighting conditions. In particular, looking in the top-right it appears to the eye that the surface swoops into the screen and then up in a jutting spike vertically behind another jutting spike horizontally. Moreover "beyond" (i.e. looking "deeper" into the page where it turns orange) the vertical spike is then followed by many smaller blue ones which also appear to continue up "behind" the horizontally-jutting part of the feature. More patterns can doubtless be discerned; but they are not so easy of description.
Now the site says something about how the "overlap" can be thought of as being due to "competing attractors", where it switches from one behavior to another. But that's not what I'm interested in so much. Rather I'm really interested in the apparent 3-dimensional behavior. Is there some surface or shape $F(x, y, z) = k$ in $\mathbb{R}^3$ such that when colored with some suitable shader (that is, a function which takes in an incident light vector and a point on the surface and produces a color) and projected from a suitable vantage point in perspective projection, produces the given image? Now there of course may be more than one such surface possible -- but is there some fairly "natural" way to extract such a surface from the mapping? I know, it's hard to put all this down into formal mathematics because it's so qualitative and perceptual. But nonetheless "is there something there"? What does this surface mean mathematically, and most interestingly of all, what do the points on the surface that are "obscured" in the picture, e.g. the points on the vertical spike "behind" the horizontal one, mean in terms of the original dynamical system? It seems almost "holographic" in some sense, in that this 3-dimensional shape is somehow encoded within the 2-dimensional map. Is there some what to "holographically" extract the shape?
More formally, what this means is we need to express the Lyapunov exponent $E$ as follows. Take $\Sigma$ to be a surface in $\mathbb{R}^3$. Now for a given point $C \in \mathbb{R}^3$, the camera point, some plane $P$, the view screen, held off from $C$, and a point $L \in \mathbb{R}^3$, the light point, we have a map
$$S: \Sigma \times \mathbb{R}^3 \rightarrow \mathbb{R}$$
a shading map which assigns a Lyapunov exponent value given a point on the surface and a vector representing an incident light ray at that point. This then creates, under the given lighting point $L$, the Lyapunov value (where $E: \Sigma \rightarrow \mathbb{R}$)
$$E(u) = S(u, \mathbf{u} - \mathbf{L})$$
where $\mathbf{u}$ and $\mathbf{L}$ are the vectors in $\mathbb{R}^3$ corresponding to the points $u$ and $L$, respectively.
Then we must have that $E(p, q)$ is the value of $E(u)$ where the point $u$ is the earliest point on the surface $\Sigma$ struck by the ray emanating from $C$ and passing through $P$ at a point $(p, q)$ upon it (take two perpendicular vectors of unit length on $P$ as bases) can be given as the value obtained under the perspective projection of the surface as viewed from the camera point $C$ looking down the $z$-axis (i.e. into the picture), that is, at $(C_x + p, C_y + q, C_z)$, which form a plane at the point $C$, when we take the perspective projection onto that plane with the nearest surface point being the one whose $E(u)$-value gets mapped to that plane, that plane then holds $E(p, q)$.
What would be suitably "natural" forms of $\Sigma$, $S$, $L$, $C$, etc. especially $\Sigma$ for a suitably standard $C$ and $P$ and coordinates on $P$ (e.g. maybe just take $C = (0, 0, 1)$ and $P$ to be the usual $xy$-plane so $(p, q)$ is $(p, q, 0)$ on that plane)?
Note that this is, of course, very ambiguous and that's why I'm not really sure how to proceed: a 2-dimensional plane held at the right distance "painted" with the picture given would also satisfy these conditions and that's obviously worthless and trivial. To eliminate that, perhaps an interesting first condition would be that $E(u)$ must be a smooth function on $\Sigma$ (thus the discontinuities created by overlap in the case of the "painted plane" would disqualify it and so a plane as being a valid candidate for $\Sigma$). What further "natural" conditions need to be applied to get uniqueness, and then what is the final $\Sigma$ and what does it mean dynamically? I do not expect any closed form for $\Sigma$ at all, rather a method or process to obtain it would be the expected answer and perhaps also, some nice pictures showing its rotation and viewing from different vantage points :) Even more bonus points could be then to find maps $f$ which have these projections as their Lyapunov graphs. How does changing the vantage point change the map $f$? How are they related to the original map? (It may also be that a point light source may not be the most natural one to use -- if not, try also a plane light (i.e. all rays parallel), etc.)
