Geometric structure of the $E(n).O(n)$ state space for the digit map $f(n)=(E(n).O(n))²$.

Consider the digit dynamical system

$$ f(n)=\bigl(E(n)\,O(n)\bigr)^2, $$

where $E(n)$ and $O(n)$ denote the sums of the even and odd decimal digits of a positive integer $n$, respectively.

The state of an integer may be represented by the ordered pair

$$ (E(n),O(n)). $$

I plotted all attainable $(E,O)$ states and coloured each point according to its eventual attractor:

• Black: the orbit eventually reaches $0$.
• Green: the orbit eventually reaches $324$.
• Red: the orbit eventually reaches $5184$.

For reference, I also plotted the curves

$$ x^2y^2=0,\qquad x^2y^2=324,\qquad x^2y^2=5184, $$

which correspond to the terminal products

$$ EO=0,\qquad EO=18,\qquad EO=72. $$ I have graphed all the iterates from 0 to 10^7

The interactive GeoGebra graph is available here (it may take a few moments to load):

https://www.geogebra.org/graphing/vyzppbua

The complete project, including the proof that the only fixed points are $0$, $324$, and $5184$, together with computational observations, is available here:

https://github.com/sahushuvjnyandeep-png/Parity-State-Dynamics

Although these patterns arise computationally, I am interested in theoretical explanations of the following observations.

1. Staircase structure. The attainable $(E,O)$ states appear to lie on distinct staircase-like bands rather than being distributed uniformly. Is there a mathematical explanation for this pattern?

2. Asymmetry. The distribution is noticeably asymmetric. In particular, near the lower-right portion of the graph, the basin of attraction of $5184$ appears much denser, whereas the basin of $0$ becomes comparatively sparse. Is this asymmetry an inherent consequence of the digit-sum map, or can it be explained theoretically?

3. Dispersed basins. The three basins of attraction are highly intermingled instead of forming large connected regions. Why can neighbouring $(E,O)$ states have completely different eventual attractors? Is there a theoretical description of this basin structure?

This question is a continuation of my previous one, in which it was proved that the only fixed points of the map are $0$, $324$, and $5184$:

How can I prove that $0$, $324$, and $5184$ are the only fixed points of $f(n)=(E(n)O(n))^2$? Where $E$ and $O$ are even and odd digit sum of a number