"Fix points" and "period-2 cycles"

**Quick summary**

Two fix points and one period-2 cycle were first found in the 3D basilica by brute force and then by manual calculation.

**Detailed**

In a first attempt to search for fix points in 3D I used Eric Baird's formula for the Basilica {x,y,z}_{new} := {x²-y²-z²,2xy,-2xz} + { -1,0,0 } in a brute-force way - so no solving of nonlinear equations for now (planned for the future).

A mandatory condition for a fix point is that at any refinement level there must be a tiny cube *A* whose bounding cube bbx(f(*A*)) after one iteration intersects back with *A*.

So I walked the 2-cube in width 2^13 * 2^13 * 2^13 pixels and checked every such tiny cube. Since there is no image construction involved, the only constraint here is time, not space.

The search returned 28 tiny cubes as possible fix points. Quite a low number compared to all 2^39 cubes present, so the approach seemed valid.

Of course, for this simple case school math and WolframAlpha sufficed to see that {0.5 +- 0.5*sqrt(5), 0, 0} are the only two fix points here.

Fortunately the brute force approach returned those too (in fact 8 tiny cubes contain the actual fix points: two fix point coordinates are 0 and lie on the grid, hence two adjacent cubes share it, i.e. 2*2 and since there are two solutions for the x coordinate, that gives

a total of 8 as found, a nice afterwards confirmation of the brute force method itself). The rest of the 28 I deem will vanish when increasing refinement level.

The image below shows the basilica and the two fix points (exaggerated), each at the crossing of three perpendicular red lines (one is in the background on the right side, dimmed quite a bit). It would be interesting to see at what refinement level those actually are identified as interior.

While fix points worked quite well I tried the same approach for period-2 cycles. Again with 2^39 cubes in refinement 13, 632 possible cubes involved in such a cycle were found.

Manual browsing through the cube coordinates revealed the two known fix points and two further points which appear to be around the origin and {-1,0,0}. And indeed, the sequence {0,0,0} -> {-1,0,0} is a period 2 cycle as seen by manual calculation.

Quite interesting is, that 532 of those possible period-2 cubes cluster around the fix-point at 0.5-0.5*sqrt(5), 100 cluster around the origin, 8 lie around the other fix point and only 4 around -1,0,0.

Is this a hint towards super-attracting (fix point at -0.61...), attracting (origin, but not the 2nd cycle point) and repelling (other fix point)? Or just a result describing the algorithmic (and not fractal) behaviour that would change at other refinements?

Since I have no idea whether something like a multiplier exists for the 3D case here to characterize cycles, I think I will just partition the interior according to orbit reachability and see what comes out.

EDIT: Correction: It must state 520 instead of 532 cubes clustering around the fix point.