Late one particular evening I was looking at the x3+5 Stopping Times Graph and its tiling,
and I began to be curious: I wonder what % of this graph is blue? What % is green? What % is orange? Recall that each color corresponds to a distinct convergent loop (blue is n=1, green is n=5, orange is n=19, etc). Really my question is: up to some fixed n value what % of the dots fall into each of the different convergent loops?
For each convergent loop let us define a function %(n) to be the percentage of hailstones ≤ n which fall to that particular loop. Graphing this for each n value up to 10000 gives us:
When I first wrote this program to test for the % memberships, I had no idea what to expect from the resultant image... But this picture startled me, perhaps primarily because the % memberships seem to be "chilling out" over time and perhaps (I hesitate to suggest this) approaching a limit? This was very unexpected.
I wanted to study this phenomenon further but the initial program I wrote for the task struggled to go much beyond 10^5. After some time and significant work, I re-engineered the machine so that it was much more efficient and streamlined and now could processes nearly arbitrarily far out given enough time. Here's a look at this same graph but much further.
It's very surprising. Given the tiling phenomenon, I guess it makes sense that the Collatz landscapes exhibit some degree of stability at large scales, but this is very stable behavior... much more so than we tend to expect from the Collatz System. Let's look at another example.
The x3.564+1 World
We can see from this graph that the % memberships actually (surprisingly!) are not coalescing down to limit at all, but rather are stabilizing to some kind of a semi-sinusoidal wiggle. If we take just a few of those wiggles and zoom in on them, they look like:
Let's look at a few more examples.
The x3.428+1 World
The x3.001+1 World
The x1.7+2.5 World
A key fact is that it's not even just the various different convergent loops which have this sinusoidally stabile "wiggle" – all branches have this property too! Non-trivial ones anyway...
Non-trivial branches are defined as follows: Consider the x3+1 world. Any particular node tends to spawn many many more nodes and branches above it which spread out and out. But there are some that don't – I call them "Barren Branches" or "Trivial Branches" – which occur whenever a node is divisible by 3. These branches are tiny and have a % membership which drops to 0 over time. While slightly more complicated, trivial branches also occur in other xa+b Collatz Worlds.
To give you an example of the sinusoidal stability of the branches of x3+1 space, let's look at the % memberships of these 6 branches circled below (2 trivial, 4 non-trivial).
It seems we have discovered some kind of global stability hidden within the Collatz system.