🚀 go-pugleaf

Interestingly, Herman Rubin says of the (integer) hailstone sequences:  

>>The chaotic nature is that a slight difference at one point
>>makes a big difference a considerable time in the future.

Most of the widely discussed "chaotic" dynamical systems, like
4x(1-x), iterate over real numbers. But real numbers
are not computable, and fractal images (presented by
their proponents) are always floating point approximations to a
platonic ideal. I find the "chaos" idea, of an iterative
system having a "deterministic aperiodicity", quite stimulating
precisely because it does not seem computable.

I played with the hailstone numbers for a couple of hours once, and
found them somewhat "unpredictable".  It seemed to me that the
sequences are the result of a non linear dynamical system over the
integers (x -> x/2 if even, x -> 3x+1 if odd).  But the hailstone
system, though sensitive to initial parameter, always seems to halt by
finding the (1-4-2-1) attractor, though I'm not sure this has been
proven.  Other non-linear operations, like modular multiplication by a
constant, (the basis for pseudo-randomness), also lead to simple
periodic cycles.

Are there any deterministic functions over the integers which are
aperiodic?  Such a function would have to traverse an infinite subset
of the integers in an "unpredictable" way, without halting or cycling.
"Get the next prime number" would seem to be a candidate but for its
monotonicity...