I'm interested in automatic sequence or substitution systems.
I focused on the simplest form with a binary alphabet without constants.
A well-known example is Thue-Morse sequence
with axiom [0] and rules [ [0] → [0, 1], [1] → [1, 0] ]
It is known that it's uniformly recurrent word( or almost periodic ) and fractal.
On the other hand, it is not normal or equidistributed.
in the sense that subwords are presented unequally.
E.g. 3-tuples [0, 1, 0],[1, 0, 1],[1, 0, 0],[0, 0, 1],[0, 1, 1],[1, 1, 0]]
the same quantity but there are no [0,0,0] and [1,1,1].
Such sequences can be visualized by turtle graphics (or AnglePath). Thue-Morse produces Koch curve for 60°
but primitive picture for 90°
Exploring various substitution rules I've found next example:
UPD Recently found a simpler example
init [0], [[0,0] -> [1], [0] → [0, 0], [1] → [1, 0]]
Turtle graphics for it looks very interesting:
60°:
90°:
Two questions are raised.
- Such properties as fractality and almost periodicity are strictly defined for infinite sequences. Is it possible to formulate a computational criterion for the degree of these properties for finite sequence?
- I’ve done some preliminary calculations, and it looks like the difference between "my" sequence and Thue-Morse is only quantitative. But turtle graphics look very different. What properties might be behind this?