That’s interesting. I like the connection that any eventually periodic sequence corresponds to a rational number and any divergent sequence (which we know from Unicity of trajectory - #3 by Failix correspond to non-periodic parity sequences) corresponds to an irrational number.
In this context, a weak version of the 3n+1 problem (are there any divergent sequences) can be reformulated as: are there any irrational fractions corresponding to parity sequences? Which seems interesting (although, I have no idea if that’s useful or not …)