# Distribution of tuples in chaotic sequences

I study infinite aperiodic sequences like Thue-Morse.

Simple substitution rules allow you to get even more complex.
I'm interested in the distribution of tuples in such sequences.
For example, in Thue-Morse $$1$$ and $$0$$ represent equivalently,
of the 2-tuples $$\{00, 01,10,11\}$$ represented all four, but not equally,
and of the eight 3-tuples present only six.

For a strictly chaotic binary sequence, all $$n$$-tuples must be represented uniformly,
each type with a probability of $$2^{-n}$$.
If you plot their distribution (sorted by descending, in Log-Log scale), it will be the following:

For sufficiently large $$n$$ we see that the plot is not horizontal,
this is obviously related to limited sample size (in this case it was only $$50000$$).
I’m interested in the real shape of these plots, especially the "tails" on the right,
to compare with quasi-random sequences.
For example, rule $$11 \to 00, 00 \to 01, 0 \to 10$$ generates a rich enough sequence with this distribution of tuples:

I’d like to know what is the effect of quasi-chaotic, and what is the limit of the sample.
To do this, I need to know the approximate shape (equation) for the tail of tuples distributions.