  The complexity $\Vert n\Vert$ of a natural number is the least number of $1$
needed to represent $n$ using the 5 symbols $(, ), *, +, 1$. A natural number
$n$ is called stable is $\Vert 3^kn\Vert =\Vert n\Vert +3k$. For each natural
number $n$, the number $3^an$ is stable for some $a\ge0$, and we define the
stable complexity of $n$ as $\Vert n \Vert _{\rm st}=\Vert 3^an\Vert -3a$. We
show that the closure of the set of all fractions $n/3^{\lfloor \Vert {n}\Vert
_{\rm st}/3\rfloor}$ has remarkable properties; self-similarity $3K'''=K$,
well-ordered, and certain arithmetical properties. We pose the question about
the unicity of this compact. This question raises some problems about the
complexity of natural numbers that we are unable to answer.
