The Fibonacci word is a binary sequence defined as follows.
We use a substitution rule $0\to 01$, $1\to 0$. Then, starting with the binary string $0$, apply the substitution rules successively. So we get $S_0=0$, $S_1=01$, $S_2=010$, $S_3=01001,\ldots$ and ultimately we get the following aperiodic sequence in $\{0,1\}^{\mathbb Z_+}$, known as the *Fibonacci word*:
$$
0100101001001\ldots
$$
For $S\in \{0,1\}^{\mathbb Z_+}$ and $n\in\mathbb Z_+$, let us define the complexity function $\sigma_n(S)$ as the number of distinct subwords of length $n$ in $S$. It is well-known that if $S$ is the Fibonacci word defined above then $\sigma_n(S)=n+1$. Here is my question:

Given a real number $x>1$. Construct a binary substitution rule $0\to A$, $1\to B$ (here $A$ and $B$ are finite binary strings) where, starting with $0$ and applying the substitution rule successively we get an infinite aperiodic sequence $S$ in $\{0,1\}^{\mathbb Z_+}$ so that $$\lim_{n\to\infty} \frac{\sigma_n(S)}{n}= x$$

Note that $A$ and $B$ must be chosen so that if $S_k$ is the binary string we get starting with $0$ and applying the substitution rule $k$ times, then $S_k$ is a prefix of $S_\ell$ for every $\ell>k$. That way the sequence $S_k$ stabilizes to an infinite binary sequence $S$ as $k$ tends to infinity.