TITLE: Subwords of the Fibonacci word
QUESTION [19 upvotes]: The Fibonacci word is the limit of the sequence of words starting with  "$0$" and satisfying rules $0 \to 01, 1 \to 0$.  It's equivalent to have initial conditions $S_0 = 0, S_1 = 01$ and then recursion $S_n= S_{n-1}S_{n-2}$.
I want to know what words cannot appear as subwords in the limit $S_\infty$.  At first I thought $000$ and $11$ were the only two that could not appear.  Then I noticed $010101$.  Is there any characterization of which words can or cannot appear as subwords of the Fibonacci word?
Loosely related, this word appears as the cut sequence of the line of slope $\phi^{-1}$ through the origin where $\phi = \frac{1 + \sqrt{5}}{2}$.

REPLY [5 votes]: There is a simple algorithm to determine the set of words that do not appear as factors (subwords) in the Fibonacci word $f$ (which comes essentially from the characterization Ale De Luca pointed out in his question).
Take the Fibonacci sequence $(F_n)_{n\geq 0}=1,1,2,3,5,8,\ldots$. For every $n$ larger than $1$ take the word of length $F_n-2$ appearing at the beginning of the Fibonacci word $f$ (i.e., the prefix of $f$ of length $F_n-2$). This gives you the sequence of words $\lambda,0,010,010010,\ldots$ Now, for each of these words $v$, if $v$ is followed by symbol $1$ then take $0v0$; otherwise, if $v$ is followed by symbol $0$, then take $1v1$. This gives you the set of words of minimal length that do not appear as factors in $f$, that is, the minimal forbidden factors of $f$: $11,000,10101,00100100,\ldots$
The words that do not appear as subwords in the Fibonacci word $f$ are precisely the words over $\{0,1\}$ that contain a minimal forbidden factor.