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 $\phi = \frac{1 + \sqrt{5}}{2}$.

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