# What does “The closure of the shift-orbit of the Fibonacci word” mean?

Im trying to translate an article about rauzy fractal. But since my English is not good enough I cant understand this paragraph:

There is also a dynamical interpretation: We can shift the Fibonacci word by
erasing its first letter; if we consider the closure of the shift-orbit
of the Fibonacci word for the natural topology on infinite words, we get a very
simple symbolic dynamical system. Its domain is a Cantor-like set,
which projects continuously to the interval, and the projection conjugates
the shift to the rotation by the golden number.


And this is the definition of a Fibonacci word:

Define a sequence of words on the alphabet of two letters a, b starting with a and at each step substituting every a by ab and every b by a. Elementary algebra shows that the lengths of the words a, ab, aba, abaab, ... are the Fibonacci numbers, the ratio of the frequencies of a and b tends to the golden number $\phi = \frac{1+\sqrt{5}}{2}$ and each word is a prefix of the next one. In this way we define the Fibonacci word, the only infinite word abaababaab... that is invariant by the substitution rule.

I specialy do not understand the meaning of the closure of the shift-orbit.

Let $X$ be the set of infinite strings of $a$s and $b$s; the Fibonacci word is then one member of $X$. If $s=s_0s_1s_2\ldots$ and $t=t_0t_1t_2\ldots$ are distinct members of $X$, we let

$$m(s,t)=\min\{k\in\Bbb Z^+;s_k\ne t_k\}\;.$$

Finally, we define a metric $d$ on $X$ as follows. For any $s,t\in X$,

$$d(s,t)=\begin{cases} 0,&\text{if }s=t\\ 2^{-m(s,t)},&\text{if }s\ne t\;. \end{cases}$$

It’s not difficult to verify that $d$ really is a metric on $X$. It generates the product topology on $\{a,b\}^{\Bbb N}$ when $\{a,b\}$ is given the discrete topology.

Now let $\sigma:X\to X$ be the left-shift function: if $s=s_0s_1s_2\ldots\,$, then $\sigma(s)=s_1s_2s_3\ldots\,$. Let $f$ be the Fibonacci word. The shift-orbit of $f$ is the set

$$\{f,\sigma(f),\sigma^2(f)=\sigma(\sigma(f))\ldots\}=\{\sigma^k(f):k\ge 0\}\;,$$

and we want its closure in the metric space $\langle X,d\rangle$.

• This is a fine answer but I'm worried that it might be over OP's head yet. An example of a word that's in the closure but not in the shift-orbit proper would go a long way, but at least at first glance I can't see any reasonably explicit way of defining such a word (IIRC this is coupled pretty closely to some of Connes' noncommutative geometry and specifically the examples on Penrose tilings) – Steven Stadnicki Nov 19 '14 at 18:45
• @Steven: I didn’t immediately see one either. I share your worry; that’s why I went with a metric rather than just the product topology. Unfortunately, I’m not familiar enough with the area to be comfortable trying to break it down more or to try to relate it to anything that might be helpful. – Brian M. Scott Nov 19 '14 at 18:50