What does "The closure of the shift-orbit of the Fibonacci word" mean?

Question

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.

Answer 1

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 $\langle X,d\rangle$.