We consider the tent map:
$T_2(x)=2x \ \ \ 0\leq x \leq 1/2$
$T_2(x)=2-2x \ \ \ 1/2\leq x \leq 1$
Note that the maximum of $T_2$ is 1 and occurs at $x=1/2$. To describe the dynamics of $T_2$ via symbolic dynamics, we thus need to modify $\Sigma_2$ somewhat since there is an ambiguity in the sequence associated to any rational number of the form $p/2^k$ where $p$ is an integer. For example, $1/2$ may be described by either $(11000...)$ or $(01000...)$. To remedy this, we identify any two sequences of the form $(s_0 ... s_k * 1000...)$, where $*=0$ or $1$. For example, the sequences $(1101000...)$ and $(1111000...)$ are to be thought of as representing the same point. Let $\Sigma_2'$ denote $\Sigma_2$ with these identifications.
The exercise required to prove that $S: I \rightarrow \Sigma_2'$ is one-to-one, where $S(x)$ is the itinerary of $x$, i.e. a sequence $S(x)=s_0s_1s_2...$ where $s_j=0$ if $T_2 ^j(x)\in [0,1/2]$, $s_j=1$ if $T_2 ^j(x)\in [1/2,1]$. I solved the exercise in this way: there are $x,y \in [0,1]$ such that $x\neq y$, $y=x+\delta$ ($\delta>0$) and $S(x)=S(y)$ (reductio ad absurdum). But $S(x)=S(y)$ occurs only if $T_2 ^n(x)$ and $T_2 ^n(y)$ belong simultaneously to the same range: $[0,1/2]$ or $[1/2,1]$. The difference between $T_2 ^n(x)$ and $T_2 ^n(y)$, in module, increasing more and more: $2^n \delta$. So that $T_2 ^n(x)$ and $T_2 ^n(y)$ continue to belong to the same interval, it must be: $\delta = 0$, but $x\neq y$. C.V.D.
Any suggestions, please? This prof can be solved in other ways?