# Prove that $S: I \rightarrow \Sigma_2'$ is one-to-one.

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?

Thanks.

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Isn't a sequence associated to a real by associating $a_1a_2a_3\dots$ to $\sum a_n2^{-n}$? If so, wouldn't the two descriptions of $1/2$ be $10000\dots$ and $01111\dots$? – Gerry Myerson Sep 7 '12 at 12:59
Excuse me for the delay in my response, but I have problems with the internet connection. Yes, $1/2$ may be described by either $(11000...)$ or $(01000...)$ in $\Sigma_2$. So the function $I\rightarrow \Sigma$ isn't one-to-one.But if we identify any two sequences of the form $(s0...sk∗1000...)$, where $∗=0$ or $1$, we define $\Sigma_2 '$ that denote $\Sigma_2$ with these identifications. – Mark Sep 10 '12 at 6:39
You've not answered my questions. I do not understand in what sense 11000... describes 1/2, and I do not understand in what sense 01000... describes 1/2. – Gerry Myerson Sep 10 '12 at 13:06
I'm sorry, I did not understand.The sequence $11000...$ describes 1/2 for the following reason:$x=1/2 \in [0,1/2]$ or $\in [1/2,1]$. In first case we associate $0$ as first bit, in the second we associate $1$ as first bit. But, if we substitute 1/2 in $2x$ or $2-2x$ we obtain $T_2(1/2)=1$ in both cases,and $T_2(1/2)=1\in [1/2,1]$ and it is associated the bit $1$.So we have, for $x=1/2$, the sequences $01$ or $11$.Now we substitute the value $x=1$ in the function $2-2x$ and we have:$T_2 ^2 (1/2)=0$ that belongs to the interval $[0,1/2]$.I.e. bit $0$.And now we have the sequences $010$ or $110$. – Mark Sep 13 '12 at 7:48
Ah. So when you say "A may be described by B," what you mean is "B is the itinerary of A under the tent map." – Gerry Myerson Sep 13 '12 at 12:31