Mapping between numbers and symbolic representations I am not a physicist but applying symbolic dynamics for information coding in signal processing. Is there any mapping between symbols and numbers?
 A: While I am not familiar with signal processing, the question of the 1-to-1 mapping may be easy to answer - assuming your alphabet of symbols is finite.
Suppose your symbol alphabet is $A = \{ a_0, a_1, \dots , a_{n-1} \} $. First define a mapping $$ f : A \rightarrow \mathbb \{ 0, 1, \dots (n-1) \} : f(a_i) \mapsto i.$$
Then for a sequence of symbols $$b = (b_!, b_2, \dots, b_m )$$ define a mapping $$ b \mapsto f(b_1) + (f(b_2) \times n) + (f(b_3) \times n^2) + \dots + (f(b_m) \times n^{m-1}).$$
In other words, the sequence $b$ maps to a base $n$ expansion where the $i'th$ element of the sequence is the $(m-i)'th$ digit of the expansion.  As such, it is clearly a bijection.
You can then biject between the base $n$ expansion and the binary (base 2) expansion.
I hope I haven't misunderstood your question.
EDIT :   As per your comment below, here is a less mathematical description.  

First, number your alphabet of symbols, say $a_1, a_2, ... a_n$.  The order in which you number them is not important, however it is essential that both the sender and receiver number the alphabet in the same way.  Next, convert the string of symbols into a numerical value written in base $n$, where $n$ is the size of your set of possible symbols.  You will treat each symbol in the sequence as a digit in a number written in base $n$.  Once you have your symbol sequence converted into a base $n$ number, you will need to convert this base $n$ number into binary (base 2) for transmission.
To go the other way, start will a binary string (base 2), convert this into a base $n$, then use the digits of the base $n$ number to read off the symbol sequence in your message.  
For convenience, let's suppose your alphabet has 10 symbols, $\{ +, -, =, 0, 1, x, y, z, <, > \} $.  Then we could convert the message $x + 1 = y$ as follows :
The first symbol in our sequence (x) is the 6'th symbol in our alphabet, so the first digit would be 6.  The + is the second symbol of the sequence and the 1'st symbol of our alphabet, so the second digit would be a 1.  Continute to obtain the numerical encoding of 61537.  Finally, convert this number into binary for transmission. In our example, this would be 1111000001100001.
Conversely, if you receive a signal (111100001100001), then you can convert (decode) it into a symbol string by following the above procedure in reverse.  First convert the binary to decimal, and then read off the symbol corresponding to each digit.
If you are unfamiliar with converting numbers between different bases, then here is a good place to start [wikipedia] (http://en.wikipedia.org/wiki/Binary_number).  Good luck.
A: I am not sure, what assumptions you can make, but as your question is right now, I can think of the following two counter-examples. I always regard the Heaviside function $θ$ as a symbolisation function.


*

*One-to-one mapping between numerical time series and symbolic time series: $\sin(t)$ and $2\sin(t)$ are obviously mapped to the same symbolic series.

*Preservation of dimension: $\sin(t)$ has dimension 1 and $\sin(t)·(\sin(αt)+2)$ has dimension 2 for $α∈ℝ\backslashℚ$, but they obviously have the same symbolic time series.

