I'm not completely sure what the following notation means:
$T: [0,1] \longrightarrow [0,1]$.
$T$ is a transformation. I guess in the context of the notation, $T$ is basically a transformation function I guess. I approximate this transformation function in a historical time period and use it for a future time period. Does this notation mean, that $T$ is defined for a closed intervall between 0 and 1 and that using $T$ on a future period doesn't change that fact? So to speak, $T$ is also defined for a closed intervall between 0 and 1 for the future period?
I thank everybody in advance.
The notation $$ f:X\to Y $$ is used to mean "$f$ is a function which takes elements from $X$ as input and gives elements of $Y$ as output". Calling it things like mapping or transformation doesn't change this (at least not without more context). $X$ is called the domain, and $Y$ is called the codomain.
So in your case, $T$ is a function which can take any element of $[0,1]$ as input and give you an element of $[0,1]$ in return. So $T(0)$ makes sense, as does $T(1), T(0.5)$ and $T(\pi/4)$. And these values are all somewhere in the codomain, i.e. somewhere between $0$ and $1$.
On the other hand, $T(-1), T(2.3)$ or $T(\sqrt 5)$ do not make sense, as $-1, 2.3$ and $\sqrt 5$ are not elements of the domain $[0,1]$