Chaotic mappings are known as highly sensitive to their initial state. It is well known that the first type of Chebyshev polynomials is chaotic, e.g. this. This mapping is defined recursively as $T_n(x) = 2xT_{n-1}(x) - T_{n-2}(x)$, $T_0(x) = 1$ and $T_1(x) = x$. It also has a closed form of $\cos(n \times \arccos (x) )$ for $x \in [-1,+1]$ and $n \in \mathbb{N}$.
It is also well known that $T_n(x)$ satisfies a so called semigroup property, i.e. $T_n(T_m(x)) = T_m(T_n(x))$. This property is useful in some applications in cryptography like this.
However, I know that in floating-point computations, e.g. IEEE 754, the numbers are rounded to their nearest representable ones. This is the place that my question arises!
How much is this chaotic mapping sensitive to a rounding error? Can we bound it? How about other chaotic mappings?
More important, is $T_n(T_m(x)) = T_m(T_n(x))$ in finite precision?
Note: I am totally new to the topic of Chaotic mappings, so even a search keyword is appreciated.