Let $(X, \mathcal{A}, \mu)$ be a probability space and $f: X \rightarrow X$ be a measurable dynamical system such that $(f,\mu)$ is an ergodic system.
Probabilistically, a $\mu$-typical orbit $(f^n(x))_{n \ge 0}$ can be easily distinguished from a sequence of $\mu$-IID draws from $X$. For example, consider an irrational rotation in the circle with Lesbegue measure.
If we remove the time-order information and consider $O = \{ f^n(x) : n \ge 0 \}$, can this set be probabilistically distinguished from a set of countable $\mu$-IID draws from $X$? I suppose not, but I don't have any argument more striking than: "due to ergodicity, the asymptotic density elements of O within every measurable set (according to any enumeration of $O$?) equals to its measure".
In fact, I'm being quite vague when referring to probabilistic distinguishable, because I'm not aware how probabilists (or statisticians?) would formalize it.
Any thoughts?
Thanks! Lucas A.