From this paper, I see one could use probability distribution to describe an ergodic system. For example, in page 12, consider logistic map $x_{t+1}=ax_{t}(1-x_{t})$ with $a=4$, its ergodic behavior can be described by a probability distribution $Beta(0.5, 0.5)$.
My question is: can we do the same thing in the opposite direction, i.e. find the underlying dynamical system given a probability distribution? If so, is the solution unique?
Thanks in advance!