I read in this article:
" Ludwig Boltzmann, coined "ergodic" as the name for a stronger but related property: starting from a random point in state space, orbits will typically pass through every point in state space. It is easy to show (with set theory) that this isn't doable,..."
Could someone please explain how this can be proved. I have seen other articles that define that an ergodic process will go to every state with some probability.
Second part of my question is, what is the relationship between stationary process and an ergodic process?
Typically, the set of states with a given energy is a manifold of dimension $>1$. Typically, the orbits are differentiable functions of time. Then one shows with topology (not with set theory) that the orbit is not equal to the whole manifold.
If you drop the condition of differentiability, then there are "space-filling" curves, so the orbit can, indeed, be everything.