Naively I would have thought that there is another definition which goes along the lines that a process is ergodic if it looses memory of its starting conditions. But, maybe I’m mixing up things here.
That's essentially equivalent except in pathological synthetic cases, as time goes to infinity. Since any sample is representative of the whole, samples that don't include the intial conditions are as representative of the whole as samples that do include the initial conditions.
Well, trivially, a dynamical system with that switches between two states is ergodic. We always know what the past state of the system was given the present.
No, ergodicity is more about visiting the each part of the phase space with some positive frequency i.e. on a long trajectory you see everything that can happen, and not too infrequently.
