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From MathOverflow, R W said:

Unfortunately, the way the term "ergodic" is used in the theory of (finite) Markov chains is completely misleading from the point of view of general ergodic theory. To be consistent, one should have called "ergodic" the chains whose state space does not admit a decomposition into non-trivial non-communicating subsets. The notion of ergodicity you are referring to would rather correspond to what is called "mixing" in ergodic theory.

  1. I wonder if The notion of ergodicity means the ergodicity concept for a Markov chain?
  2. If yes, how does or is the ergodicity concept for a Markov chain correspond to or the same as "mixing" in ergodic theory?

Thanks and regards!

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The problem I see is, that you cannot have ergodic systems that are not mixing in a discrete case (ie markov chains). It is very well possible to differentiate between the two in general dynamical systems though. –  example Apr 28 '12 at 22:09
    
ok, no. in markov chains ergodic systems actually might be decomposable... but then they are not mixing in the dynamical-system sense either... –  example Apr 28 '12 at 22:21

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