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seen Apr 7 '12 at 17:58

Apr
7
comment Having trouble understanding the concept of “mixing” in dynamical systems.
OK, I see your point. Now that I think about, it's obvious that the limit won't exist for any bounded state space if the flow is measure-preserving. Thanks for pointing out my mistake!
Apr
6
awarded  Student
Apr
6
comment Having trouble understanding the concept of “mixing” in dynamical systems.
I edited my question to match the above comment.
Apr
6
revised Having trouble understanding the concept of “mixing” in dynamical systems.
deleted 34 characters in body
Apr
6
comment Having trouble understanding the concept of “mixing” in dynamical systems.
As for the second part, I agree that I've put it in an unnecessarily confusing manner. The point is that in the infinite limit, $\phi^{t}(B)$ and $A$ are the same set of points, so their intersection is just going to be that set. So I should have said $\lim_{t \rightarrow \infty} \mu(\phi^{t}(B) \cap A) = \mu(A) = \mu(B)$. The rest of the argument follows. Does this work?
Apr
6
comment Having trouble understanding the concept of “mixing” in dynamical systems.
$\lim_{t \rightarrow \infty} \phi^{t}(B)$ is the set of points to which $B$ evolves under the flow in the infinite time limit.
Apr
6
revised Having trouble understanding the concept of “mixing” in dynamical systems.
edited tags
Apr
6
comment Having trouble understanding the concept of “mixing” in dynamical systems.
It just occurred to me that maybe the problem is that for a mixing system my choice of $A$ does not correspond to a measurable set. Is this the case? If so, how do we square that with the fact that the dynamics is measure-preserving?
Apr
6
awarded  Editor
Apr
6
revised Having trouble understanding the concept of “mixing” in dynamical systems.
added 38 characters in body
Apr
6
asked Having trouble understanding the concept of “mixing” in dynamical systems.