Tarun
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 Apr7 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! Apr6 awarded Student Apr6 comment Having trouble understanding the concept of “mixing” in dynamical systems. I edited my question to match the above comment. Apr6 revised Having trouble understanding the concept of “mixing” in dynamical systems. deleted 34 characters in body Apr6 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? Apr6 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. Apr6 revised Having trouble understanding the concept of “mixing” in dynamical systems. edited tags Apr6 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? Apr6 awarded Editor Apr6 revised Having trouble understanding the concept of “mixing” in dynamical systems. added 38 characters in body Apr6 asked Having trouble understanding the concept of “mixing” in dynamical systems.