Timeline for Silly question about mixing
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 22, 2013 at 19:06 | comment | added | Noah Stein | You're welcome. I've been away from Ergodic Theory for a while and it is always nice to get to return occasionally. | |
| Mar 22, 2013 at 17:01 | vote | accept | Etienne | ||
| Mar 22, 2013 at 15:58 | comment | added | Etienne | Many thanks for this answer and the reference! | |
| Mar 22, 2013 at 15:22 | comment | added | Ian Morris | Here is a correct modification of Lemma 2.4: if $\mu(T^{-n}A \cap B)$ is eventually nonzero whenever $\mu(A)$ and $\mu(B)$ are both nonzero then $T$ is light mixing. Proof: suppose that $T$ is not light mixing. Choose $A,B$ with $\mu(A),\mu(B)>0$ and $\liminf_{n \to \infty} \mu(T^{-n}\cap B)=0$. Choose a strictly increasing sequence $(n_k)$ such that $\mu(T^{-n_k}A \cap B)<3^{-k}\mu(B)$ for all $k \geq 1$. Let $C:=B \setminus \bigcup_{k=1}^\infty \left(T^{-n_k}A \cap B\right)$. Then $\mu(C)>\frac{1}{2}\mu(B)>0$ and $\mu(T^{-n_k}A \cap C)=0$ for all $k$, a contradiction. | |
| Mar 22, 2013 at 15:13 | comment | added | Ian Morris | You're right: if $T$ is not lightly mixing then there is no reason why we should be able to find $E$ such that $\liminf_{n \to \infty}\mu(T^{-n}E\cap E)=0$. I think that the authors err when they state that in order to check light mixing it is sufficient to check the case $A=B$: this is fine for weak, strong and probably mild mixing because in those cases the relevant expressions are linear in $\chi_A$ and $\chi_B$, but lim inf is of course not linear. | |
| Mar 22, 2013 at 15:05 | comment | added | Noah Stein | Actually now I am suspicious of Lemma $2.4$. In particular, the identity transformation satisfies the condition which is listed there as equivalent to lightly mixing. But the identity is definitely not lightly mixing. I haven't looked to see how this is involved with the details of the paper, but I'll see if I can find another source for a result like this. | |
| Mar 22, 2013 at 14:17 | comment | added | Ian Morris | What a fascinating paper! If I'm not wrong, when $T$ is invertible, Lemma 2.4 in that article shows that Étienne's condition (with positive-measure rather than nonempty intersections) is precisely light mixing. | |
| Mar 22, 2013 at 13:00 | history | answered | Noah Stein | CC BY-SA 3.0 |