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I would like to know if there is a notion or an example related to the following situation: a transform $T$ on a space $E$, which is equipped with an infinite measure $\mu$, satisfies $\mu(A\cap T^{-n}B)\sim \mu(A)\mu(B)c_n$ as $n$ tends to infinity, for any measurable subets $A$ and $B$ in some finite-measure subset of $E$, where $c_n$ is regularly varying with an index between -1 and 0.

This may be related to the notion pointwise dual ergodicity in infinite ergodic theory （Aaronson 1997）.

Sorry for not being more precise since I'm very ignorant in this area.

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    $\begingroup$ Probably not very interesting as this would say that if $A$ and $B$ don't intersect initially, they never will. (The identity transformation has this property, but not much else). $\endgroup$
    – Anthony Quas
    Commented Aug 20, 2015 at 20:00
  • $\begingroup$ Ah sorry. There was a typo in my previous question. It is $\mu(A)\mu(B)$, not $\mu(A\cap B)$. $\endgroup$
    – Uchiha
    Commented Aug 20, 2015 at 21:57
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    $\begingroup$ Then taking $A=B=E$ would make this not make sense. $\endgroup$
    – Anthony Quas
    Commented Aug 20, 2015 at 23:10
  • $\begingroup$ Probably I should not say for any A and B, but any A and B in some finite measure set. $\endgroup$
    – Uchiha
    Commented Aug 21, 2015 at 6:56
  • $\begingroup$ @Ray, when most people say ergodic theory, they mean dynamics on finite measure space, hence the measure limitation is not very interesting either. Moreover, try to mimic the proof of ergodicity out of mixing, and it fails in your definition. Perhaps you should tell us what notation exactly do you have in mind, and do you know a reasonable interesting system which have this property (most favorably, homogeneous one, on which such a statement can be analyzed in algebraic tools). $\endgroup$
    – Asaf
    Commented Aug 21, 2015 at 9:46

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A theorem of Hajian, Ito and Kakutani shows that every infinite measure preserving transformation always has weakly wandering sets. That is a set $A$ and a subsequence $n_k\to\infty$ such that $\left\{ T^{-n_k}A\right\}_{k=1}^\infty$ are pairwise disjoint. See Aaronsons book or the new book of Eigen, Hajian, Ito and Prasad. If you are willing to change the mixing condition of sets for decay of correlations of "nice" functions in smooth dynamical systems then the paper of Melbourne and Terhesiu http://mat.univie.ac.at/~terhesiu/papers/invent_rev.pdf is a formulation similar to what you want.

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  • $\begingroup$ Thanks a lot. The paper looks relevant. I guess the existence of weakly wandering set does not exclude the property I want if the set $E$ in my question avoids the weakly wandering set? $\endgroup$
    – Uchiha
    Commented Aug 24, 2015 at 22:17
  • $\begingroup$ The existence of one weakly wandering set doesn't. However the theorem of Hajian and Kakutani actually says that the space is a union of weakly wandering sets (modulo measure zero subsets). $\endgroup$
    – user78465
    Commented Aug 25, 2015 at 8:52

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