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）

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.

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 a 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.

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）

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 a 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 $E$, where $c_n$ is regularly varying with an index between -1 and 0.

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 a 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.