given dynamic system $(X, \mathcal{B}, F, \mu), \mu \circ F^{-1}=\mu, F $ is mixing, $ A \in \mathcal{B}, s.t. \mu(A) >0 $.

consider dynamic system $(X\times X, \mathcal{B}\otimes \mathcal{B}, F\times F, \mu \times \mu)$, then $ F \times F $ is mixing too, hence ergodic, $(\mu \times \mu )(A \times A )>0$.

fixed delay time $n_0 \in \mathbb{N}$, define $R(x)=\min \{n: F^n(x) \in A \} $, and stopping times recursively: for all $ (x, x') \in X \times X$,

$ \tau_1(x,x')= n_0 +R(F^{n_0}x) $

$ \tau_2(x,x')= \tau_1+n_0 +R(F^{n_0+\tau_1}x') $

$ \tau_3(x,x')= \tau_2+n_0 +R(F^{n_0+\tau_2}x) $

$ \tau_4(x,x')= \tau_3+n_0 +R(F^{n_0+\tau_3}x') $ ect.

then we know $ F^{\tau_1} \in A \times X, F^{\tau_2} \in X \times A$ ect.

can we prove:

for almost $ (x, x') $ wrt $ \mu\times \mu$, $ \exists \tau_n=\tau_n(x,x'), s.t. F^{\tau_n} (x,x') \in A \times A$? Thanks!