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The return times set and mixing for measure preserving transformations

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  • In this paper the relationship between the return times set andseveral mixing properties in measure-theoretical dynamical systems(MDS) is investigated. For an MDS $T$ on a Lebesgue space$(X,$ß,$\mu)$, let ß$^+=\{B\in$ ß$:\mu(B)>0\}$ and$N(A,B)=\{n\in Z_+: \mu(A\cap T^{-n}B)>0\}$ for $A, B\in$ß$^+$. It turns out that $T$ is ergodic iff$N(A,B)$≠$\emptyset$ iff $N(A,B)$ is syndetic; $T$ is weaklymixing iff the lower Banach density of $N(A,B)$ is $1$ iff $N(A,B)$is thick; and $T$ is mildly mixing iff $N(A,B)$ is an $ IP^ * $-set iff$N(A,B)$ is an $(IP-IP)^*$-set for all $A,B\in$ ß$^+$ ifffor each $IP$-set $F$ and $A\in$ß$^+$, $\mu(\bigcup_{n\in{F}}T^{-n}A)=1$. Finally, it is shown that $T$ is intermixing iff$N(A,B)$ is cofinite for all $A,B\in$ß$^+$.
    Mathematics Subject Classification: 37A25,37A05; Secondary: 54H20.


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