Non-wandering sets of the powers of maps of a star
Song Shao Xiangdong Ye
Let $T$ be a star and $\Omega(f)$ be the set of non-wandering points of a continuous map $f:T\rightarrow T$. For two distinct prime numbers $p$ and $q$, we prove: (1) $\Omega(f^p)\cup \Omega(f^q)=\Omega(f)$ for each $f \in C(T,T)$ if and only if $pq > End(T)$, (2) $\Omega(f^p)\cap \Omega(f^q)=\Omega(f^{p q})$ for each $f\in C(T,T)$ if and only if $p+q \ge End(T)$, where $End(T)$ is the number of the ends of $T$. Using (1)-(2) and the results in [3], we obtain a complete description of non-wandering sets of the powers of maps of 3-star and 4-star.
keywords: Non-wandering set star. tree
The return times set and mixing for measure preserving transformations
Rui Kuang Xiangdong Ye
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$^+$.
keywords: density intermixing ergodicity mild mixing weak mixing strong mixing set of return times.
Recurrence properties and disjointness on the induced spaces
Jie Li Kesong Yan Xiangdong Ye
A topological dynamical system induces two natural systems, one is on the hyperspace and the other one is on the probability measures space. The connection among some dynamical properties on the original space and on the induced spaces are investigated. Particularly, a minimal weakly mixing system which induces a $P$-system on the probability measures space is constructed and some disjointness result is obtained.
keywords: weakly mixing minimal Induced spaces disjointness.

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