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### Open Access Journals

DCDS

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.

DCDS

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.

DCDS

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.

## Year of publication

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