J. Xiong
Shipai, 510631 Guangzhou, China
Department of Mathematics,
South China Normal University
Publications:
Wang H., Xiong J.
Sweep out and chaos
2005, vol. 10, no. 1, pp. 113118
Abstract
Let $X$ be a compact metric space and let $\mathscr{B}$ be a $\sigma$algebra of all Borel subsets of $X$. Let $m$ be a probability outer measure on $X$ with the properties that each nonempty open set has nonzero $m$measure and every open set is $m$measurable. And for every subset $Y$ of $X$ there is a Borel set $B$ of $X$ such that $Y \subset B$ and $m(Y)=m(B)$. We prove that $f : (X, \mathscr{B}, m) \to (X, \mathscr{B}, m)$ sweeps out if and only if for any increasing sequence $J$ of positive integers, there is a finitely chaotic set $C$ for $f$ with respect to $J$ such that $m(C)=1$.
