Non-compact and Non-trivial Minimal Sets of a Locally Compact Flow
Shigeo Kono · Tokyo Journal of Mathematics · 1982
a metric 8pace with metric $d$ .A flow, or a dynamical system, on $X$ is the triplet (X, $R,$ $f$ ), where $f$ is a map of $X\times R$ onto $X$ such that a) $f(x, O)=x$ for every $xeX$, b) $f(f(x, s),$ $t$ ) $=f(x, s+t)$ for every $xeX$ and every $s,$ $teR$,The trajectory of $xeX$ in the flow (X, $R,$ $f$ ) is defined to be the set invariant, and no proper subsets of $M$ have these properties.$L^{+}(x)$ denotes the set { $yeX$; there exists a sequence $\{t_{n}\}$ in $R$ with $ t_{n}\rightarrow+\infty$ and $f(x, t_{n})\rightarrow y$ }.$L^{-}(x)$ denotes the set { $y\in X$ ; there exists a 8equence $\{t_{n}\}$ in $R$ with $ t_{n}\rightarrow-\infty$ and $f(x, t.)\rightarrow y$ }.$L^{+}(x)(L^{-}(x))$ is called the positive (negative) limit set of $x$ .A point $x\in X$ or the trajectory $C(x)$ is called positively (negatively)receding, if $L^{+}(x)(L^{-}(x))$ is empty; receding, if $x$ is receding both positively and negatively; positively (negatively) asymptotic, ifPoisson stable, if $x$ is both positively and negatively Poisson stable.$J^{+}(x)$ denotes the set { $y\in X$ ; there exist a sequence $\{x_{n}\}$ in $X$ and a sequence $\{t_{n}\}$ in $R$ such that $x_{n}\rightarrow x,$ $ t_{n}\rightarrow+\infty$ , and $f(x_{n}, t_{n})\rightarrow y$ }.$J^{-}(x)$ denotes the set { $y\in X$ ; there exist a sequence $\{x_{n}\}$ in $X$ and a sequence $\{t_{n}\}$ in $R$ such that $x_{n}\rightarrow x,$ $ t_{n}\rightarrow-\infty$ , and $f(x_{n}, t_{n})\rightarrow y$ }.$J^{+}(x)(J^{-}(x))$ is called the first positive (negative) prolongational limit set of $x$ .A point $x\in X$ is called non-wandering, if $x\in J^{+}(x)$ or $xeJ^{-}(x)$ .($x\in J^{+}(x)$ and $x\in J^{-}(x)$ are equivalent [1, p. 35, Theorem 2.12]).