Wandering point

A point $q$ in the phase space $R$ of a dynamical system $f(p,t)$ with a neighbourhood $U(q)$ for which there exists a moment in time $T$ such that $f(U(q),t)$ has no common points with $U(q)$ for all $t\geq T$ (all points of $U(q)$, from some moment on, leave the neighbourhood $U(q)$). A point $q$ without such a neighbourhood is said to be non-wandering. This property of a point — to be wandering or non-wandering — is two-sided: If $f(U(q),t)$ has no common points with $U(q)$, then $U(q)$ has no common points with $f(U(q),-t)$. A wandering point may become non-wandering if the space $R$ is extended. For instance, if $R$ is a circle with one rest point $r$, all points of $R\setminus r$ are wandering points. They become non-wandering if the points of some spiral without rest points, winding itself around this circle from the outside or from the inside, are added to $R$.

Comments

A set $A\subset R$ is positively recursive with respect to a set $B\subset R$ if for all $T$ there is a $t>T$ such that $f(B,t)\cap A\neq\emptyset$. Negatively recursive is defined analogously. A point $x$ is then non-wandering if every neighbourhood of it is positively recursive with respect to itself (self-positively recursive). A point $x$ is positively Poisson stable (negatively Poisson stable) if every neighbourhood of it is positively recursive (negatively recursive) with respect to $\{x\}$. A point is Poisson stable if it is both positively and negatively Poisson stable. If $P\subset R$ is such that every $x\in P$ is positively or negatively Poisson stable, then all points of $\bar P$ are non-wandering. See also Wandering set.

References

[a1]	N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) pp. 30–36