# Wandering set

The set $W$ of all wandering points (cf. Wandering point) of some dynamical system $f(p,t)$. Since for each point $q$ the set $W$ contains all points of the neighbourhood $U(q)$, it is open in the phase space $R$. Accordingly, the set $M=R\setminus W$ of all non-wandering points is closed. The sets $W$ and $M$ are invariant, i.e. with each of their points $q$ they contain the point $f(q,t)$ for an arbitrary $t$. In a compact space $R$ each wandering point $f(q,t)$ tends to $M$ both when $t\to\infty$ and when $t\to-\infty$.

#### References

 [1] G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927) [2] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) [3] K.S. Sibirskii, "Introduction to topological dynamics" , Noordhoff (1975) (Translated from Russian)