# Wandering set

From Encyclopedia of Mathematics

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) |

#### Comments

#### References

[a1] | S. Smale, "Differentiable dynamical systems" Bull. Amer. Math. Soc. , 73 (1967) pp. 747–817 |

[a2] | N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) pp. 30–36 |

**How to Cite This Entry:**

Wandering set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Wandering_set&oldid=31719

This article was adapted from an original article by K.S. Sibirskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article