# Periodic point

A point on a trajectory of a periodic motion of a dynamical system $f ^ { t }$( $t \in \mathbf R$ or $t \in \mathbf Z$) defined on a space $S$, i.e. a point $x \in S$ such that there is a number $T > 0$ for which $f ^ { T } x = x$ but $f ^ { t } x \neq x$ for $t \in ( 0, T)$. This number $T$ is called the period of the point $x$( sometimes, the name period is also given to all integer multiples of $T$).
The trajectory of a periodic point is called a closed trajectory or a loop. When the latter terms are used, one frequently abandons a concrete parametrization of the set of points on the trajectory with parameter $t$ and considers some class of equivalent parametrizations: If $f ^ { t }$ is a continuous action of the group $\mathbf R$ on a topological space $S$, a loop is considered as a circle that is topologically imbedded in $S$; if $f ^ { t }$ is a differentiable action of the group $\mathbf R$ on a differentiable manifold $S$, a loop is considered as a circle that is smoothly imbedded in $S$.
If $x$ is a periodic point (and $S$ is a metric space), then the $\alpha$- limit set $A _ {x}$ and the $\omega$- limit set $\Omega _ {x}$( cf. Limit set of a trajectory) coincide with its trajectory (as point sets). This property, to a certain extent, distinguishes a periodic point among all points that are not fixed, i.e. if the space in which the dynamical system $f ^ { t }$ is given is a complete metric space and if a point $x$ is such that $\Omega _ {x} = \{ f ^ { t } x \} _ {t \in \mathbf R }$, then $x$ is a fixed or a periodic point of $f ^ { t }$.