Limit point of a trajectory

$\{f^tx\}$ of a dynamical system $f^t$

A point

$$x_\alpha=\lim_{k\to\infty}f^{t_k}x\label{1}\tag{1}$$

(an $\alpha$-limit point) or

$$x_\omega=\lim_{k\to\infty}f^{t_k}x\label{2}\tag{2}$$

(an $\omega$-limit point), where $\{t_k\}_{k\in\mathbf N}$ is a sequence such that $t_k\to-\infty$ as $k\to\infty$ in \eqref{1}, or $t_k\to+\infty$ as $k\to\infty$ in \eqref{2}, and for which the limits in \eqref{1} or \eqref{2} exist.

For a trajectory $\{f^tx\}$ of a dynamical system $f^t$ (or, in other words, for $f(t,x)$, cf. [1]), an $\alpha$-limit point ($\omega$-limit point) is the same as an $\omega$-limit point ($\alpha$-limit point) of the trajectory $\{f^{-t}x\}$ of the dynamical system $f^{-t}$ (the system with reverse time). The set $\Omega_x$ ($A_x$) of all $\omega$-limit points ($\alpha$-limit points) of a trajectory $\{f^tx\}$ is called the $\omega$-limit set ($\alpha$-limit set) of this trajectory (cf. Limit set of a trajectory).

References

[1]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)

Comments

For a dynamical system with discrete time (or, a cascade) similar definitions and the same terminology are used (now in the above the sequences $\{t_k\}$ have to be in $\mathbf Z$).