$𝛼$ - and $𝜔$ -limit sets

The $𝜔$ -limit set $𝐿 𝜔 (𝑥 0)$ ( $𝛼$ -limit set $𝐿 𝛼 (𝑥 0)$ ) of $𝑥 0 \in 𝑀$ is the set of all $𝑥 \in 𝑀$ for which there exists a strictly increasing (decreasing), unbounded sequence of times $(𝑡 𝑛) \infty 𝑛 = 1$ such that $l i m 𝑛 \to \infty Φ 𝑡 𝑛 𝑥 0 = 𝑥$ .¹ #m/def/dynamics