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$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system.

Prove: $w(x_{0})$ is closed.

Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as $$\omega(x_0) = \{y \mid \exists (n_k)\, n_k \to \infty, f^{n_k}(x_0) \to y\}$$

Eric
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  • Do you mean $\omega(x_0) = {y \mid \exists (n_k), n_k \to \infty, f^{n_k}(x_0) \to y}$? – martini Jul 08 '14 at 13:55
  • @martini yes. It is the definition of $w(x_{0})$. – Eric Jul 08 '14 at 14:02
  • If $X$ is a metric space, the title "Holomorphic Dynamics" was out of place. Actually, titles like "Holomorphic Dynamics" are never a good choice: the title should be a lot more specific than the name of some area of mathematics. –  Jul 08 '14 at 19:09
  • @Thisismuchhealthier. You are right, thx very much. – Eric Jul 12 '14 at 12:59

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