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If $\gamma(x^0)$ is a periodic orbit then $\omega(x^0)=\alpha(x^0)=\gamma(x^0)$

I am very confused solving this problem, here I put how each of these sets is defined:

$\omega(x^0)=\{y:\text{$y$ is a $\omega$-limit point of } \gamma^{+}(x^0)\}$,

$\alpha(x^0)=\{y:\text{$y$ is a $\alpha$-limit point of } \gamma^{-}(x^0)\}$,

$\gamma(x^0)=\{\varphi(t,x^0):t\in I_{x^0}\}$

$\gamma^{+}(x^0)=\{\varphi(t,x^0):t\geq0\}$

$\gamma^{-}(x^0)=\{\varphi(t,x^0):t\leq0\}$

How can I use the orbit to be periodic to conclude this equality? Thank you very much.

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1 Answer 1

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The periodic orbit $\gamma(x_{0})$ is complete and invariant set (if we talk about nice dynamical systems). Hence, if you start on $\gamma(x_{0})$, you will stay there $\forall t$. Therefore the limit sets of any point $x\in \gamma(x_{0})$ (even $x_{0}$) are exactly $\gamma(x_{0})$. The equality follows.

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