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If $y_0\in\langle x_0\rangle$, from some orbit $\langle x_0\rangle$, then the eventual behavior of $\langle y_0\rangle$ is the same as the eventual behavior of $\langle x_0\rangle$. In fact, if $y_0=x_k$ for some $k\in\mathbb{N}$, then $\langle y_0\rangle = \langle x_k\rangle$. Explain why this is so.

I understand this intuitively where $\langle x_0\rangle = \langle x_0,x_1,x_2,...,y_0,y_1,y_2,...\rangle$ and so $\langle y_0\rangle$ follows the behavior of $\langle x_0\rangle$.

But is there a more technical way to explain this?

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Although you don't define "eventual behavior", this usually refers to the asymptotic behavior or long-time behavior of the orbit. This does not depend on any finitely many initial terms (in the same way as a limit of a sequence does not depend on any finitely many initial terms of the sequence).

For example, if the orbit of $x_0$ converges to a point $p$, then any point on its orbit has an orbit that converges to the same point $p$, simply because (as you write) the orbit of that new point is a subset of the orbit of $x_0$ (notice that it can even coincide with the original orbit, when $x_0$ is a fixed or periodic point).

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