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In the book of An introduction to chaotic dynamical systems (2nd edition) by Devaney, there is a example that says;

There is a function such that $f^3[3,4]\subset [1,5]$ so that $f^3$ has at least one fixed point in $[3,4]$. Then he claims that the point is unique, and therefore must be the fixed point for $f$, not the period 3 point. (then showing it is unique).

I'm confused in the unique part, what do we know about unique fixed points? that i'm missing, thanks.

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A point $p$ is a periodic point of period 3 for map $f$ iff $f^3(p) = p$ and all points $p$, $f(p)$ and $f^2(p)$ are different points. Actually, $f^2(p)$ and $f^3(p)$ are also points of period 3. If map $f$ has a point of period 3, then $f^3$ has three different fixed points: $p$, $f(p)$ and $f^2(p)$. So...

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