# Why exists a periodic point?

by HeMan   Last Updated August 13, 2018 03:20 AM

In Brin's book "Introduction to dynamical systems", page 10, he defines a cuadratic function $q_\mu$ for $\mu>4$. He observes that $[\frac{1}{\mu}, \frac{1}{2}] \subset q_{\mu}^2([\frac{1}{\mu}, \frac{1}{2}])$ and then he says that by the intermediate value theorem, exists a fixed point of $q_{\mu}^2$, but I don't have any idea about why this is true. Any hint would be appreciated.

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$q_\mu^2$ attains maximum at some point of $x_M\in[1/\mu,1/2]$ and minimun at some point $x_m\in[1/\mu,1/2]$.
Since $[1/\mu,1/2]\subset q_\mu^2([1/\mu,1/2])$, then $q_\mu^2(x_M)-x_M\geq1/2-x_M\geq0$ and $q_\mu^2(x_m)-x_m\leq1/mu-x_m\leq0$.
Therefore $q_mu^2(x)-x$ vanishes at some point of $[1/\mu,1/2]$.