So far, I have to read about quadratic map in "Introduction to Dynamical Systems" by Brin and Stuck.
The quadratic map is $$q_{\mu}(x) = \mu x(1-x), \quad \mu > 0.$$
Now, I need to show when $\mu > 1$ and $x \notin [0,1]$, the limit of iterations $q^n_{\mu}(x)$ tends to $-\infty$.
I read several topics on Stack discussing about quadratic map, but none of them discussed this one. The only thing I know is I can write quadratic map as $$q^{n+1}_{\mu}(x) = \mu q^n_{\mu}(x)(1 - q^n_{\mu}(x)).$$
But, how can I continue? Please give me some advice. Thanks so much.
Okay, thanks to the hint from Will Jagy, I got my answers.
Fix $\mu > 0$, let $q(x) = \mu x(1-x)$ and let $x < 0$. We shall show by induction $$q^n(x) < \mu^n x, \text{ for all } n \in \mathbb{N}.$$
For the case $x>1$, we do the same since $(1-x)<0$.