Conditions necessary for a function to contain multiple attractive fixed points Given a function $f(x)$, what are the necessary conditions for $f(x)$ to contain multiple attractive fixed points?
Another key condition is I want the construction of the fixed point $x^*$ to be repeated application of $f$: $\lim\limits_{x \rightarrow n} f^n(x) = x^*$
One approach I looked at was forcing $f$ to be a contraction mapping therefore having a Lipshitz constant be less than one. This will force the function to contain a fixed point (via Banach's fixed point theorem), but I don't think a contraction map is capable of having multiple attractive fixed points. 
Any ideas on how to approach this construction?

All attracting fixed points must be fixed points first of all. So, the function $f(x)$ must have multiple fixed points, or solutions for $f(x^*)=x^*$. If function $f(x)$ is differentiable in open neighbourhoods of those fixed points $\{x^*\}$ and $|f'(x^*)|<1$, then those points are also attracting. Let's use these statements to construct a function $f(x)$ with multiple contracting fixed points. 
Consider $\left\{\frac{1}{4},\frac{2}{4},\frac{3}{4} \right\}\subset (0,1)$, the polynomial 
$$P_{3}(x)=\left(x-\frac{1}{4}\right)\left(x-\frac{2}{4}\right)\left(x-\frac{3}{4}\right)$$
and 
$$f(x)=x-P_{3}(x)$$
We have
$$f\left(\frac{1}{4}\right)=\frac{1}{4},f\left(\frac{2}{4}\right)=\frac{2}{4},f\left(\frac{3}{4}\right)=\frac{3}{4}$$
$$f'(x)=1-\left(x-\frac{2}{4}\right)\left(x-\frac{3}{4}\right)-\left(x-\frac{1}{4}\right)\left(x-\frac{3}{4}\right)-\left(x-\frac{1}{4}\right)\left(x-\frac{2}{4}\right)=\\
-3x^2+3x+\frac{5}{16}$$
And
$$\left|f'\left(\frac{1}{4}\right)\right|=\frac{7}{8}<1$$
$$\left|f'\left(\frac{2}{4}\right)\right|=\frac{17}{16}>1$$
$$\left|f'\left(\frac{3}{4}\right)\right|=\frac{7}{8}<1$$
So $f(x)$ has 2 attracting fixed points.
Now, try with $\left\{\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6} \right\}\subset (0,1)$ ... and continue with $\left\{\frac{1}{2n},\frac{2}{2n},...,\frac{2n-1}{2n} \right\}\subset (0,1)$ to obtain a function with $n$ attracting fixed points.
