Find the set of convergence of iterative sequence

Question

Qustion

Suppose $f(x)=\frac{1}{4}+x-x^2$.

1. Show that the iterative sequence $0,f(0),f^2(0),\cdots$ converges to some $L$.
2. Find all $x\in \mathbb R$ such that $x,f(x),f^2(x),\cdots$ also converges to $L$.

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I know that $L=1/2$. Also, since $|f'(x)|=|1-2x|<1\iff 0<x<1$, $x,f(x),f^2(x),\cdots$ converges to $1/2$ if $0<x<1$. But how do I prove that $0\leq x\leq 1$ are all $x$ which converge to $1/2$?

Answer 1

Cobweb diagrams could be useful to answer such a question:

(Here is the link to the interactive graph: https://www.desmos.com/calculator/np8skcskmz)

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Here is a sketch of how to make the heuristics coming from the cobweb diagram more rigorous:

• First compute the fixed points of $f$ to be $-\dfrac{1}{2}$ and $\dfrac{1}{2}$.
• $\dfrac{1}{2}$ is also a critical point of $f$, and it's a global maximum.
• $f$ is 2-to-1 except at its global maximum, so there is a unique number $\alpha$ different than $-\dfrac{1}{2}$ such that $f(\alpha)=-\dfrac{1}{2}$. It's easily computed that $\alpha = \dfrac{3}{2}$.
• The cobweb diagram suggests to partition the real line as follows: put $I_0 = ]-\infty,-1/2[$, $I_1 = ]-1/2,1/2[$, $I_2 = ]1/2,3/2[$, $I_3 = ]3/2,\infty[$. Then

$$\mathbb{R} = I_0\,\uplus\,\{-1/2\}\,\uplus\,I_1\,\uplus\,\{1/2\}\,\uplus\,I_2\,\uplus\,\{3/2\}\,\uplus\,I_3.$$

• It suffices to then verify the limits of orbits, depending on the cell the initial condition is taken from. If $x\in I_0$, $f(x)<x$, so $\lim_{n\to\infty}f^n(x)=-\infty$. If $x\in I_1$, $x<f(x)<1/2$, and the limit ought to be a fixed point, so $\lim_{n\to\infty} f^n(x) = 1/2$. If $x\in I_2$, then $f(x)\in I_1$ and we are in the previous case. If $x\in I_3$, then $f(x)\in I_0$ and we are in the first case.