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Prove that $f(x)=x(1-x)$ on $I$ is conjugate to $g(x)=x^2-\frac{3}{4}$ on a certain interval in $\mathbb{R}$. Determine this interval.

Suppose $I$ and $J$ are intervals and function $f$ from $I$ to $I$ and $g$ from $J$ to $J$ we say that $f$ and $g$ are conjugate if there is a homeomorphism $h$ from $I$ to $J$ such that $h$ satisfies the conjugacy equation $h\circ f = g\circ h$.

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Can you give more context as to where this question came from? – Jonah Sinick Oct 29 '12 at 6:59

$f$ has a unique fixed point at $x=0$ with derivative $1$, $g$ has fixed points at $-1/2$ and $3/2$ with derivatives of $-1$ and $3$, resp. The local dynamics near these fixed points are not conjugate, so any conjugacy interval would not contain a neighborhood of any of these fixed points, or the critical point for that matter (because the orbit of the critical point for $g$ approaches $-1/2$ from both sides).

In other words, any conjugacy between these two would not capture any of the interesting dynamics. However, there are certainly intervals on which these maps are conjugate, e.g., intervals of the form $[a,\infty)$ with large $a$.

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