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{\bf Question}

Let $f:\ \bf{R} \longrightarrow \bf{R}$ and $g:\ \bf{R}
\longrightarrow \bf{R}$ be two diffeomorphisms, each having the
origin as an attracting fixed point (no flips) with basin of
attraction the whole of $\bf{R}$.  Choose some $p>0$ and let
$I=[f(p),p],\ J=[g(p),p]$. Construct $h:\ I \longrightarrow J$ of
the form $h(x)=ax+b$ so that $h(p)=p$ and $h(f(p))=g(p)$.  Use
this to construct a conjugacy between $f,\ g$ on $\bf{R}$.

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{\bf Answer}

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$\underline{\rm{Note}}$: This requires $f$, $g$ to be invertible.
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Consider the intervals $f(I)=[f^2(p),f(p)]$ and
$g(J)=[g^2(p),g(p)]$.

Define $h_1:\ f(I) \longrightarrow g(J)$ by $h_1(x)=g \cdot h
\cdot f^{-1}(x)$ (if a conjugacy exists then on $f(I)$ it
\underline{has} to be this.) . Then if $x=f(u)$ (say) where $u \in
I$ then $h_1f(u)=gh(u)$, i.e. $h_1 \circ f=g \circ h:\ I
\longrightarrow g(J)$.

Next define $h_2:\ f^2(I) \longrightarrow g^2(J)$ by $h_2(x)=g
\cdot h_1 \cdot f^{-1}(x)$: this gives $h_2 \circ f=h \circ h_1:\
f(I) \longrightarrow g^2(J)$. Continue indefinitely, defining
$h_n:f^n(I) \longrightarrow g^n(J)$ by $h_n(x)=g \cdot
h_{n-1}f^{-1}(x)$. Likewise define $h_{-1}:f^{-1}(I)
\longrightarrow g^n(J)$ by $h_{-1}(x)=g^{-1}hf(x)$, so
$gh_{-1}=hf:f^{-1}(I) \longrightarrow J_1$ and inductively define
$h_{-n}:f^{-n}(I) \longrightarrow g^{-n}(J)$ by $h_{-n}(x):g^{-1}
\cdot h_{-n+1}f^{-1}(x)$ ($n=1,2,3, \cdots$). Then (writing
$h=h_0$) the family of maps $\{h_m\}_{m \in \bf{Z}}$ defines a
continuous (both ways) bijection $\bf{R}^+ \longrightarrow
\bf{R}^+$ conjugating $f,\ g$. Do likewise for $\bf{R}^-$.
Finally, map 0 to 0.
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