Consider $[0,1]$ with the usual euclidean topology. Now $G$ be the set of homeomorphisms from $[0,1]$ onto $[0,1]$. $G$ forms a group under composition. Now, Let $F=\{ f\in G | f(0)=0 \}\ $. Now $F$ is a normal subgroup of $G$.
The problem is the following. Suppose $g\in F$ has exactly one fixed interior point. I need a homeomorphism $h \in F$ which is conjugate to $g$ in $F$ such that either of the following happens:
$$ h(x)>g(x) \hspace{4 cm} \forall x\in(0,1)$$ $$ h(x)>g^{-1}(x) \hspace{3.5 cm} \forall x\in (0,1)$$
I have no idea how to come up with such a homeomorphism satifying either of the two properties and also being conjugate to $g$ in $F$. I suppose that $g$ having exactly one fixed interior point is crucial but I am not sure how to use it!
Thanks in advance for any kind of help!