Determine whether mappings are topologically conjugate

Question

Let $f$ and $g: \mathbb{R} \to \mathbb{R}$ be given by $f(x)= \frac{1}{2}x$ and $g(x)= 3x$. I want to determine if they are topologically conjugate, i.e., that there exists a homeomorphism such that $\phi(f(x))=g(\phi(x))$. A necessary condition for conjugacy is that they have the same number of fixed points, which turns out to be just $x=0$ and implies that $\phi(0)=0$. But how can I find a sufficient condition? I've tries $\phi(x)=x^a$ and similar but I'm not sure if I actually need to find $\phi(x)$ to prove conjugacy.

Answer 1

The maps are not topologically conjugate. If they were, both fixed points would have the same type of stability, but one is stable and the other is unstable.