Question about topological conjugation

Question

Are topological conjugation the dynamical systems (in discrete time) :

$f,g:\mathbb{R}\rightarrow\mathbb{R}$ with $f(x)=\dfrac{1}{2}x$ and $g(x)=-\dfrac{1}{4}x$.

Answer 1

No, they are not topologically conjugate. Here's why.

Suppose there exists a homeomorphism $h : \mathbb{R} \to \mathbb{R}$ that is a topological conjugacy from $f$ to $g$, so $hfh^{-1}=g$.

Each of $f,g$ has a unique fixed point at $x=0$, and so $h(0)=0$.

Consider any $x \ne 0$, and so $y=h(x) \ne 0$. The point $0$ is not between the points $x$ and $f(x)=\frac{1}{2}x$. It follows that the point $h(0)=0$ is not between the points $y=h(x)$ and $g(y)=g(h(x))=h(f(x))$. But $0$ is between $y$ and $g(y)=-\frac{1}{4}y$.