Geometric description of all orbits of the action $a ^ {(x,y)} = (ax,y)$

Question

What are the orbits of the following group action of $(\mathbb{R}^+,\times)$ on the plane $\mathbb{R^2}$ where $a ^ {(x,y)} = (ax,y)$.

I have found

• $Orb(1,0) = (r,0)$ for $r \geq 0,$
• $Orb(0,a) = (0,a)$ for $a \in \mathbb{R},$
• $Orb(-1,0) = (-r,0)$ for $r \geq 0,$

These three orbits indeed cover the entire plane $\mathbb{R}^2$ but are supposed to be disjoint while first orbit and last one are not $(0,0)$ appears in both.

What am I doing wrong?

Answer 1

$(0,0)$ is a fixed point so it is not in any orbit.

The orbits you describe does not fully cover $\mathbb R^2$, for example $(2,2)$ is not in any of your orbits. Maybe a picture is helpful.

A total description of the orbits is $O_{+,y} = \{ (a,y) : a > 0 \}$, $O_{-,y} = \{ (a,y) : a < 0 \}$ and $ O_{0,y} = \{(0,y)\}$. The orbits $O_{+,y}$ $O_{-,y}$ are (open) horizontal half-lines in opposite directions and the orbits $O_0$ represent the point between these two half-lines.