Today I revisited the concept of a group action with someone. I recalled the definition of a "flow" which is a group action of the additive group of real numbers on the set $X:$
$$\varphi: X \times \Bbb R \to X$$
such that:
$\varphi(x,0)=x$
$\varphi(\varphi(x,t)),s)=\varphi(x,s+t).$
I wondered about changing $\Bbb R$ to the multiplicative group of real numbers like so:
$$\psi: X \times \Bbb R^*_+ \to X$$
satisfying the conditions:
$\psi(x,1)=x$
$\psi(\psi(x,e^t)),e^s)=\psi(x,e^{s+t}).$
Does the $\psi$ group action have a name?
Since $(\Bbb R,+)$ is isomorphic to $(\Bbb R_{\gt0},\times)$ by exponentiation, I think these two group actions must be nearly the same.