Let $d$ be a positive integer. Let $f:\mathbb{R}^{d+1}\rightarrow \mathbb{R}^d$ and $g:\mathbb{R}^d\rightarrow \mathbb{R}^d$, be once continuously differentiable functions and define the solution map $\Phi(t,x)$ to the (family of) initial value problems $$ \begin{aligned} \partial_t \Phi(t,x) &= f(t,\Phi(t,x))\\ \Phi(0,x)&\triangleq g(x). \end{aligned} $$
Is there a theorem characterizing $f$, such that the set $X$ defined by: $$ X\triangleq \left\{ g(x) : (\exists t \in [0,\infty)) g(x)=\Phi(t,x) (\forall x \in \mathbb{R}^d) \right\}, $$ is dense in $C(\mathbb{R}^d,\mathbb{R}^d)$ and $\Phi$ satisfies $$ \Phi(t,x)\circ \Phi(s,x)= \Phi(t+s,x) , (\forall x \in \mathbb{R}^d). $$
If this is not possible, is there a theorem discussing a "reasonable" Borel-measure $\mu$ on $C(\mathbb{R}^d;\mathbb{R}^d)$ such that $\{f^n\}$ is $\mu$-almost everywhere dense?
Note: $C(\mathbb{R}^d;\mathbb{R}^d)$ is considered with the compact-open topology.