I am studying Homegeneity lemma. I am not understanding the following paragraph:
Given any fixed unit vector $c \in S^n$, consider the differential equations
$\frac{dx_i}{dt} = c f(x_1,x_2,\ldots,x_n)$ for $i=1,2,\ldots,n$, where $f$ is a smooth function from $\mathbb{R}^n$ to $\mathbb{R}$ with $f(x)=0$ for outside the unit sphere and on sphere and $f(x)>0$ for inside the unit interval.
For any $y \in \mathbb{R}^n$ these equations have a unique solution $x = x(t)$, defined all real numbers which satisfies the initial condition $x(0)=y$. We will use the notation $x ( t ) = F_t (y)$ for this solution. Then clearly
- $F_t(y)$ is defined for all $t$ and $y$ and depends smoothly on $t$ and $y$,
- $F_0(y) =y$,
- $F_(t+s)(y) = F_t \circ F_s(y)$.