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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

  1. $F_t(y)$ is defined for all $t$ and $y$ and depends smoothly on $t$ and $y$,
  2. $F_0(y) =y$,
  3. $F_(t+s)(y) = F_t \circ F_s(y)$.
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