Is there a closed smooth manifold $M$ such that for each real $x\neq 0$ there is a nowhere vanishing vector field $v$ on $M$ and a diffeomorphism $\phi:M\to M$ such that $\phi\_\*v=xv$?



Diffeomorphisms pushing forward vector field to any non-zero multiple
Such a manifold exists. First let's construct a non-compact example.


Take $PSL(2,\mathbb R)$ and take two $1$-parameter subgroups, given by $$\begin{pmatrix}
e^{t} & 0 \\
0 & e^{-t} 
\end{pmatrix}, \begin{pmatrix}
1 & t \\
0 & 1 
\end{pmatrix}$$
Consider actions on $PSL(2,\mathbb R)$ of these two groups by multiplication on the left. Then $v$ is the vector field tangent to the second flow, while the first flow will give you a $1$-parameter family of diffeos that will dilate $v$ by any positive constant.


Now, to get the compact example, quotient $PSL(2,\mathbb R)$ from the right by a cocompact action of the fundamental group $\Gamma$ of a compact hyperbolic surface.


It remains to understand how to reverse the sign of $v$. For this recall, that we can identify $PSL(2,\mathbb R)/\Gamma$ with the unit tangent bundle of a hyperbolic surface (whose $\pi\_1$ is equal to $\Gamma$). Now, the flow given by $v$ is the horocyclic flow. In order to reverse it, we can take a hyperbolic surface that admits and orientation reversing isometric involution. Such an involution clearly lifts to the unit tangent bundle and it sends the horocyclic flow to its inverse.


