Fix an algebraic integer $x\neq 0$. Does there exist a closed smooth manifold $M$ with a class $\rho\in H^{1}\_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^\*\rho=x\rho$?



Smooth covers pulling back a cohomology class to any algebraic multiple
Every nonzero algebraic integer $x$ in $\mathbb R$ is an eigenvalue of an $\mathbb R$-diagonalizable integer matrix $A \in M\_n(\mathbb Z)$ with $\det(A) \neq 0$ for some $n$. So take the map of tori $A^t: \mathbb R^n /\mathbb Z^n \to \mathbb R^n/\mathbb Z^n$. This acts by $A^t$ on $\mathbb Z^n = H\_1(\mathbb R^n /\mathbb Z^n)$ and hence by $A$ on $H^1(\mathbb R^n /\mathbb Z^n)$.


