In this article we prove that the fibration of $L^2(S^1)$ by potentials which are isospectral for the 1-dimensional periodic Schrödinger equation, is trivial. This result can be applied, in particular, to $N$-gap solutions of the Korteweg-de Vries equation (KdV) on the circle: one shows that KdV, a completely integrable Hamiltonian system, has global action-angle variables.

Let $T_a$ be the tent map with slope a. Let c be its turning point, and ${\mu }_a$ the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function g, it is shown that for almost every a, $ʃgd{\mu }_a=li{m}_{n\to \infty }\frac{1}{n}{\sum }_{i=0}^{n-1}g\left(T_a^i\left(c\right)\right)$. As a corollary, we deduce that the critical point of a quadratic map is generically not typical for its absolutely continuous invariant probability measure, if it exists.