With reference to the problem of heat conduction in one-dimensional systems, I discuss the structure of the invariant measure, when the system is put in contact with two thermal reservoirs at different temperatures. By solving a Hamiltonian & stochastic model, I am able to show that the invariant measure can be effectively described as the product of independent Gaussians centered along suitable
"normal modes", that are (power-law) localized in real space. Moreover, I show that such a representation applies also to a strictly deterministic model: the purely quartic Fermi-Pasta-Ulam chain.