In this talk, I will first review some recent results on
self-consistent transfer operators which are nonlinear operators
describing the time evolution of weakly coupled maps in the
thermodynamic limit where the number of coupled maps goes to infinity.
In particular, I will focus on results concerning existence of fixed
states, their stability, and persistence under perturbations.
Then I will show how to use self-consistent operators to describe the
evolution of measures for weakly coupled maps where the number of
coupled maps is very large, but finite, leading to the definition of
self-sustaining measures. These are  "almost" invariant  measures for
the dynamic, and although they might be very different from the
asymptotic equilibrium states of the system, they describe its
statistical behavior for stretches of time that are exponentially large
in the number of coupled maps.