  Chaotic dynamics is always characterized by swarms of unstable trajectories,
unpredictable individually, and thus generally studied statistically. It is
often the case that such phase-space densities relax exponentially fast to a
limiting distribution, that rules the long-time average of every observable of
interest. Before that asymptotic timescale, the statistics of chaos is
generally believed to depend on both the initial conditions and the chosen
observable. I show that this is not the case for a widely applicable class of
models, that feature a phase-space (`field') distribution common to all
pushed-forward or integrated observables, while the system is still relaxing
towards statistical equilibrium or a steady state. This universal profile is
determined by both leading and first subleading eigenfunctions of the transport
operator (Koopman or Perron-Frobenius) that maps phase-space densities forward
or backward in time.
