Dynamical ensembles equivalence in fluid mechanics

An attempt to put togheter various theoretical, mathematical, or experimental results recently developed in apparently unrelated subjects. Namely Ruelle's approach to turbulence, [1], the body of Nos\'e--Hoover type of molecular dynamics experiments, [2],[3],[4], mathematical results on Lyapunov exponents (the pairing rule, [5],[6]) and experimental results on them ([7],[8])), theoretical as well as mathematical results on fluctuations ([9] (multifractality), and [10] (chaotic hypothesis)). The key idea that we try to clarify is that of "dynamical ensembles", as a generalization of the classical "equilibrium ensembles", arguing that they should be identified with the SRB distributions and that they share several properties with the classical ensembles. Most of the results invoked here did not deal directly with the Navier Stokes equations and yet they seem to have a lot to do with them (as we shall argue): here the discussion will focus on the Navier Stokes and dissipative Euler equations with the aim of proposing several experiments apt to test the equivalence of dynamical ensembles and the chaotic hypothesis. The ideas developed, to a great extent, from the efforts put in interpreting the experimental results in [4].