The paper is concerned with volume-preserving three-dimensional mappings of \(\mathbf T^3\), of the general form: \(x'=x+F(y,z)\), \(y'=y +G(x',z)\), \(z'=z+H(x',y')\). As the authors widely and clearly explain, these mappings appear quite naturally in connection with the dynamics of medium-size particles (”passive scalars”) suspended in incompressible flows. Both for this hydrodynamical motivation and for the intrinsic mathematical interest, the authors consider the case of one or two “slow coordinates”, called actions, and two or one “fast coordinates”, called angles; in such an odd-dimensional system they look for the possible existence of KAM-like phenomena. The investigation technique is a combination of numerical study and low order perturbation theory. The main results are the following: in the case of a single action, the behavior is quite similar to the behavior of two-dimensional mappings, like the “standard mapping”; in particular, for slow enough action (\(H\) of order \(\varepsilon\), \(F\) and \(G\) of order one), one finds two-dimensional invariant surfaces (tori), which topologically obstruct the phase space. The authors explicitly conjecture that a KAM-like theorem holds. The typical system with a single action which is studied, is a discretized version of the so-called ABC model. Instead, in the case of two actions (\(F\), \(G\) of order \(\varepsilon\), \(H\) of order one), KAM theory does not appear to apply as well. A new kind of Arnold-like diffusion is found.
The paper is completely self-contained, very clear and illuminating in all the questions it touches.