  We describe classes of ergodic dynamical systems for which some statistical
properties are known exactly. These systems have integer dimension, are not
globally dissipative, and are defined by a probability density and a two-form.
This definition generalizes the construction of Hamiltonian systems by a
Hamiltonian and a symplectic form. Some low dimensional examples are given, as
well as a discretized field theory with a large number of degrees of freedom
and a local nearest neighbor interaction. We also evaluate unequal-time
correlations of these systems without direct numerical simulation, by Pad\'e
approximants of a short-time expansion. We briefly speculate on the possibility
of constructing chaotic dynamical systems with non-integer dimension and
exactly known statistics. In this case there is no probability density,
suggesting an alternative construction in terms of a Hopf characteristic
function and a two-form.
