title: On exact statistics and classification of ergodic systems of integer dimension.
abstract: 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 Padé 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.
authors.affiliation: Department of Physics, Brown University, Providence, Rhode Island 02912, USA.
authors.forename: Zachary
authors.initials: Z
authors.lastname: Guralnik
authors.affiliation: Harvard University, Center for Brain Science, Cambridge, Massachusetts 02138, USA.
authors.forename: Cengiz
authors.initials: C
authors.lastname: Pehlevan
authors.affiliation: Department of Physics, Brown University, Providence, Rhode Island 02912, USA.
authors.forename: Gerald
authors.initials: G
authors.lastname: Guralnik