Construction of dynamics in one-dimensional systems of statistical mechanics

It is well known that in one-dimensional systems the microcanonical, small canonical, and grand canonical distributions have the same thermodynamic limit. This limit can be regarded as a measure on the phase space of an infinite system of particles. Under the assumption that the binary interaction potential has compaet support, it is shown that one can find a one- parametric group of transformations in the phase space that preserve this measure and are related in a natural manner to the infinite system of Hamiltonian equations that describe the motion of the particles. This result has been previously proved by Lanford under the assumption that the potential has bounded modulus and finite range.